OMA – Vol 1 – Issue 1 (2017) – PISRT

Forgotten polynomial and forgotten index of certain interconnection networks

pisrt — Sun, 31 Dec 2017 04:20:03 +0000

Open Journal of Mathematical Analysis

Vol. 1 (2017), Issue 1, pp. 44–59
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2017.0005

Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks

Hajra Siddiqui$^{1}$, Mohammad Reza Farahani
Department of Mathematics and Statistics University of Lahore Pakistan.; (H. S)
Department of Applied Mathematics of Iran University of Science and Technology, (IUST) Narmak, Tehran 16844, Iran.; (M. R. F)
$^{1}$Corresponding Author; hajraasiddiqui3@gmail.com

Copyright © 2017 Hajra Siddiqui, Mohammad Reza Farahani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: September 21, 2017 – Accepted: October 15, 2017 – Published: December 31, 2017

Abstract

Chemical reaction network theory is an area of applied mathematics that attempts to model the behavior of real world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved. It is experimentally proved that many properties of the chemical compounds and their topological indices are correlated. In this report, we compute closed form of forgotten polynomial and forgotten index for interconnection networks. Moreover we give graphs to see dependence of our results on the parameters of structures.

Keywords:

Forgotten polynomial; Topological index; Interconnection networks.

1. Prelimaniries

In mathematical chemistry, mathematical tools such as polynomials and numbers predict properties of compounds without using quantum mechanics. These tools, in combination, capture information hidden in the symmetry of molecular graphs. Most commonly known invariants of such kinds are degree-based topological indices. These are the numerical values that correlate the structure with various physical properties, chemical reactivity and biological activities [1, 2, 3, 4, 5]. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity and fracture toughness of a molecule are strongly connected to its graphical structure and this fact plays a synergic role in chemical graph theory.

The butterfly graphs are considered as the principal graphs of FFT(Fast Fourier Transforms) networks and it is efficient in performing FFT. In butterfly networks the series of switch stages and interconnection patterns that allow $n$ inputs and $n$ outputs to be inter linked with each other. The Benes network is comprises of continues butterflies and is known as permutation routing whereas butterfly network is identified as Fast Fourier transforms [6]. Butterfly network and Benes network contains important multistage interconnection networks that comprises of attractive topologies for communication networks [7]. Further these topologies are helpful in parallel computing systems which are IBM,SP1/SP2, MIT transit project, internal structures of optical couplers [8, 9 ].

For the connection of numerous homogenously replicated processes multiprocessor interconnection networks are important which is also called a processing node. Message passing is mostly used for the management and communication between processing nodes for program execution. Planning and using of multiprocessor interconnection networks have significant consideration due to the availability of powerful microprocessors and memory chips [10]. For the extreme parallel computing, multipurpose interconnection mesh networks are widely known. This is the fact that these networks having topologies which reproduce the communication pattern of a wide variety regarding natural problems. Mesh networks have recently received a lot of attention for better scalability to larger networks as compared to hyper cubes [11].

A graph with vertex set $V(G)$ and edge set $E(G)$ is connected, if there exist a connection between any pair of vertices in $G$. The distance between two vertices $u$ and $v$ is denoted as $d(u,v)$ and is the length of shortest path between $u$ and $v$ in graph $G$. The number of vertices of $G$, adjacent to a given vertex $v$, is the \emph{degree} of this vertex, and will be denoted by $d_{v}$. For details on basics of graph theory, any standard text such as [12] can be of great help.

The topological index of a molecule structure can be considered as a non-empirical numerical quantity which quantifies the molecular structure and its branching pattern in many ways. In this point of view, the topological index can be regarded as a score function which maps each molecular structure to a real number and is used as a descriptor of the molecule under testing. Topological indices [13, 14, 15, 16, 17, 18, 19, 20, 21, 22] gives a good predictions of variety of physico-chemical properties of chemical compounds containing boiling point, heat of evaporation, heat of formation, chromatographic retention times, surface tension, vapor pressure etc. Since the 1970s, two degree based graph invariants have been extensively studied. These are the first Zagreb index $M_{1}$ and the second Zagreb index $M_{2}$ [23, 24] and defined as: $M_{1}(G)=\sum\limits_{v\in V(G)}(d_{v})^2$ and $M_{2}(G)=\sum\limits_{usv\in V(G)}d_{u}d_{v}$.

In this article, we compute closed form of the forgotten polynomial for interconnection networks. We also computed forgotten index of these networks. For detailed study about degree-based topological indices, we refer [25, 26, 27, 28, 29, 30] and the references therein.

2. Mesh derived Networks

There are various open problems suggested for various interconnection networks. To quote Stojmenovic [27]: Designing new architectures remains an area of intensive investigation given that there is no clear winner among existing ones. Let the graph $G$ shown in figure $1$

Figure 1.Graph $G$

The stellation of $G$ is denoted by $St(G)$ and can be obtained by adding a vertex in each face of $G$ and then by join these vertices to all vertices

Figure 2. Stellation of G(dotted)

The dual $Du(G)$ of a graph $G$ is a graph that has a vertex for each face of $G$. The graph has an edge whenever two faces of $G$ are separated from each other, and a self-loop when the same face appears on both sides of an edge, see Figure 3. Hence, the number of faces of a graph is equal to the number of edges of its dual

Figure 3. Dual of graph G

In dual graph, if we delete the vertex corresponding to the bounded face of planer graph, which is unique in it, we get bounded dual $Bdu(G)$ (see Figure 4).

Figure 4. Bounded dual of graph G (dotted)

Given a connected plane graph $G$, its medial graph $M(G)$ has a vertex for each edge of $G$ and an edge between two vertices for each face of $G$ in which their corresponding edges occur consecutively (see Figure 5).

Figure 5. Medial of G (dotted)

Under this section two new architectures are introduced by using $m \times n$ mesh network, the defining parameters $m$ and $n$ are the number of vertices in any column and row. The restricted dual of Mesh $m \times n$ is $m-1 \times n-1$ can be easily noticed. By applying medial operation on mesh $m{\rm \; \times\; n}$ and deleting the vertex of unrestrained face we found the restricted medial of mesh $m{\rm \; \times\; n}$. By taking union $m{\rm \; \times\; n}$ mesh and its restricted medial in a way that the vertices of restricted medial are placed in the middle of each edge of $m{\rm \; x\; n}$ mesh, the resulting architecture will be the planar named as mesh derived network of first type $MDN1[m,n]$ network as depicted in Fig. 6. The vertex and edge cardinalities of $MDN1[m,n]$ network are $3mn-m-n$ and $8mn-6(m+n){\rm +}4$ respectively. The second architecture is obtained from the union of $m{\rm \; \times\; n}$ mesh and its restricted dual $m-1{\rm \; }\times{\rm \;}n-1$ mesh by joining each vertex of $m-1{\rm \; }\times{\rm \; }n-1$ mesh to each vertex of parallel face of $m{\rm \; \times\; n}$ mesh. The resulting architecture will be mesh derived network of second type $MDN2[m,n]$ network as depicted in Figure 17. The number of vertices and edges of this non planar graph are $2mn-m-n+1$ and $8(mn-m-n+1)$ respectively. Some other types of mesh derived networks are well-defined and considered in [31]. The main graph parameter which is discussed in [31] for mesh derived networks is the metric dimension of networks. Now we compute topological indices of these mesh derived networks

Figure 6. MDN1[5, 5] (dotted)

Table 1. Edge partition $MDN1[m, n]$

$(d_{u}, d_{v})$	Number of edges
$(2,4)$	$8$
$(3,4)$	$4(m+n-4)$
$(3,6)$	$2(m+n-4)$
$(4,6)$	$4(mn-n-m)$
$(4,4)$	$4$
$(6,6)$	$4(mn-2m-2n+4)$

Theorem 2.1. Let $MDN1 [m, n]$ is mesh derived networks. Then the Forgotten polynomial and forgotten index of $MDN1 [m, n]$ are respectively \begin{eqnarray*}F(MDN1[m,n],x)&=&\; 8x^{20} +4(m+n-4)x^{25} +2(m+n-4)x^{45}\\ &&+ 4(mn-m-n)x^{52} + 4x^{32} \\ &&+4(mn-2m-2n+4)x^{72}\end{eqnarray*} \begin{eqnarray*} F(MDN1[m,n])&=&\; 20^{8} \times 25^{4(m+n-4)} \times 45^{2(m+n-4)}\\ &\times& 52^{4(mn-m-n)} \times 32^{4} \times 72^{4(mn-2m-2n+4)} \end{eqnarray*} Proof. From the definition of forgotten polynomial, we have \begin{eqnarray*} F(MDN1[m,n],x)&=&\sum\limits _{uv\in E(MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &=&\sum\limits _{uv\in E_{1} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]}\\ &&+\sum\limits _{uv\in E_{2} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+\sum\limits _{uv\in E_{3} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+\sum\limits _{uv\in E_{4} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+\sum\limits_{uv\in E_{5} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]}\\ &&+\sum\limits_{uv\in E_{6} (MDN1[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &=& \left|E_{1} (MDN1[m,n])\right|x^{20} +\left|E_{2} (MDN1[m,n])\right|x^{25}\\ && +\left|E_{3} (MDN1[m,n])\right|x^{45}+\left|E_{4} (MDN1[m,n])\right|x^{52}\\ &&+ \left|E_{5} (MDN1[m,n])\right|x^{32} +\left|E_{6} (MDN1[m,n])\right|x^{72} \\ &=&8x^{20} +4(m+n-4)x^{25} +2(m+n-4)x^{45}\\ &&+4(mn-m-n)x^{52}+4x^{32}\\ &&+4(mn-2m-2n+4)x^{72} \end{eqnarray*}

Figure 7. 3D plot of Forgotten Polynomial

From the definition of forgotten index we have \begin{eqnarray*} F(MDN1[m,n])&=&\prod _{uv\in E(MDN1[m,n])}[du^{2} +dv^{2} ] \\ &=& \prod _{uv\in E_{1} (MDN1[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{2} (MDN1[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{3} (MDN1[m,n])}[du^{2} +dv^{2} ] \\ &&\times \prod _{uv\in E_{4} (MDN1[m,n])}[du^{2} +dv^{2} ] \\ &&\times \prod _{uv\in E_{5} (MDN1[m,n])}[du^{2} +dv^{2} ]\\ && \times \prod _{uv\in E_{6} (MDN1[m,n])}[du^{2} +dv^{2} ] \\ &=& 20^{\left|E_{1} (MDN1[m,n])\right|} \times 25^{\left|E_{2} (MDN1[m,n)\right|}\\ &&\times 45^{\left|E_{3} (MDN1[m,n])\right|} \times 52^{\left|E_{4} (MDN1[m,n])\right|}\\ && \times 32^{\left|E_{5} (MDN1[m,n])\right|} \times 72^{\left|E_{6} (MDN1[m,n])\right|} \\ &=& 20^{8} \times 25^{4(m+n-4)} \times 45^{2(m+n-4)} \times 52^{4(mn-m-n)}\\ && \times 32^{4} \times 72^{4(mn-2m-2n+4)} \end{eqnarray*}

Figure 8. 3D plot of Forgotten Index

3. Mesh derived Networks 2

Figure 9. MDN2[m,n]

Table 2. Edge partition MDN2[m, n]

$(d_{u}, d_{v})$	Number of edges
$(3,6)$	$4$
$(3,5)$	$8$
(5,6)	$8$
$(5,5)$	$2(m+n-6)$
$(6,8)$	4
$(5,8)$	$2(m+n-4)$
$(5,7)$	$4(m+n-6)$
$(7,7)$	$2(m+n-8)$
$(6,7)$	$8$
$(7,8)$	$6(m+n-6)$
$(8,8)$	$8mn-24(m+n)+72$

Theorem 3.1. Let $MDN2 [m, n]$ is mesh derived networks. Then the Forgotten Polynomial and Forgotten Index of $MDN2 [m, n]$ are \begin{eqnarray*} F(MDN2[m,n],x)&=&4x^{45} +8x^{34} +8x^{61} +2(m+n-6)x^{50} +4x^{100} \\ &&+2(m+n-4)x^{89} + 4(m+n-6)x^{74} +2(m+n-8)x^{98}\\ &&+ 8x^{85} +6(m+n-6)x^{113}+(8mn-24(m+n)+72)x^{128} \end{eqnarray*} \begin{eqnarray*} F(MDN2[m,n])&=& 45^{4} \times 34^{8} \times 61^{8} \times 50^{2(m+n-6)} \times 100^{4} \times 89^{2(m+n-4)} \times 74^{4(m+n-6)}\\ &&\times 98^{2(m+n-8)} \times 85^{8} \times 113^{6(m+n-6)} \times 128^{(8mn-24(m+n)+72)} \end{eqnarray*}

Proof. From the definition of Forgotten Polynomial we have \begin{eqnarray*} F(MDN2[m,n],x)\\ &=&\sum _{uv\in E(MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &=& \sum _{uv\in E_{1} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ && +\sum _{uv\in E_{2} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{3} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{4} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{5} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{6} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{7} (MDN2[m,n])}x^{[du^{2} +dv^{2} ]}\\ &&+\sum _{uv\in E_{8} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{9} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ && +\; \sum _{uv\in E_{10} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]} \\ &&+ \sum _{uv\in E_{11} (MDN2[m,n])}x^{[du^{2} \; +dv^{2} ]}\\ &=& \left|E_{1} (MDN2[m,n])\right|x^{45} +\left|E_{2} (MDN2[m,n])\right|x^{34}\\ &&+ \left|E_{3} (MDN2[m,n])\right|x^{61} +\left|E_{4} (MDN2[m,n])\right|x^{50}\\ &&+\left|E_{5} (MDN2[m,n])\right|x^{100} +\left|E_{6} (MDN2[m,n])\right|x^{89} \\ &&+ \left|E_{7} (MDN2[m,n])\right|x^{74}+\left|E_{8} (MDN2[m,n])\right|x^{98}\\ &&+\left|E_{9} (MDN2[m,n])\right|x^{85} +\left|E_{10} (MDN2[m,n])\right|x^{113}\\ &&+ \left|E_{11} (MDN2[m,n])\right|x^{128} \\ &=& 4x^{45} +8x^{34} +8x^{61} +2(m+n-6)x^{50} \\ &&+4x^{100} +2(m+n-4)x^{89}+ 4(m+n-6)x^{74} \\ &&+2(m+n-8)x^{98}+ 8x^{85} +6(m+n-6)x^{113} \\ &&+(8mn-24(m+n)+72)x^{128} \end{eqnarray*}

Figure 10. 3D plot of forgotten polynomial

\begin{eqnarray*} F(MDN2[m,n])&=&\prod _{uv\in E(MDN2[m,n])}[du^{2} +dv^{2} ] \\ &=&\prod _{uv\in E_{1} (MDN2[m,n])}[du^{2} +dv^{2} ] \\ &&\times \prod _{uv\in E_{2} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{3} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{4} (MDN2[m,n])}[du^{2} +dv^{2} \\ &&\times \; \; \prod _{uv\in E_{5} (MDN2[m,n])}[du^{2} +dv^{2} ] \\ &&\times \prod _{uv\in E_{6} (MDN2[m,n])}[du^{2} +dv^{2} ] \\ &&\times\prod _{uv\in E_{7} (MDN2[m,n])}[du^{2} +dv^{2} ] \\ &&\times \prod _{uv\in E_{8} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{9} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ &&\times \prod _{uv\in E_{10} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ && \times \prod _{uv\in E_{11} (MDN2[m,n])}[du^{2} +dv^{2} ]\\ &=& 45^{\left|E_{1} (MDN2[m,n])\right|} \times 34^{\left|E_{2} (MDN2[m,n])\right|} \times 61^{\left|E_{3} (MDN2[m,n])\right|}\\ &\times& 50^{\left|E_{4} (MDN2[m,n])\right|} \times 100^{\left|E_{5} (MDN2[m,n])\right|} \\ &\times& 89^{\left|E_{6} (MDN2[m,n])\right|} \times \; 74^{\left|E_{7} (MDN2[m,n])\right|} \times 98^{\left|E_{8} (MDN2[m,n])\right|}\\ &\times& 85^{\left|E_{9} (MDN2[m,n])\right|} \times 113^{\left|E_{10} (MDN2[m,n])\right|} \\ &\times& 128^{\left|E_{11} (MDN2[m,n])\right|}\\ &=& 45^{4} \times 34^{8} \times 61^{8} \times 50^{2(m+n-6)} \\ &\times& 100^{4} \times 89^{2(m+n-4)} \times 74^{4(m+n-6)} \times 98^{2(m+n-8)} \times 85^{8} \\ &\times& 113^{6(m+n-6)} \times 128^{(8mn-24(m+n)+72)} \end{eqnarray*}

Figure 11. 3D plot of forgotten index

4. Butterfly Network

Butterfly network is the most common bounded-degree derivative network of the hypercube. The set $V$ of vertices of an $r$-dimensional butterfly network resemble to pairs $[w, i]$ where $i$ is the dimension or level of a node $(0\le i\le r)$ and $w$ is an $r$-bit binary number that represents the row of the node. Two nodes $[w,i]$ and $[w',i']$ are connected by an edge if and only if $\acute{i}=i+1$ and either: \noindent 1. $w$ and $w'$ are duplicate, or \noindent 2. $w$ and $w'$ differ in specifically the $i$-th bit. \noindent Undirected edges are found in the network. An $r$-dimensional butterfly network is represented by $BF(r)$. Manuel et el. [26] offered the diamond representations of these networks. The normal and diamond representations of 3-dimensional butterfly network are given in Figure $12$. The vertex and edge cardinalities are $2^{r} (r+1)$ and $r2^{r+1}$ respectively

Figure 12. Normal representation of butterfly BF(3)

Figure 13. Diamond representation of butterfly BF(3)

Let $G=BF(r)$ be the butterfly network. From figure 12, we see that the graph has $2r(r+1)$ number of vertices and $r2r+1$ number of edges. The edge partition of graph $G$ is shown in table 3.

Table 3. Edge partition $BF(r)$

$(d_{u}, d_{v})$	Number of edges
$(2,4)$	$2^{r+2}$
$(4,4)$	$2^{r+1}(r-2)$

Theorem 4.1. Let $BF(r)$ is butterfly network then the Forgotten Polynomial and Forgotten Index of $BF(r)$ are

$F(BF(r),x)$= $2^{r+2} x^{20} +2^{r+1} (r-2)x^{32} $
$F(BF(r))$= $20^{2^{r+2} } \; \times 32^{2^{r+1} (r-2)} $

Proof. From the definition of forgotten polynomial we have \begin{eqnarray*} F(BF(r),x)&=&\sum _{uv\in E(BF(r))}x^{[du^{2} \; +dv^{2} ]} \\ &=&\sum _{uv\in E_{1} (BF(r))}x^{[du^{2} \; +dv^{2} ]} +\sum _{uv\in E_{2} (BF(r))}x^{[du^{2} \; +dv^{2} ]} \\ &=& \left|E_{1} (BF(r))\right|x^{20} +\left|E_{2} (BF(r))\right|x^{32} \\ &=& 2^{r+2} x^{20} +2^{r+1} (r-2)x^{32} \end{eqnarray*}

Figure 14. Plot of forgotten polynomial

From the definition of forgotten index we have \begin{eqnarray*} F(BF(r))&=&\prod _{uv\in E(BF(r))}[du^{2} +dv^{2} ] \\ &=& \prod _{uv\in E_{1} (BF(r))}[du^{2} +dv^{2} ] \; \times \prod _{uv\in E_{2} (BF(r))}[du^{2} +dv^{2} ]\\ &=& 20^{\left|E_{1} (BF(r))\right|} \times 32^{\left|E_{2} (BF(r))\right|} \\ &=& 20^{2^{r+2} } \; \times 32^{2^{r+1} (r-2)} \end{eqnarray*}

Figure 15. Plot of forgotten index

5. Benes Network

The $r$-dimensional Benes network is continuous butterflies which are represented by $B_{(r)}$. This $r$-dimensional Benes network has levels, each level with $2r$ nodes. The level $0$ to level $r$ nodes in the network form an $r$-dimensional butterfly. The middle level of the network is shared by these butterflies. Manuel et al. suggested the diamond representation of the Benes network [32]. Figure 16 shows the normal representation of $B_{(3)}$ network, while diamond representation of $B_{(3)}$ is described in Figure 17. The number of vertices and number of edges in an $r$-dimensional Benes network are $2^r(2r+1)$ and $r2^{r+2}$ respectively.

Figure 16. Normal representation of Bens Network $B_{(3)}$

Figure 17. Diamond representation of Bens Network $B_{(3)}$

Let be the Benes network. From figure 1, we see that the graph has $2r(2r+1)$ number of vertices and $r2^{r+2}$ number of edges. Table 4 shows the edge partition of graph $G$.

Table 4. Edge partition $BF(r)$

$(d_{u}, d_{v})$	Number of edges
$(2,4)$	$2^{r+2}$
$(4,4)$	$2^{r+2}(r-1)$

Theorem 5.1. Let $B_{(r)}$ is Benes network then the Forgotten Polynomial $$F(B_{(r)} ,x)= 2^{r+2} x^{20} +2^{r+2} (r-1)x^{32} $$ and Forgotten Index of $B_{(r)}$ are $$F(B_{(r)} )= 20^{2^{r+2} } \; \times 32^{2^{r+2} (r-1)} $$

Proof. From the definition of forgotten polynomial we have \begin{eqnarray*} F(B_{(r)} ,x)&=&\sum _{uv\in E(B_{(r)} )}x^{[du^{2} \; +dv^{2} ]} \\ &=&\sum _{uv\in E_{1} (B_{(r)} )}x^{[du^{2} \; +dv^{2} ]} +\sum _{uv\in E_{2} (B_{(r)} )}x^{[du^{2} \; +dv^{2} ]} \\ &=&\left|E_{1} (B_{(r)} )\right|x^{20} +\left|E_{2} (B_{(r)} )\right|x^{32} \\ &=& 2^{r+2} x^{20} +2^{r+2} (r-1)x^{32} \end{eqnarray*}

Figure 18. Plot of forgotten polynomial

From the definition of forgotten index we have \begin{eqnarray*} F(B_{(r)})&=&\prod\limits_{uv\in E(B_{(r)} )}[du^{2} +dv^{2} ] \\ &=& \prod\limits _{uv\in E_{1} (B_{(r)} )}[du^{2} +dv^{2} ] \times \prod\limits _{uv\in E_{2} (B_{(r)} )}[du^{2} +dv^{2} ] \\ &=& 20^{\left|E_{1} (B_{(r)} )\right|} \times 32^{\left|E_{2} (B_{(r)} )\right|} \\ &=& 20^{2^{r+2}} \times 32^{2^{r+2} (r-1)} \end{eqnarray*}

Figure 19. Plot of forgotten index

6. Conclusion

In this article, we have computed Forgotten polynomial and forgotten index for interconnected networks. These indices are actually functions of chemical graphs and encode many chemical properties as viscosity, strain energy, and heat of formation.

Competing Interests

The author do not have any competing interests in the manuscript.

References

Rücker, G., & Rücker, C. (1999). On topological indices, boiling points, and cycloalkanes. Journal of chemical information and computer sciences 39(5), 788-802. [Google Scholor]
Klavžar, S., & Gutman, I. (1996). A comparison of the Schultz molecular topological index with the Wiener index. Journal of chemical information and computer sciences, 36(5), 1001-1003. [Google Scholor]
Brückler, F. M., Došlic, T., Graovac, A., & Gutman, I. (2011). On a class of distance-based molecular structure descriptors. \emph{Chemical physics letters,} 503(4), 336-338. [Google Scholor]
Deng, H., Yang, J., & Xia, F. (2011). A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes. Computers & Mathematics with Applications, 61(10), 3017-3023. [Google Scholor]
Zhang, H., \& Zhang, F. (1996). The Clar covering polynomial of hexagonal systems I. Discrete applied mathematics, 69(1-2), 147-167. [Google Scholor]
Beneš, V. E. (Ed.). (1965). Mathematical theory of connecting networks and telephone traffic (Vol. 17). Academic press. [Google Scholor]
Manuel, P. D., Abd-El-Barr, M. I., Rajasingh, I., & Rajan, B. (2008). An efficient representation of Benes networks and its applications. Journal of Discrete Algorithms, 6(1), 11-19. [Google Scholor]
Liu, X. C., & Gu, Q. P. (2002). Multicasts on WDM all-optical butterfly networks. Journal of Information Science and Engineering, 18(6), 1049-1058. [Google Scholor]
Xu, J. (2013). Topological structure and analysis of interconnection networks (Vol. 7). Springer Science & Business Media. [Google Scholor]
Chen, M. S., Shin, K. G., & Kandlur, D. D. (1990). Addressing, routing, and broadcasting in hexagonal mesh multiprocessors. IEEE Transactions on Computers, 39(1), 10-18. [Google Scholor]
Cynthia, V. J. A. (2014). Metric dimension of certain mesh derived graphs. Journal of Computer and Mathematical Sciences, Vol, 5(1), 1-122. [Google Scholor]
West, D. B. (2001). Introduction to graph theory (Vol. 2). Upper Saddle River: Prentice hall . [Google Scholor]
Gutman, I. (1993). Some properties of the Wiener polynomial. Graph Theory Notes of New York, 25, 13-18. [Google Scholor]
Deutsch, E., & Klavžar, S. (2015). M-Polynomial and Degree-Based Topological Indices. Iranian Journal of Mathematical Chemistry, 6(2), 93-102. [Google Scholor]
Munir, M., Nazeer, W., Nizami, A. R., Rafique, S., & Kang, S. M. (2016). M-Polynomials and Topological Indices of Titania Nanotubes. Symmetry, 8(11), 117. [Google Scholor]
Munir, M., Nazeer, W., Rafique, S., & Kang, S. M. (2016). M-polynomial and degree-based topological indices of polyhex nanotubes. Symmetry, 8(12), 149. [Google Scholor]
Ajmal, M., Kang, S. M., Nazeer, W., Munir, M., & Jung, C. Y. (2016). Some Topological Invariants of the Möbius Ladders. Global Journal of Pure and Applied Mathematics, 12(6), 5317-5327. [Google Scholor]
Munir, M., Nazeer, W., Rafique, S., Nizami, A. R., & Kang, S. M. (2017). Some computational aspects of triangular Boron nanotubes. Symmetry , 9(1), 6. [Google Scholor]
Javaid, M., & Jung, C. Y. (2017). M-Polynomials and Topological Indices of Silicate and Oxide Networks. International Journal of Pure and Applied Mathematics , 115 (1) 129-152. [Google Scholor]
Kwun, Y. C., Munir, M., Nazeer, W., Rafique, S., & Kang, S. M. (2017). M-Polynomials and topological indices of V-Phenylenic Nanotubes and Nanotori. Scientific reports , 7(1), 8756. [Google Scholor]
Gutman, I., & Polansky, O. E. (2012). Mathematical concepts in organic chemistry. Springer Science & Business Media. [Google Scholor]
Wiener, H. (1947). Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1), 17-20. [Google Scholor]
Gutman, I., & Das, K. C. (2004). The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem , 50, 83-92. [Google Scholor]
Das, K. C., & Gutman, I. (2004). Some properties of the second Zagreb index. MATCH Commun. Math. Comput. Chem, 52(1), 3-1. [Google Scholor]
Sardar, M. S, Zafar, S., & Farahani, M. R. (2017). The generalized zagreb index of capra-designed planar benzenoid series $ca_k(c_6)$, Open J. Math. Sci., 1(1), 44-51. [Google Scholor]
Rehman, H. M, Sardar, R., Raza, A. (2017). computing topological indices of hex board and its line graph. Open J. Math. Sci., 1(1), 62- 71. [Google Scholor]
Farahani, M. R., Gao, W., Baig, A. Q., & Khalid, W. (2017). Molecular description of copper (II) oxide. Macedonian Journal of Chemistry and Chemical Engineering, 36(1), 93-99. [Google Scholor]
Gao, W., Wang, Y., Wang, W., & Shi, L. (2017). The first multiplication atom-bond connectivity index of molecular structures in drugs. Saudi Pharmaceutical Journal, 25(4), 548-555. [Google Scholor]
Gao, W., Farahani, M. R., Wang, S., & Husin, M. N. (2017). On the edge-version atom-bond connectivity and geometric arithmetic indices of certain graph operations. Applied Mathematics and Computation, 308, 11-17. [Google Scholor]
Zomaya, A. Y. (1996). Parallel and distributed computing handbook. [Google Scholor]
Cynthia, V. J. A. (2014). Metric dimension of certain mesh derived graphs. Journal of Computer and Mathematical Sciences, 5(1), 1-122. [Google Scholor]
Manuel, P. D., Abd-El-Barr, M. I., Rajasingh, I., & Rajan, B. (2008). An efficient representation of Benes networks and its applications. Journal of Discrete Algorithms, 6(1), 11-19 [Google Scholor] .

Mapping properties of integral operator involving some special functions

pisrt — Sun, 31 Dec 2017 04:10:00 +0000

Open Access Full-Text PDF

Open Journal of Mathematical Analysis

Vol. 1 (2017), Issue 1, pp. 34 – 43
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2017.0004

Mapping properties of integral operator involving some special Functions

Muhey U Din$^{1}$, Mohsan Raza, Saddaf Noreen
Department of Mathematics, Government College University Faisalabad, Pakistan.; (M.U.D & M.R & S.N)
$^{1}$Corresponding Author; muheyudin@yahoo.com

Copyright © 2017 Muhey U Din, Mohsan Raza, Saddaf Noreen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: September 21, 2017 – Accepted: November 25, 2017 – Published: December 31, 2017

Abstract

In this article, we are mainly interested to find some sufficient conditions for integral operator involving normalized Struve and Dini function to be in the class $N\left( \mu \right)$. Some corollaries involving special functions are also the part of our investigations.

Keywords:

Analytic functions; Open unit disc; Bessel function; Struve function; Dini function; Salagean derivative.

1. Introduction

Let $\mathcal{A}$ denote the class of functions $f$ of the form

\begin{equation} f(z)=z+\sum\limits_{n=2}^{\infty }a_{n}z^{n}, \label{1.1} \end{equation}

(1)

which are analytic in the open unit disc $\mathcal{U}=\left\{ z:\left\vert z\right\vert <1\right\} $ and $\mathcal{S}$ denote the class of all functions in which are univalent in $\mathcal{U}$. Let $\mathcal{S}^{\ast }\left( \alpha \right) ,\ \mathcal{C}\left( \alpha \right) $ and $\mathcal{K} \left( \alpha \right) $ denote the classes of starlike, convex and close-to-convex functions of order $\alpha $ and are defined as: \begin{equation*} \mathcal{S}^{\ast}\left( \alpha \right)=\left\{ f:f\in \mathcal{A}\text{ and }{Re}\left( \frac{zf^{\prime }\left( z\right) }{f\left( z\right) }% \right) >\alpha ,\ \ z\in \mathcal{U}\text{, }\alpha \in \left[ 0,1\right) \right\} \end{equation*} \begin{equation*} \mathcal{C}\left( \alpha \right) =\left\{ f:f\in \mathcal{A}\text{ and } {Re}\left( 1+\frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }\right) >\alpha ,\ \ z\in \mathcal{U}\text{, }\alpha \in % \left[ 0,1\right) \right\} , \end{equation*} and \begin{equation*} \mathcal{K}\left( \alpha \right) =\left\{{ f:f\in \mathcal{A}\text{ and }{Re}}\left( \frac{zf^{\prime }\left( z\right) }{g\left( z\right)}\right) >\alpha ,\ \ z\in \mathcal{U}\text{, }\alpha \in \left[ 0,1\right),\ g\in \mathcal{S}^{\ast }\right\} . \end{equation*} It is clear that \begin{equation*} \mathcal{S}^{\ast }\left( 0\right) =\mathcal{S}^{\ast },\ \mathcal{C}\left( 0\right) =\mathcal{C\ }\text{and\ }\mathcal{K}\left( 0\right) =\mathcal{K}. \end{equation*} Here, we introduce another class $N(p,\gamma )$, $1<\gamma <\frac{2^{p}+1}{ 2^{p-1}+1},$ consisting of the function of the form of (1) satisfying the relation \begin{equation*} Re\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right) <\gamma , \end{equation*} where $D^{p}$ stands for the Salagean operator introduced by Salagean [ 1] in 1983. For $p = 0$, $p = 1$ the class $N(p,\gamma )$ reduce to the classes $M(\gamma )$ and $N(\gamma )$, respectively. These classes were generalized by many researchers. For the study of these classes we refer [ 2]. In geometric function theory, special functions play an important role. Special functions have their own importance in pure and applied mathematics. The widely use of these functions have attracted many researchers to work on the different directions. Geometric properties of special functions such as Hypergeometric functions, Bessel functions, Struve functions, Mittage Lefller functions, Wright functions and some other related functions is an ongoing part of research in geometric function theory. We refer for some geometric properties of these functions [3, 4, 5, 6, 7, 8, 9, 10] and references therein. The Struve functions $H_{v}$ and $L_{v}$ appeared as special solutions of the second order inhomogeneous differential equations of the form

\begin{equation} z^{2}w^{\prime \prime }(z)+zw(z)+(z^{2}-v^{2})w(z)=\frac{4\left( \frac{1}{2} z\right) ^{v+1}}{\sqrt{\pi }\Gamma \left( v+\frac{1}{2}\right) }, \label{1.2} \end{equation}

(2)

\begin{equation} z^{2}w^{\prime \prime }(z)+zw(z)-(z^{2}-v^{2})w(z)=\frac{4\left( \frac{1}{2} z\right) ^{v+1}}{\sqrt{\pi }\Gamma \left( v+\frac{1}{2}\right) }, \label{1.3} \end{equation}

(3)

known as inhomogeneous Bessel differential equations. Both equations (2) and (3) are similar and can be converted into each other by changing $z$ into $iz$. In the solution of equation (2), a function appeared in an article by Struve [11], was later ascribed Struve's name and the special notation $H_{v}.\ $ It is defined as

\begin{equation} H_{v}(z)=\overset{\infty }{\underset{n=0}{\sum }}\frac{\left( -1\right) ^{n}\left( \frac{z}{2}\right) ^{2n+v+1}}{\Gamma \left( n+\frac{3}{2}\right) \Gamma \left( n+v+\frac{3}{2}\right) }. \label{1.4} \end{equation}

(4)

The modified Struve functions $L_{v}$ of order $v$ was introduced by J. W. Nicholson in $1911.\ $ It is defined as

\begin{equation} L_{v}(z)=-ie^{-iv\frac{\pi }{2}}H_{v}(iz)=\overset{\infty }{\underset{n=0}{% \sum }}\frac{\left( \frac{z}{2}\right) ^{2n+v+1}}{\Gamma \left( n+\frac{3}{2}% \right) \Gamma \left( n+v+\frac{3}{2}\right) }, \label{1.5} \end{equation}

(5)

where $\Gamma (z)$ is the gamma function. Applications of Struve functions occur in water-wave and surface-wave problems, unsteady aerodynamics, resistive MHD instability theory and optical diffraction. More recently, Struve functions have appeared in many particle quantum dynamical studies of spin decoherence and nanotubes. Now consider the second order inhomogeneous differential equation

\begin{equation} z^{2}w^{\prime \prime }(z)+bzw^{\prime }(z)+\left[ cz^{2}-v^{2}+(1-b)v\right] w(z)=\frac{4\left( \frac{z}{2}\right) ^{v+1}}{\sqrt{\pi }\Gamma \left( v+% \frac{b}{2}\right) }, \label{1.6} \end{equation}

(6)

where $b,\ c,\ v\in \mathbb{C}$. The equation (6) generalizes the equation (2) and (3). In particular for $b=1, c=1$, we obtain (2). For $b=1, c=-1 $ we get (3). Its particular solution has the series form

\begin{equation} w_{v}(z)=\overset{\infty }{\underset{n=0}{\sum }}\frac{\left( -1\right) ^{n}c^{n}\left( \frac{1}{2}z\right) ^{2n+v+1}}{\Gamma \left( n+\frac{3}{2}% \right) \Gamma \left( n+v+\frac{b+2}{2}\right) }. \label{1.7} \end{equation}

(7)

It is known as generalized Struve functions of order $v$. Consider the transformation

\begin{eqnarray} u_{v,b,c}(z) &=&2^{v}\sqrt{\pi }\Gamma (v+(b+2)/2)z^{(-v-1)/2}w_{v,b,c}(% \sqrt{z}) \notag \\ &&\overset{\infty }{\underset{k=0}{=\sum }}\frac{\left( -c/4\right) ^{n}z^{n}% }{\left( 3/2\right) _{n}\left( k\right) _{n}}, \label{1.8} \end{eqnarray}

(8)

where $k=v+(b+2)/2\neq 0,-1,-2,-3,...$and \begin{equation*} \left( \gamma \right) _{n}=\frac{\Gamma \left( \gamma +n\right) }{\Gamma \left( \gamma \right) }=\left\{ \begin{array}{c} 1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n=0,\ \gamma \in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion \backslash \left\{ 0\right\} , \\ \gamma \left( \gamma +1\right) \ldots \left( \gamma +n-1\right) ,\ n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,\ \gamma \in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .% \end{array}% \right. \end{equation*} The function $u_{v,b,c}$ is analytic in $\mathcal{U\ }$ and is the solution of the differential equation

\begin{equation} 4z^{2}u^{\prime \prime }\left( z\right) +2\left( 2v+b+3\right) zu^{\prime }\left( z\right) +\left( cz+2v+b\right) u\left( z\right) =2v+b. \label{1.9} \end{equation}

(9)

The Bessel function of the first kind $J_{v}$ is defined by \begin{equation} J_{v}(z)=\sum\limits_{m=1}^{\infty }\frac{(-1)^{m}}{m!\Gamma (v+m+1)}\left( \frac{z}{2}\right) ^{2m+v}, \label{1.10} \end{equation}

(10)

where $\Gamma $ stands for Euler gamma function. It is a particular solution of the second order linear homogeneous differential equation \begin{equation*} z^{2}w^{\prime \prime }(z)+zw^{\prime }(z)+(z^{2}-v^{2})w(z)=0, \end{equation*} where $v\in\mathbb{C}$. We consider the normalized Dini functions $q_{\upsilon }:\mathcal{U} \rightarrow\mathbb{C}$ defined as

\begin{eqnarray} q_{v}\left( z\right) &=&2^{v-1}\Gamma \left( v+1\right) z^{1-\frac{v}{2}% }\left( \left( 2-v\right) J_{v}\left( \sqrt{z}\right) +\sqrt{z}J_{v}^{\prime }\left( \sqrt{z}\right) \right) \notag \\ &=&z+\sum\limits_{m=1}^{\infty }\frac{\left( -1\right) ^{m}\left( m+1\right) \Gamma \left( v+1\right) }{4^{m}m!\Gamma \left( v+m+1\right) }z^{m+1},\text{ }z\in \mathcal{U}\text{.} \label{1.11} \end{eqnarray}

(11)

For normalized Dini functions we refer [12, 13] In the present paper, we are mainly interested about the integral operator involving some special functions defined as

\begin{equation} F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta_{m}}(z)=\lim\limits_{0}^{z}\overset{n}{\underset{i=1}{\prod }}\left( \frac{% g(t)}{t}\right) ^{\alpha _{i}}\overset{n}{\underset{i=1}{\prod }}\left( \frac{D^{p}f_{j}(t)}{t}\right) ^{\beta _{j}}dt. \label{mohi} \end{equation}

(12)

This integral operator generalizes many operators due to this reason it has great importance. By assigning different values to parameters we get differenr operators defined in [14, 15, 16, 17]. Recently Porwal and kumar [18] studied the mapping properties of integral operators involvig Bessel functions.

2. Main Lemmas

The following lemmas play an very important role to derive our main results.

Lemma 2. 1. If $b,v \in \mathbb{R}$, and $c\in \mathbb{C}$, $k=v+\frac{b+2}{2}$ are so constrained that \begin{equation*} k>\max \left\{ 0,\frac{7\left\vert c\right\vert }{24}\right\} , \end{equation*} then the function $u_{v,b,c}:\mathcal{U}\rightarrow \mathbb{C}$ defined by (8) satisfies the following inequality: \begin{equation*} \left\vert \frac{zu_{v,b,c}^{\prime }(z)}{u_{v,b,c}(z)}-1\right\vert \leq \frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{% 3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\ \ \ \ \left( z\in \mathcal{U}\right) . \end{equation*}

Lemma 2.2 . Let $v\in \mathbb{R}$ and consider the normalized Dini function $q_{v}(z):\mathcal{U}\rightarrow \mathbb{C},$ defined by \begin{equation*} q_{v}\left( z\right) =2^{v-1}\Gamma \left( v+1\right) z^{1-\frac{v}{2}% }\left( \left( 2-v\right) J_{v}\left( \sqrt{z}\right) +\sqrt{z}J_{v}^{\prime }\left( \sqrt{z}\right) \right) , \end{equation*} where $J_{v}(z)$ is the Bessel function of first kind. Then \begin{equation*} \left\vert \frac{zq_{v}^{\prime }(z)}{q_{v}(z)}-1\right\vert \leq \frac{4v+9 }{2\left( 4v^{2}+9v+3\right) },\text{ \ }v>\frac{-9+\sqrt{33}}{8}. \end{equation*} The main objective of this paper is to give sufficient conditions for integral operator involving some special functions. The main results are given below.

3. Sufficient Conditions of Integral Operator Defined by Normalized Struve Function

Theorem 3.1 . Let $v_{1},\ldots ,v_{n},$ $b\in \mathbb{R}$, $c\in\mathbb{C}$ and $k_{i}>\frac{7\left\vert c\right\vert }{24}$ with $k_{i}=v_{i}+\left( b+2\right) /2,\ i=1,\ldots ,n.$ Let $u_{v_{i},b,c}:\mathcal{U}\rightarrow\mathbb{C}$ be defined as \begin{equation*} u_{v_{i},b,c}(z)=2^{v}\sqrt{\pi }\Gamma \left( v+\frac{b+2}{2}\right) z^{% \frac{\left( -1-v\right) }{2}}w_{v_{i},b,c}(\sqrt{z}). \end{equation*} Suppose $k=\min \left\{ k_{1},k_{2},\ldots ,k_{n}\right\} $, $\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}$ $(i=1,2,...,n)$ are positive real numbers and let $f_{j}(z),$ $(j=1,2,...,m)$ be of the form of (1) is in class $N\left( p,\gamma _{j}\right)$. More over\ these numbers satisfy the relation \begin{equation*} 1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1}, \end{equation*} then the function $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}$ defined by (12) is in $N\left( \mu \right) $, where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*}

Proof. We easily observe that $u_{v_{i},b,c},\forall i=1,2,\cdots n$ are analytic and normalized of the form of $u_{v_{i},b,c}\left( 0\right) =u_{v_{i},b,c}^{\prime }\left( 0\right) -1=0.$ Clearly $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}$ also analytic and normalized form of the form of $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\left( 0\right) =F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( 0\right) -1=0.$ On the other hand, it is easy to see that \begin{equation*} F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\left( z\right) =\overset{n}{\underset{i=1}{\prod }}\left( \frac{% u_{v_{i},b,c}}{z}\right) ^{\alpha _{i}}\overset{n}{\underset{i=1}{\prod }}% \left( \frac{D^{p}f_{j}(z)}{z}\right) ^{\beta _{j}}. \end{equation*} Differentiating logarithmically, we get \begin{eqnarray*} &&\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{% zu_{v_{i},b,c}^{\prime }(z)}{u_{v_{i},b,c}\left( z\right) }-1\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}% -1\right) , \end{eqnarray*} or equivalently, \begin{eqnarray*} &&1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) } \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{zu_{v_{i},b,c}^{\prime }(z)}{% u_{v_{i},b,c}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right)\\ &+&1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}. \end{eqnarray*} This implies that \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}Re\left( \frac{zu_{v_{i},b,c}^{% \prime }(z)}{u_{v_{i},b,c}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}Re\left( \frac{D^{p+1}f_{j}(z)}{% D^{p}f_{j}(z)}\right)\\ &+&\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) . \end{eqnarray*} Now, by using Lemma 2.1 for each $v_{i}$, where $i=1,2,\cdots n,$ we obtain \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq &\sum\limits_{i=1}^{n}\alpha _{i}\left( 1+\frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\right) \\ &+& \sum\limits_{j=1}^{m}\beta _{j}\gamma _{j}+\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) \\ &=&1+\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Now consider the funnction \begin{equation*} \tau :\left( \frac{7\left\vert c\right\vert }{24},\infty \right) \rightarrow\mathbb{R}, \end{equation*} defined by \begin{equation*} \tau \left( k\right) =\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }, \end{equation*} is decreasing function \begin{equation*} \frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\leq \frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }. \end{equation*} Therefore \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq& 1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Since $1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1},$ therefore $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\in N\left( \mu \right) $, where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*} which completes the proof.

By setting $\beta _{1}=\beta _{2}=...=\beta _{n}=0$ and $p=1$ in Theorem 3.1 we will obtain the result given below

Corollary 3.2. Let $v_{1},\ldots ,v_{n},$ $b\in\mathbb{R}$, $c\in\mathbb{C}$ and $k_{i}>\frac{7\left\vert c\right\vert }{24}$ with $k_{i}=v_{i}+\left( b+2\right) /2,\ i=1,\ldots ,n.$ Let $u_{v_{i},b,c}:\mathcal{U}\rightarrow\mathbb{C}$ be defined as \begin{equation*} u_{v_{i},b,c}(z)=2^{v}\sqrt{\pi }\Gamma \left( v+\frac{b+2}{2}\right) z^{% \frac{\left( -1-v\right) }{2}}w_{v_{i},b,c}(\sqrt{z}). \end{equation*} Suppose $v=\min \left\{ v_{1},v_{2},...,v_{n}\right\}$. Let $\alpha _{1},\ldots ,\alpha _{n}$ $(i=1,2,...,n)$ are positive real numbers and $f_{j}(z),$ $(j=1,2,...,m)$ be of the form of (1) is in class $N\left( p,\gamma _{j}\right)$. More over these numbers satisfy the relation

\begin{equation*} 1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}<\frac{3}{2}, \end{equation*} then the function $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}:\mathcal{U}\rightarrow\mathbb{C}$ defined by (12) is in $N\left( \mu \right) ,$ where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}. \end{equation*} The next theorem gives other sufficient conditions for the integral operator defined in (12). The key tool in the proof is Lemma 2.2.

4. Sufficient Conditions of Integral Operator Defined by Normalized Dini Function

Theorem 4.1 . Let $v_{1},\ldots ,v_{n}>\frac{-9+\sqrt{33}}{8},$ where $n\in \mathbb{N}$. Let $q_{v_{i}}:\mathcal{U}\rightarrow \mathbb{C}$ be defined as \begin{equation*} q_{v_{i}}\left( z\right) =2^{v_{i}-1}\Gamma \left( v_{i}+1\right) z^{1-\frac{% v_{i}}{2}}\left( \left( 2-v_{i}\right) J_{v_{i}}\left( \sqrt{z}\right) + \sqrt{z}J_{v_{i}}^{\prime }\left( \sqrt{z}\right) \right) . \end{equation*} Suppose $k=\min \left\{ k_{1},k_{2},\ldots ,k_{n}\right\} $, $\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}$ are positive real numbers and let $f_{j}(z),$ $(j=1,2,...,m)$ be of the form of (1) is in class $N\left( p,\gamma _{j}\right) .$ More over these numbers satisfy the relation \begin{equation} 1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1}, \label{m16} \end{equation} then the function $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}$ defined by (12) is in $N\left( \mu \right) ,$ where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*}

Proof. We easily observe that $q_{v_{i}}\forall i=1,2,\cdots n$ are analytic and normalized of the form of $q_{v_{i}}\left( 0\right) =q_{v_{i}}^{\prime }\left( 0\right) -1=0.$ Clearly $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}$ also analytic and normalized form of the form of $$F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\left( 0\right) =F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( 0\right) -1=0.$$ On the other hand, it is easy to see that \begin{equation*} F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\left( z\right) =\overset{n}{\underset{i=1}{\prod }}\left( \frac{ q_{v_{i}}}{z}\right) ^{\alpha _{i}}\overset{n}{\underset{i=1}{\prod }}\left( \frac{D^{p}f_{j}(z)}{z}\right) ^{\beta _{j}}. \end{equation*} Differentiating logarithmically, we get \begin{eqnarray*} &&\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{ zq_{v_{i}}^{\prime }(z)}{q_{v_{i}}\left( z\right) }-1\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)} -1\right) , \end{eqnarray*} or equivalently, \begin{eqnarray*} &&1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) } \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{q_{v_{i}}^{\prime }(z)}{% q_{v_{i}}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right) +1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}. \end{eqnarray*} This implies that \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{ F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}Re\left( \frac{zq_{v_{i}}^{\prime }(z)}{q_{v_{i}}\left( z\right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j} Re\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right) +\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) . \end{eqnarray*} Now, by using Lemma 2.2 for each $v_{i}$, where $i=1,2,\cdots n,$ we obtain \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{ F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq &\sum\limits_{i=1}^{n}\alpha _{i}\left( 1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\gamma _{j}+\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) \\ &=&1+\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\right)\\ &+& \sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Now as it is clear that the function \begin{equation*} \phi (v):\left( \frac{-9+\sqrt{33}}{8},\infty \right) \rightarrow \mathbb{R}, \end{equation*} defined by \begin{equation*} \phi (v)=\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }, \end{equation*} is decreasing function. Therefore \begin{equation*} \frac{4v_{i}+9}{2\left( 4v_{i}^{2}+9v_{i}+3\right) }\leq \frac{4v+9}{2\left( 4v^{2}+9v+3\right) }. \end{equation*} Therefore \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}^{\prime }\left( z\right) }\right\}\\ &\leq&1+\frac{4v+9}{2\left(4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha_{i}\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Since $1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n} \alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) < \frac{2^{p}+1}{2^{p-1}+1},$ therefore $$F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}\in N\left( \mu \right) $$ , where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*} which completes the proof.

By setting $\beta _{1}=\beta _{2}=...=\beta _{n}=0$ and $p=1$ in Theorem 4.1 we will obtain the result given below

Corollary 4.2. Let $v_{1},\ldots ,v_{n}>\frac{-9+\sqrt{33}}{8},$ where $n\in\mathbb{N}$. Let $q_{v_{i}}:\mathcal{U}\rightarrow \mathbb{C}$ be defined as \begin{equation*} q_{v_{i}}\left( z\right) =2^{v_{i}-1}\Gamma \left( v_{i}+1\right) z^{1-\frac{% v_{i}}{2}}\left( \left( 2-v_{i}\right) J_{v_{i}}\left( \sqrt{z}\right) +% \sqrt{z}J_{v_{i}}^{\prime }\left( \sqrt{z}\right) \right) . \end{equation*} Suppose $v=\min \left\{ v_{1},v_{2},...,v_{n}\right\}$. Let $\alpha _{1},\ldots ,\alpha _{n}$ $(i=1,2,...,n)$ are positive real numbers and $f_{j}(z),$ $(j=1,2,...,m)$ be of the form of (1) is in class $N\left( p,\gamma _{j}\right).$ More over these numbers satisfy the relation \begin{equation*} 1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}<\frac{3}{2}, \end{equation*} then the function $F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},...,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}$ defined by (12) is in $N\left( \mu \right)$, where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}. \end{equation*}

Referance

Salagean, G. S. (1983). Subclasses of univalent functions, Complex analysis-fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), 362-372. Lecture Notes in Math, 1013.[Google Scholor]
Uralegaddi, B. A., Ganigi, M. D., & Sarangi, S. M. (1994). Univalent functions with positive coefficients. Tamkang J. Math, 25(3), 225-230. [Google Scholor]
Baricz, À. R. P. À. D. (2006). Bessel transforms and Hardy space of generalized Bessel functions. Mathematica, 48(71), 127-136. [Google Scholor]
Baricz, À. (2010). Generalized Bessel functions of the first kind. Springer. [Google Scholor]
Bansal, D., & Prajapat, J. K. (2016). Certain geometric properties of the Mittag-Leffler functions. Complex Variables and Elliptic Equations, 61(3), 338-350. [Google Scholor]
Orhan, H., & Yagmur, N. (2014). Geometric properties of generalized Struve functions. Annals of the Alexandru Ioan Cuza University-Mathematics.[Google Scholor]
Raza, M., Din, M. U., & Malik, S. N. (2016). Certain geometric properties of normalized Wright functions. Journal of Function Spaces, 2016,Article ID 1896154, 8 pages. [Google Scholor]
Srivastava, H. M., Din, M. U.,& Raza, M., Univalence of Certain Integral Operators Involving Generalized Struve functions. (accepted). [Google Scholor]
Baricz, À., Deniz, E., & Yağmur, N. (2016). Close-to-convexity of normalized Dini functions. Mathematische Nachrichten, 289(14-15), 1721-1726. [Google Scholor]
Baricz, À ., Ponnusamy, S., & Singh, S. (2016). Modified Dini functions: monotonicity patterns and functional inequalities. Acta Mathematica Hungarica, 149(1), 120-142. [Google Scholor]
Breaz, D. (2008). Certain Integral Operators on the Classes $\mathcal{M}(\beta_{i})$ and $\mathcal{N}(\beta_{i})$. Journal of Inequalities and Applications, 2008(1), 719354. [Google Scholor]
Din, M. U, Raza, M, & Deniz, E. Sufficient Conditions for Univalence of Integral Operators Involving Dini Functions. (Submitted). [Google Scholor]
Deniz, E., Orhan, H., & Srivastava, H. M. (2011). Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions. Taiwanese J. Math, 15(2), 883-917. [Google Scholor]
Struve, H. (1882). Beitrag zur Theorie der Diffraction an Fernröhren. Annalen der Physik, 253(13), 1008-1016. [Google Scholor]
Pascu, N. N., & Pescar, V. (1990). On the integral operators of Kim-Merkes and Pfaltzgraff. Mathematica (Cluj), 32(55), 185-192. [Google Scholor]
Porwal, S. (2011). Mapping properties of an integral operator. Acta Universitatis Apulensis, (27), 151-155. [Google Scholor]
Porwal, S., & Breaz, D. (2014). Mapping properties of an integral operator involving Bessel functions. In Analytic Number Theory, Approximation Theory, and Special Functions (pp. 821-826). Springer New York. [Google Scholor]
Porwal, S., & Kumar, M. (2017). Mapping properties of an integral operator involving Bessel functions. Afrika Matematika, 28(1-2), 165-170.[Google Scholor]

Integral inequalities for differentiable harmonically $(s,m)$-preinvex functions

pisrt — Sun, 31 Dec 2017 04:05:16 +0000

Open Access Full-Text PDF

Open Journal of Mathematical Analysis

Vol. 1 (2017), Issue 1, pp. 25–33
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2017.0003

Integral Inequalities for Differentiable Harmonically $(s,m)$-preinvex Functions

Imran Abbas Baloch$^{1}$, Imdat İşcan

Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan.; (I.A.B)
Department of Mathematics, Faculty of Arts and Sciences, Giresun University, 28200, Giresun, Turkey.; (I.I)
$^{1}$Corresponding Author; iabbasbaloch@gmail.com, iabbasbaloch@sms.edu.pk
e-mails : iabbasbaloch@gmail.com, iabbasbaloch@sms.edu.pk ˙

Copyright © 2017 Imran Abbas Baloch, Imdat İşcan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: February 6, 2018 – Accepted: June 26, 2018 – Published: June 29, 2018

Abstract

In this paper, we define a new generalized class of preinvex functions which includes harmonically $(s,m)$-convex functions as a special case and establish a new identity. Using this identity, we introduce some new integral inequalities for harmonically $(s,m)$-preinvex functions.

Keywords:

Convex function; Harmonically (s, m)-convex functions; Preinvex functions; Harmonically preinvex functions.

1. Introduction

In this section, we recall some basic concepts, properties and results in the convex analysis. For more details, see [1, 2] and the references therein. Let $K$ be a set in the finite dimensional Euclidean space $\mathbb{R}^{n}$,whose inner product and norm are denoted by $\langle.,.\rangle$ and $\|.\|$ respectively.

Definition 1.1 . A set $K$ in $\mathbb{R}^{n}$ is said to be a convex set, if and only if, $$ (1 - t) u + tv \in K,\;\; for\; all\;u,v\;\in K\;,\;t\;\in[0,1]. $$

Definition 1.2. A function $f$ on the convex set $K$ is said to be a convex function if and only if $$ f((1 - t)u + tv) \leq (1 - t)f(u) + t f(v),\;\; for\; all \;u,v\;\in K\;,\;t\;\in[0,1]. $$

For the differentiable convex function, we have the following interesting result.

Theorem 1.3 .[3] Let $K$ be a nonempty convex set in $\mathbb{R}^{n}$, and let $f$ be a differentiable convex function on the set $K$. Then $u \in K$ is the minimum of $f$ if and only if $u \in K$ satisfies the inequality $$ \langle f'(u), v - u \rangle \geq 0, \; for \;all\;v \in K. $$

Definition 1.4 . [4] A set $K_{\eta} \subseteq \mathbb{R}$ is said to be invex set with respect to the bifunction $\eta(.,.)$ if and only if $$ x + t \eta(y,x) \in K_{\eta},\; for\; all\; x, y \in K_{\eta},\; t \in [0,1]. $$

The invex set $K_{\eta}$ is also called $\eta$-connected set. Note that, if $\eta(b,a) = b - a$, then invex set becomes the convex set. Clearly, every convex set is an invex set but converse is not true in general.

Definition 1.5 . [5] Let $K_{\eta}$ be an invex set in $\mathbb{R}$. Then, a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be preinvex function with respect to the bifunction $\eta(.,.)$ if and only if $$ f(x + t \eta(y,x)) \leq (1 - t) f(x) + t f(y) \;\;for\; all\; x, y \in K_{\eta},\; t \in [0,1]. $$

Theorem 1.6 . [6] Let $K_{\eta}$ be an invex set in $\mathbb{R}$ and let $f$ be a differentiable preinvex function on set $K_{\eta}$. Then $u \in K_{\eta}$ is the minimum of $f$ if and only if $u \in K_{\eta}$ satisfies the inequality $$ \langle f'(u), \eta(v , u )\rangle \geq 0, \; for \;all\;v \in K_{\eta}. $$

Definition 1.7 . [7] A set $K_{h} \subset \mathbb{R}/ \{0\} \rightarrow \mathbb{R}$ is said to be a harmonically convex set if and only if $$ \frac{uv}{v + t (u - v)} \in K_{h}, \;\; for \; all\; u, v \in K_{h},\; t \in [0,1].$$

Definition 1.8 . [7] A function $f : K_{h} \subset \mathbb{R}/ \{0\} \rightarrow \mathbb{R}$ is said to be harmonically convex function if and only if $$ f \left(\frac{xy}{tx + (1 - t)y} \right) \leq (1 - t)f(x) + t f(y),\;\;for \;all \;x,y\in K_{h},\;t\in[0,1].$$

Definition 1.9 . [8] The function $f: I \subset (0, \infty) \rightarrow \mathbb{R}$ is said to be harmonically $(s,m)$-convex in second sense, where $s \in (0,1]$ and $m \in (0,1]$ if $$f \big(\frac{mxy}{mty + (1 - t)x}\big) = f \big( (\frac{t}{x} + \frac{1 - t}{my})^{-1} \big) \leq t^{s} f(x) + m (1 - t)^{s} f(y)$$ $\forall x, y \in I$ and $t \in [0,1]$.

Remark 1.1 . Note that for $s = 1$, $(s,m)$-convexity reduces to harmonically $m$-convexity and for $m = 1$, harmonically $(s,m)$-convexity reduces to harmonically $s$-convexity in second sense and for $s,m = 1$, harmonically $(s,m)$-convexity reduces to ordinary harmonically convexity .

Definition 1.10. A set $I = [a, a + \eta(b,a)] \subseteq \mathbb{R}/ \{0\}$ is said to be a harmonic invex set with respect to the bifunction $\eta(,)$ if and only if $$ \frac{x(x + \eta(y,x))}{x + (1 - t)\eta(y,x)} \in I,\;\; for\;all\;x,y\in I,\;t\in[0,1]$$

Definition 1.11. Let $h:[0,1] \subseteq J \rightarrow \mathbb{R}$ be a non-negative function. A function $f: I \rightarrow [a, a + \eta (b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}$ is relative harmonic preinvex function with respect to an arbitrary nonnegative function $h$ and an arbitrary bifunction $\eta(,)$ if $$ f\left(\frac{x(x + \eta(y,x))}{x + (1 - t)\eta(y,x)}\right) \leq h(1 - t)f(x) + h(t)f(y), \;\; for\;all\;x,y\in I,\;t\in[0,1] $$

2. Main Results

Now, we define the class of harmonically $(s,m)$-preinvex functions which is motivated by the definition of harmonically $(s,m)$-convex functions defined by I. A. Baloch et al.[8].

Definition 2.1. A function $f: [a, a + \eta (b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}$ is said to be harmonically $(s,m)$-preinvex functions with respect to the bifunction $\eta(,)$, if $$ f\left( \frac{x(x + \eta(my, x))}{x + t \eta(my, x)}\right) = f \left( \frac{t}{x} + \frac{1 - t}{x + \eta (my,x)} \right)^{-1} \leq t^{s} f(x) + m(1 - t)^{s} f(y) $$ for all $x , y \in [a, a+ \eta(b , a)]$, with $x < my$, $t \in [0, 1]$, $s \in (0, 1]$, $m \in (0, 1]$.

Note: if $\eta(y,x) = y - x$, then harmonic $(s,m)$-preinvexity reduce to harmonic $(s,m)$-convexity. We need the following identity, which plays an important role in the derivations of our main results.

Lemma 2.2. Let $f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}$ be a differentiable function on the interior of $I^{\circ}$ of $I$. If $f' \in [a, a + \eta (mb, a)]$ and $\lambda \in [0,1]$, then \begin{eqnarray*} &&\Upsilon_{f}(a,a + \eta(mb,a);\lambda)\\ &=& \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left[\int_{0}^{\frac{1}{2}}\frac{\lambda - 2t}{(a + t\eta(mb,a))^{2}} f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\right.\\ &&\left.+ \int^{1}_{\frac{1}{2}}\frac{2 - 2t - \lambda}{(a + t\eta(mb,a))^{2}} f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\right] \end{eqnarray*} where $$ \Upsilon_{f}(a,a + \eta(mb,a);\lambda) $$ $$= (1 - \lambda) f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \lambda\left[\frac{f(a) + f(a + \eta(mb,a))}{2}\right]$$ $$ -\frac{2a(a + \eta(mb,a))}{\eta(mb,a)} \int_{a}^{a + m\eta(b,a)} \frac{f(x)}{x^{2}}dx $$

Proof. Integrating by parts, we have \begin{eqnarray*} &&I_{1} = \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{\lambda - 2t}{(a + t\eta(mb,a))^{2}} f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\\ &&=\frac{1 - \lambda}{2} f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \frac{\lambda}{2} f(a + \eta(mb,a)) \\ &&- \int_{0}^{\frac{1}{2}}f\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt, \end{eqnarray*} and $$ I_{2} = \frac{a(a,a + \eta(mb,a))\eta(b,a)}{2}\int^{1}_{\frac{1}{2}}\frac{2t - 2 + \lambda}{(a + t\eta(mb,a))^{2}} f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt$$ $$= \frac{1 - \lambda}{2} f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \frac{\lambda}{2} f(a )- \int^{1}_{\frac{1}{2}}f\left( \frac{a(a + \eta(mb,a))}{x + t\eta(mb,a)}\right)dt$$ Thus $$I_{1} + I_{2}$$ $$= (1 - \lambda) f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \lambda\left[\frac{f(a) + f(a + \eta(mb,a))}{2}\right]$$ $$ -\frac{2a(a + \eta(mb,a))}{\eta(mb,a)} \int_{a}^{a + m\eta(b,a)} \frac{f(x)}{x^{2}}dx$$ which is the required result.

Theorem 2.3. Let $f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}$ be a differentiable function on the interior $I^{\circ}$ of $I$. If $f' \in [a, a + \eta (mb, a)]$ and $|f'|^{q}$ is harmonic $(s,m)$-preinvex function on $I$ for $q \geq 1$ and $\lambda \in[0,1]$, then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\sigma_{1}(a,b;\lambda)^{1 - \frac{1}{q}}\{\sigma_{2}(a,b;\lambda, s)|f'(a)|^{q}$$ $$ + m \sigma_{3}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}}$$ $$ + \sigma_{4}(a,b;\lambda)^{1 - \frac{1}{q}}\{\sigma_{5}(a,b;\lambda, s)|f'(a)|^{q} + m \sigma_{6}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}},$$ where one can evaluate these integrals using any mathematical software (i.e maple). $$\sigma_{1}(a,b;\lambda) = \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} dt , $$ $$ \sigma_{2}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|(1 - t)^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$ \sigma_{3}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|t^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$\sigma_{4}(a,b;\lambda) = \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}} dt $$ $$ \sigma_{5}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|2 - 2t - \lambda|(1 - t)^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$ \sigma_{6}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|2 - 2t - \lambda|t^{s}}{(a + t\eta(mb,a))^{2}} . $$

Proof. Using Lemma 2.2 and the power mean inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$\leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a +t\eta(mb,a)}\right)\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left( \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} dt\right)^{1 - \frac{1}{q}}$$ $$ \times \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}}$$ $$+ \left( \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}}\right)^{ 1 - \frac{1}{q}}$$ $$ \times \left(\int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt \right)^{\frac{1}{q}}$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(b,a)}{2}\left( \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} dt\right)^{1 - \frac{1}{q}}$$ $$ \times \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} \{t^{s}|f'(a)|^{q} + m(1 - t)^{s} |f'(b)|^{q} \}dt\right)^{\frac{1}{q}}$$ $$+ \left( \int^{1}_{\frac{1}{2}}\frac{| 2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}}\right)^{ 1 - \frac{1}{q}}$$ $$ \times \left(\int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}} \{t^{s}|f'(a)|^{q} + m(1 - t)^{s} |f'(b)|^{q} \}dt\right)^{\frac{1}{q}}$$ $$ = \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\sigma_{1}(a,b;\lambda)^{1 - \frac{1}{q}}\{\sigma_{2}(a,b;\lambda, s)|f'(a)|^{q} $$ $$+ m \sigma_{3}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}}$$ $$ + \sigma_{4}(a,b;\lambda)^{1 - \frac{1}{q}}\{\sigma_{5}(a,b;\lambda, s)|f'(a)|^{q} + m \sigma_{6}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}},$$ which is the required result.

If $q=1$, then Theorem 1.6 reduces to the following result, which appears to be a better new one than already exists.

Corollary 2.4 . Let $f: [a, a + \eta (m b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}$ be a differentiable function on the interior of $I^{\circ}$ of $I$. If $f' \in [a, a + \eta (mb, a)]$ and $|f'|$ is harmonic $(s,m)$-preinvex function on $I$ and $\lambda \in[0,1]$, then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\{\sigma_{2}(a,b;\lambda, s) + m \sigma_{3}(a,b;\lambda, s)\}|f'(a)|$$ $$ + \{\sigma_{5}(a,b;\lambda, s) + m\sigma_{6}(a,b;\lambda, s)\}|f'(b)|,$$ where $\sigma_{2}(a,b;\lambda, s)$, $\sigma_{3}(a,b;\lambda, s)$, $\sigma_{5}(a,b;\lambda, s)$, $\sigma_{6}(a,b;\lambda, s)$ are given as in Theorem 1.6.

Theorem 2.5. Let $f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R} $ be a differentiable function on the interior of $I^{\circ}$ of $I$. If $f' \in [a, a + \eta (mb, a)]$ and $|f'|^{q}$ is harmonic $(s,m)$-preinvex function on $I$ for $p,q > 1$, $\frac{1}{p} + \frac{1}{q} = 1$ and $\lambda \in[0,1]$, then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$\leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} (\sigma_{7}(a,b;\lambda, p))^{\frac{1}{p}}$$$$\times \left(\{(1 - \frac{1}{2^{s + 1}} )|f'(a)|^{q} + \frac{m}{2^{s + 1}} |f'(b )|^{q} \}\right)^{\frac{1}{q}} $$ $$ + (\sigma_{8}(a,b;\lambda, p))^{\frac{1}{p}} \left(\{m(1 - \frac{1}{2^{s + 1}} )|f'(b)|^{q} + \frac{1}{2^{s + 1}} |f'(a )|^{q} \}\right)^{\frac{1}{q}} , $$ where $$ \sigma_{7}(a,b;\lambda, p) = \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt, $$ $$\sigma_{8}(a,b;\lambda, p) = \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt . $$

Proof. Using Lemma 2.2 and Holder's integral inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|}{(a + t\eta(mb,a))^{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ \leq \frac{a(a,a + \eta(mb,a))\eta(mb,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$$$\times \left(\int_{0}^{\frac{1}{2}}\left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}} \left(\int^{1}_{\frac{1}{2}} \left|f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(b,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda -2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$ $$\times \left(\int_{0}^{\frac{1}{2}}\{t^{s} |f'(a)|^{q} + m(1 - t)^{s} |f'(b)|^{q}\}dt\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}} \left(\int^{1}_{\frac{1}{2}} \{t^{s} |f'(a)|^{q} + m(1 - t)^{s} |f'(b)|^{q}\}dt\right)^{\frac{1}{q}} $$ $$= \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda - 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$ $$\times \left(\{(1 - \frac{1}{2^{s + 1}} )|f'(a)|^{q} + m \frac{1}{2^{s + 1}} |f'(b )|^{q} \}\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 - 2t - \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}}$$$$ \times \left(\{m(1 - \frac{1}{2^{s + 1}} )|f'(b)|^{q} + \frac{1}{2^{s + 1}} |f'(a )|^{q} \}\right)^{\frac{1}{q}} $$ The proof completes.

Theorem 2.6 . Let $f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R} $ be a differentiable function on the interior $I^{\circ}$ of $I$. If $f' \in [a, a + \eta (mb, a)]$ and $|f'|^{q}$ is harmonic $(s,m)$-preinvex function on $I$ for $p,q > 1$, $\frac{1}{p} + \frac{1}{q} = 1$ and $\lambda \in[0,1]$, then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \times \left( \frac{\lambda^{p + 1} + ( 1 - \lambda)^{ p + 1}}{2(p + 1)} \right)^{\frac{1}{p}} $$ $$ \times (\sigma_{9}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{10}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}}$$ $$ + (\sigma_{11}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{12}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}} ,$$ where $$ \sigma_{9}(a,b;\lambda, p) = \int^{\frac{1}{2}}_{0}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt$$ $$ \sigma_{10}(a,b;\lambda, p) = \int^{\frac{1}{2}}_{0}\frac{(1 - t)^{s}}{(a + t\eta(mb,a))^{2q}}dt $$ $$ \sigma_{11}(a,b;\lambda, p) = \int_{\frac{1}{2}}^{1}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt$$ $$ \sigma_{12}(a,b;\lambda, p) = \int_{\frac{1}{2}}^{1}\frac{(1 - t)^{s}}{(a + t\eta(mb,a))^{2q}}dt $$

Proof. Using Lemma 2.2 and the Holder's integral inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}|\lambda - 2t| \left|\frac{f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}| 2 - 2t - \lambda| \left|\frac{f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|dt $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\left( \int_{0}^{\frac{1}{2}}| \lambda- 2t|^{p}dt\right)^{\frac{1}{p}} $$ $$\times \left(\int_{0}^{\frac{1}{2}}\left|\frac{f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|^{q}dt \right)^{\frac{1}{q}} $$ $$ + \left(\int^{1}_{\frac{1}{2}}|2 -2t - \lambda|^{p}dt\right)^{\frac{1}{p}} \left( \int^{1}_{\frac{1}{2}}\left|\frac{f'\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t)\eta(mb,a))^{2}}\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\left( \int_{0}^{\frac{1}{2}}|\lambda - 2t|^{p}dt\right)^{\frac{1}{p}} $$ $$ \times \left( |f'(a)|^{q}\int^{\frac{1}{2}}_{0}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt + m|f'(b)|^{q} \int^{\frac{1}{2}}_{0}\frac{(1 - t)^{s}}{(a + t\eta(mb,a))^{2q}}dt \right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}|2 - 2t - \lambda|^{p}dt\right)^{\frac{1}{p}} $$ $$\times \left( |f'(a)|^{q}\int^{1}_{\frac{1}{2}}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt + m |f'(b)|^{q} \int^{1}_{\frac{1}{2}}\frac{(1 - t)^{s}}{(a + t\eta(mb,a))^{2q}}dt \right)^{\frac{1}{q}} $$ $$= \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \times \left( \frac{\lambda^{p + 1} + ( 1 - \lambda)^{ p + 1}}{2(p + 1)} \right)^{\frac{1}{p}} $$ $$\times (\sigma_{9}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{10}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}}$$ $$ + (\sigma_{11}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{12}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}} .$$ This completes the proof.

3. Conclusion

In this paper, we have studied the class of Harmonically $(s,m)$-preinvex functions which is generalization of Harmonically preinvex functions and have established similar results to Hermite-Hadamard inequalities for this class. Class of functions defined in this paper may stimulate further research in this field. The interested researchers are encouraged to find the particular examples of new class of functions presented in this paper.

Competing Interests

The author do not have any competing interests in the manuscript.

References

Mohan, S. R., & Neogy, S. K. (1995). On invex sets and preinvex functions. Journal of Mathematical Analysis and Applications, 189(3), 901-908. [Google Scholor]
Noor, M.A., Noor, K.I., Iftikhar, S., (2016), Hermite-Hadamard inequalities for harmonic preinvex functions, Saussurea, 6(2), 34-53. [Google Scholor]
Stampacchia, G. (1964). Formes bilinéaires coercitives sur les ensembles convexes.CR Acad. Sci. Paris, 258(1), 964. [Google Scholor]
Hanson, M.A., (1981), Hanson, M. A. (1981). On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 80(2), 545-550. [Google Scholor]
Weir, T., & Mond, B. (1988). Pre-invex functions in multiple objective optimization. Journal of Mathematical Analysis and Applications, 136(1), 29-38. [Google Scholor]
Aslam Noor, M. (1994). Variational-like inequalities. Optimization, 30(4), 323-330. [Google Scholor]
Iscan, I., (2014). Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions. Journal of Mathematics, 2014., 2014: Article ID 346305, 10 pages. [Google Scholor]
I. A. Baloch, I.İşcan, Some Ostrowski Type Inequalities For Harmonically $(s,m)$-convex functoins in Second Sense, International Journal of Analysis,Volume 2015, Article ID 672675, 9 pages. [Google Scholor]
Noor, M. A., Noor, K. I., & Iftikhar, S. (2016). Fractional ostrowski inequalities for harmonic $h$-preinvex functions. Facta Universitatis, Series: Mathematics and Informatics, 31(2), 417-445. [Google Scholor]

K-Banhatti and K-hyper Banhatti indices of dominating David Derived network

pisrt — Sun, 31 Dec 2017 04:00:54 +0000

Open Access Full-Text PDF

Open Journal of Mathematical Analysis

Vol. 1 (2017), Issue 1, pp. 13–24
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2017.0002

K-Banhatti and K-hyper Banhatti indices of dominating David Derived network

Wei Gao, Batsha Muzaﬀar$^{1}$, Waqas Nazeer

School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China.; (W.G)
Department of Mathematics and Statistics, University of Lahore, Lahore-54590, Pakistan.; (B.M)
Division of Science and Technology, University of Education, Lahore-54590, Pakistan.; (W.N)
$^{1}$Corresponding Author; batshamuzaffar41@gmail.com

Copyright © 2017 Wei Gao, Batsha Muzaﬀar, Waqas Nazeer . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: September 21, 2017 – Accepted: October 15, 2017 – Published: December 31, 2017

Abstract

Let $G$ be connected graph with vertex $V(G)$ and edge set $E(G)$. The first and second $K$-Banhatti indices of $G$ are defined as $B_{1}(G)=\sum\limits_{ue}[d_{G}(u)+d_{G}(e)]$ and $B_{2}(G)=\sum\limits_{ue}[d_{G}(u)d_{G}(e)]$ ,where $ue$ means that the vertex $u$ and edge $e$ are incident in $G$. The first and second $K$-hyper Banhatti indices of $G$ are defined as $HB_{1}(G)=\sum\limits_{ue}[d_{G}(u)+d_{G}(e)]^{2}$ and $HB_{2}(G)=\sum\limits_{ue}[d_{G}(u)d_{G}(e)]^{2}$. In this paper, we compute the first and second $K$-Banhatti and $K$-hyper Banhatti indices of Dominating David Derived networks.

Keywords:

K-Banhatti index; K-hyper Banhatti index; Dominating David Derived networks.

Introduction

Chemical graph theory is a branch of graph theory in which a chemical compound is represented by simple graph called molecular graph in which vertices are atoms of compound and edges are the atomic bounds. A graph is connected if there is atleast one connection between its vertices. Throughout this paper we take $G$ a connected graph. If a graph does not contain any loop or multiple edges then it is called a network. Between two vertices $u$ and $v$, the distance is the shortest path between them and is denoted by in graph $G$. For a vertex $v$ of $G$ the degree is number of vertices attached with it. The edge connecting the vertices $u$ and $v$ will be denoted by $uv$. Let $d_{G}(e)$ denote the degree of an edge $e$ in $G$, which is defined by $d_{G}(e)=d_{G}(u) + d_{G}(v)- 2$ with $e=uv$. The degree and valence in chemistry are closely related with each other. We refer the book [1] for more details. Now a day another emerging field is Cheminformatics, which helps to predict biological activities with the relationship of Structure-property and quantitative structure-activity. Topological indices and Physico-chemical properties are used in prediction of bioactivity if underlined compounds are used in these studies [2,3 ].

A number that describe the topology of a graph is called topological index. In 1947, the first and most studied topological index was introduced by Weiner [4]. For more details about this index can be found in [5, 6]. In 1975, Milan Randic introduced the Randic index [7]. Bollobas et al. [8] and Amic et al. [9] in 1998, working independently defined the generalized Randic index. This index was studied by both mathematicians and chemists [10]. For details about topological indices, we refer [11,12 ] The first and second $K$-Banhatti indices of $G$ are defined as:

$$B_{1}(G)=\sum\limits_{ue}[d_{G}(u)+d_{G}(e)]$$ and $$B_{2}(G)=\sum\limits_{ue}[d_{G}(u)\times d_{G}(e)]$$ where ue means that the vertex $u$ and edge e are incident in $G$. The first and second K-hyper Banhatti indices of G are defined as $$HB_{1}(G)=\sum\limits_{ue}[d_{G}(u)+d_{G}(e)]^{2}$$ and $$HB_{2}(G)=\sum\limits_{ue}[d_{G}(u)\times d_{G}(e)]^{2}.$$ We refer [13] for details about these indices. The David derived and dominating David derived network of dimension $n$ can be constructed as follows [14]: consider a n dimensional star of David network . Insert a new vertex on each edge and split it into two parts, we will get David derived network $DD(n)$ of dimension $n$.

Figure 1. Dominating David derived network of the first type $D_{1}(2)$

The dominating David derived network of the first type of dimension $n$ which can be obtained by connecting vertices of degree $2$ of $DDD(n)$ by an edge that are not in the boundary and is denoted by $D_{1}(n)$ [14]. The dominating David derived network of the second type of dimension n can be obtained by subdividing the new edges in $D_{1}(n)$ [14] and is denoted by $D_{1}(2)$.

Figure 2. Dominating David derived network of the second type $D_{2}(2)$

The dominating David derived network of the second type of dimension $n$ denoted by $D_{3}(n)$ can be obtained from $D_{1}(n)$ by introducing a parallel path of length 2 between the vertices of degree two that are not in the boundary [14, 15].

Figure 3. Dominating David derived network of the third type $D_{3}(2)$

In this article, we compute first and second $K$-Banhatti index and first and second hyper $K$-Banhatti index of Dominating David derived networks of first, second and thord type. Throughout this paper $E_{m,n}=\{e=uv\in E(G); d_{u}=m, d_{v}=n\}$ and $|E_{m,n}(G)|$ is the number of elements in $E_{m,n}(G)$.

Theorem 1.1 Let $G=D_{1}(n)$ be the dominating David derived network of $1^{st}$ type. Then the first and the second $K$-Banhatti indices of $D_{1}(n)$ are \begin{eqnarray*} B_{1}[D_{1}(n)]&=&1485n^{2}+1624n-1002\\ B_{2}[D_{1}(n)]&=&3204n^{2}+764n-3292 \end{eqnarray*}

Proof. Let $G=D_{1}(n)$ be the dominating David derived network of $1^{st}$ type. From figure 1, the edge partition of dominating David derived network of $1^{st}$ type $D_{1}(n)$ based on degrees of end vertices of each edge is give in table 1.

Edge partition of Dominating David derived network of first type

$(d_{u}, d_{v})$	Number of edges	Degree of Edges
$(2,2)$	$4n$	$2$
$(2,3)$	$4n-4$	$3$
$(2,4)$	$28n-16$	$4$
$(3,3)$	$9n^{2}-13n+24$	$4$
$(3,4)$	$36n^{2}-56n+24$	$5$
$(4,4)$	$36n^{2}-56n+20$	$6$

First $k$-Banhatti index of $D_{1}(n)$ is calculated as \begin{eqnarray*} B_{1}[D_{1}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(e)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,3}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &=& 4n[(2+2)+(2+2)]+(4n-4)[(2+3)+(3+3)]\\ &&+(28n-16)[(2+4)+(4+4)]\\ &&+(9n^{2}-13n+24)[(3+4)+(3+4)]\\ &&+(36n^{2}-56n+24)[(3+5)+(4+5)]\\ &&+(36n^{2}-56n+20)[(4+6)+(4+6)]\\ &=& 1458n^{2}+1624n-1002. \end{eqnarray*} Second $K$-Banhatti index of $D_{1}(n)$ is calculated as \begin{eqnarray*} B_{2}[D_{1}(n)]&=&\sum\limits_{ue}[d_{G}(u)d_{G}(v)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,3}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &=& 4n[(2+2)+(2.2)]+(4n-4)[(2+3)+(3.3)]\\ &&+(28n-16)[(2.4)+(4.4)]\\ &&+(9n^{2}-13n+24)[(3.4)+(4.4)]\\ &&+(36n^{2}-56n+24)[(3.5)+(4.5)]\\ &&+(36n^{2}-56n+20)[(4.6)+(4.6)]\\ &=& 3204n^{2}+764n-3292. \end{eqnarray*}

Theorem 1.2. Let $G=D_{1}(n)$ be the dominating David derived network of $1^{st}$ type. Then the first and the second $K$-hyper Banhatti indices of $D_{1}(n)$ are \begin{eqnarray*} HB_{1}[D_{1}(n)]&=&13302n^{2}-16623n+6146\\ HB_{2}[D_{1}(n)]&=&66564n^{2}-89092n+33892 \end{eqnarray*}

Proof Let $G=D_{1}(n)$ be the dominating David derived network of $1^{st}$ type. Then first $K$-hyper Banhatti index of $D_{1}(n)$ is calculated as \begin{eqnarray*} HB_{1}[D_{1}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,3}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &=& 4n[(2+2)^{2}+(2+2)^{2}]+(4n-4)[(2+3)^{2}+(3+3)^{2}]\\ &&+(28n-16)[(2+4)^{2}+(4+4)^{2}]\\ &&+(9n^{2}-13n+24)[(3+4)^{2}+(3+4)^{2}]\\ &&+(36n^{2}-56n+24)[(3+5)^{2}+(4+5)^{2}]\\ &&+(36n^{2}-56n+20)[(4+6)^{2}+(4+6)^{2}]\\ &=& 13302n^{2}-16623n+6146 \end{eqnarray*} Second $K$-hyper Banhatti index of $D_{1}(n)$ is calculated as \begin{eqnarray*} HB_{2}[D_{1}(n)]&=&\sum\limits_{ue}[d_{G}(u)d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,3}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &=& 4n[4^{2}+4^{2}]+(4n-4)[6^{2}+9^{2}]\\ &&+(28n-16)[8^{2}+16^{2}]\\ &&+(9n^{2}-13n+24)[12^{2}+12^{2}]\\ &&+(36n^{2}-56n+24)[15^{2}+20^{2}]\\ &&+(36n^{2}-56n+20)[24^{2}+24^{2}]\\ &=& 66564n^{2}-89092n+33892. \end{eqnarray*}

Theorem 1.3. Let $G=D_{2}(n)$ be the dominating David derived network of $2^{nd}$ type. Then the first and the second $K$-Banhatti indices of $D_{2}(n)$ are \begin{eqnarray*} B_{1}[D_{2}(n)]&=&1530n^{2}-1810n+650\\ B_{2}[D_{2}(n)]&=&32584n^{2}-4127n+1506 \end{eqnarray*}

Proof Let $G=D_{2}(n)$ be the dominating David derived network of $2^{nd}$ type. Table $2$ shows the edge partition of dominating David derived network of $2^{nd}$ type $D_{2}(n)$ based on degrees of end vertices of each edge

Edge partition of Dominating David derived network of second type

$(d_{u}, d_{v})$	Number of edges	Degree of Edges
$(2,2)$	$4n$	$2$
$(2,3)$	$18n^{2}-22n+6$	$3$
$(2,4)$	$28n-16$	$4$
$(3,4)$	$36n^{6}-56n+24$	$5$
$(4,4)$	$336n^{6}-56n+20$	$6$

First $K$-Banhatti index of $D_{2}(n)$ is calculated as \begin{eqnarray*} B_{1}[D_{2}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &=& 4n[(2+2)+(2+2)]+(18n^{2}-22n+6)[(2+3)+(3+3)]\\ &&+(28n-16)[(2+4)+(4+4)]\\ &&+(36n^{6}-56n+24)[(3+5)+(4+5)]\\ &&+(36n^{6}-56n+20)[(4+6)+(4+6)]\\ &=& 1530n^{2}-1810n+650. \end{eqnarray*} Second $K$-Banhatti index of $D_{2}(n)$ is calculated as \begin{eqnarray*} B_{1}[D_{2}(n)]&=&\sum\limits_{ue}[d_{G}(u)d_{G}(v)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &=& 4n[(2.2)+(2.2)]+(18n^{2}-22n+6)[(2.3)+(3.3)]\\ &&+(28n-16)[(2.4)+(4.4)]\\ &&+(36n^{6}-56n+24)[(3.5)+(4.5)]\\ &&+(36n^{6}-56n+20)[(4.6)+(4.6)]\\ &=& 32584n^{2}-4127n+1506. \end{eqnarray*}

Theorem 1.4. Let $G=D_{2}(n)$ be the dominating David derived network of $2^{nd}$ type. Then the first and the second $K$-hyper Banhatti indices of $D_{2}(n)$ are \begin{eqnarray*} HB_{1}[D_{1}(n)]&=&1351n^{2}-1693n+6246\\ HB_{2}[D_{1}(n)]&=&22606n^{2}-28486n+10582 \end{eqnarray*}

Proof Let $G=D_{2}(n)$ be the dominating David derived network of $2^{nd}$ type. First $K$- hyper Banhatti index of $D_{2}(n)$ is calculated as \begin{eqnarray*} HB_{1}[D_{2}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &=& 4n[(2+2)^{2}+(2+2)^{2}]+(18n^{2}-22n+6)[(2+3)^{2}+(3+3)^{2}]\\ &&+(28n-16)[(2+4)^{2}+(4+4)^{2}]\\ &&+(36n^{6}-56n+24)[(3+5)^{2}+(4+5)^{2}]\\ &&+(36n^{6}-56n+20)[(4+6)^{2}+(4+6)^{2}]\\ &=& 1351n^{2}-1693n+6246. \end{eqnarray*} Second $K$-hyper Banhatti index of $D_{2}(n)$ is calculated as \begin{eqnarray*} HB_{2}[D_{2}(n)]&=&\sum\limits_{ue}[d_{G}(u)d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,3}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{3,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e)^{2})+(d_{G}(v)d_{G}(e))^{2}]\\ &=& 4n[(2.2)^{2}+(2.2)^{2}]+(18n^{2}-22n+6)[(2.3)^{2}+(3.3)^{2}]\\ &&+(28n-16)[(2.4)^{2}+(4.4)^{2}]\\ &&+(36n^{6}-56n+24)[(3.5)^{2}+(4.5)^{2}]\\ &&+(36n^{6}-56n+20)[(4.6)^{2}+(4.6)^{2}]\\ &=& 22606n^{2}-28486n+10582. \end{eqnarray*}

Theorem 1.5 Let $G=D_{3}(n)$ be the dominating David derived network of $3^{rd}$ type. Then the first and the second $K$-Banhatti indices of $D_{3}(n)$ are \begin{eqnarray*} B_{1}[D_{3}(n)]&=&1944n^{2}-2128n+600\\ B_{2}[D_{3}(n)]&=&4320n^{2}-8224n+2112 \end{eqnarray*}

Proof Let $G=D_{3}(n)$ be the dominating David derived network of $3^{rd}$ type. Table $3$ shows the edge partition of dominating David derived network of $3^{rd}$ type $D_{3}(n)$ based on degrees of end vertices of each edge.

Edge partition of Dominating David derived network of second type

$(d_{u}, d_{v})$	Number of edges	Degree of Edges
$(2,2)$	$4n$	$2$
$(2,4)$	$36n^{2}-20n$	$4$
$(4,4)$	$72n^{2}-108n+44$	$6$

First $K$-Banhatti index of $D_{3}(n)$ is calculated as \begin{eqnarray*} B_{1}[D_{3}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))+(d_{G}(v)+d_{G}(e))]\\ &=& 4n[(2+2)+(2+2)]+(36n^{2}-20n)[(2+4)+(4+4)]\\ &&+(72n^{2}-108n+44)[(2+6)+(4+6)]\\ &=& 1944n^{2}-2128n+600 \end{eqnarray*} Second $K$-Banhatti index is calculated as \begin{eqnarray*} B_{2}[D_{3}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+ \sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e))+(d_{G}(v)d_{G}(e))]\\ &=& 4n[(2.2)+(2.2)]+(36n^{2}-20n)[(2.4)+(4.4)]\\ &&+(72n^{2}-108n+44)[(2.6)+(4.6)]\\ &=& 4320n^{2}-8224n+2112 \end{eqnarray*}

Theorem 1.6. Let $G=D_{3}(n)$ be the dominating David derived network of $3^{rd}$ type. Then the first and the second $K$-hyper Banhatti indices of $D_{3}(n)$ are \begin{eqnarray*} HB_{1}[D_{3}(n)]&=&18000n^{2}-23472n+8800\\ HB_{2}[D_{3}(n)]&=&94464n^{2}-130688n+50688 \end{eqnarray*}

Proof. Let $G=D_{3}(n)$ be the dominating David derived network of $3^{rd}$ type. Then the first $K$-hyper Banhatti index is calculated as \begin{eqnarray*} HB_{1}[D_{3}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)+d_{G}(e))^{2}+(d_{G}(v)+d_{G}(e))^{2}]\\ &=& 4n[(2+2)^{2}+(2+2)^{2}]+(36n^{2}-20n)[(2+4)^{2}+(4+4)^{2}]\\ &&+(72n^{2}-108n+44)[(2+6)^{2}+(4+6)^{2}]\\ &=& 18000n^{2}-23472n+8800 \end{eqnarray*} Second $K$-hyper Banhatti index of $D_{3}(n)$ is calculated as \begin{eqnarray*} HB_{1}[D_{3}(n)]&=&\sum\limits_{ue}[d_{G}(u)+d_{G}(v)]^{2}\\ &=&\sum\limits_{ue\in E_{2,2}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+ \sum\limits_{ue\in E_{2,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &&+\sum\limits_{ue\in E_{4,4}}[(d_{G}(u)d_{G}(e))^{2}+(d_{G}(v)d_{G}(e))^{2}]\\ &=& 4n[(2.2)^{2}+(2.2)^{2}]+(36n^{2}-20n)[(2.4)^{2}+(4.4)^{2}]\\ &&+(72n^{2}-108n+44)[(2.6)^{2}+(4.6)^{2}]\\ &=& 94464n^{2}-130688n+50688 \end{eqnarray*}

Conclusion

In the present report, we have computed First and second K-Banhatti and K-hyer Banhatti indices of Dominating David derived networks of first, second and third type.

2. Competing Interests

The author do not have any competing interests in the manuscript.

References

West, D. B. (2001). Introduction to graph theory (Vol. 2). Upper Saddle River: Prentice hall. [Google Scholor]
Rücker, G., & Rücker, C. (1999). On topological indices, boiling points, and cycloalkanes. Journal of chemical information and computer sciences, 39(5), 788-802. [Google Scholor]
Klavžar, S., & Gutman, I. (1996). A comparison of the Schultz molecular topological index with the Wiener index. Journal of chemical information and computer sciences, 36(5), 1001-1003. [Google Scholor]
Wiener, H. (1947). Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1), 17-20. [Google Scholor]
Dobrynin, A. A., Entringer, R., & Gutman, I. (2001). Wiener index of trees: theory and applications. Acta Applicandae Mathematicae, 66(3), 211-249. [Google Scholor]
Gutman, I., & Polansky, O. E. (2012). Mathematical concepts in organic chemistry. Springer Science & Business Media. [Google Scholor]
Randic, M. (1975). Characterization of molecular branching. Journal of the American Chemical Society, 97(23), 6609-6615. [Google Scholor]
Bollobás, B., & Erdös, P. (1998). Graphs of extremal weights. Ars Combinatoria, 50, 225-233. [Google Scholor]
Amic, D., Bešlo, D., Lucic, B., Nikolic, S., & Trinajstic, N. (1998). The vertex-connectivity index revisited. Journal of chemical information and computer sciences, 38(5), 819-822. [Google Scholor]
Hu, Y., Li, X., Shi, Y., Xu, T., & Gutman, I. (2005). On molecular graphs with smallest and greatest zeroth-order general Randic index. MATCH Commun. Math. Comput. Chem, 54(2), 425-434. [Google Scholor]
Sardar, M. S, Zafar, S., & Farahani, M. R. (2017). The generalized zagreb index of capra-designed planar benzenoid series $ca_k(c_6)$, Open J. Math. Sci., 1(1), 44-51.[Google Scholor]
Rehman, H. M, Sardar, R., Raza, A. (2017). Computing topological indices of hex board and its line graph. Open J. Math. Sci., 1(1), 62- 71.[Google Scholor]
Kulli, V. R., Chaluvaraju, B., & Boregowda, H. S. (2017). Connectivity Banhatti indices for certain families of benzenoid systems. Journal of Ultra Chemistry, 13(4), 81-87. [Google Scholor]
Imran, M., Baig, A. Q., & Ali, H. (2015). On topological properties of dominating David derived networks. Canadian Journal of Chemistry, 94(2), 137-148. [Google Scholor]
Simonraj, F., & George, A. (2012). Embedding of poly honeycomb networks and the metric dimension of star of david network. International Journal on Applications of Graph Theory in Wireless Ad Hoc Networks and Sensor Networks, 4(4), 11. [Google Scholor]

An implicit viscosity technique of nonexpansive mapping in CAT(0) spaces

pisrt — Sun, 31 Dec 2017 03:56:13 +0000

Open Access Full-Text PDF

Open Journal of Mathematical Analysis

Vol. 1 (2017), Issue 1, pp. 01 – 12
ISSN: 2616-8111 ()nline) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2017.0001

An implicit viscosity technique of nonexpansive mapping in CAT(0) spaces

Iftikhar Ahmad$^{1}$, Maqbool Ahmad
Department of Mathematics and Statistics, University of Lahore, Lahore Pakistan.; (I.A)
Department of mathematics and statistics, The university of Lahore, Lahore Pakistan.; (M.A)
$^{1}$Corresponding Author; iftikharcheema1122@gmail.com

Copyright © 2017 Iftikhar Ahmad and Maqbool Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: September 21, 2017 – Revised: November 25, 2017 – Published: December 31, 2017

Abstract

In this paper, we present a new viscosity technique of nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorems of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this paper extend and improve some recent announced in the current literature.

Keywords:

Viscosity rule; CAT(0) space; Nonexpansive mapping; Variational inequality.

1. Introduction

The study of spaces of nonpositive curvature originated with the discovery of hyperbolic spaces, and flourished by pioneering works of J. Hadamard and E. Cartan in the first decades of the twentieth century. The idea of nonpositive curvature geodesic metric spaces could be traced back to the work of H. Busemann and A. D. Alexandrov in the 50's. Later on M. Gromov restated some features of global Riemannian geometry solely based on the so-called CAT(0) inequality (here the letters C, A and T stand for Cartan, Alexandrov and Toponogov, respectively). For through discussion of CAT(0) spaces and of fundamental role they play in geometry , we refer the reader to Bridson and Haefliger [1]. As we know, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [2, 3, 4, 5, 6, 7, 8, 9, 10] and the references therein. One of the difficulties in carrying out results from Banach space to complete CAT(0) space setting lies in the heavy use of the linear structure of the Banach spaces. Berg and Nikolaev [4] introduce the noton of an inner product-like notion( quasilinearization) in complete CAT(0) spaces to resolve these difficulties. Fixed-point theory in CAT(0) spaces was frst studied by Kirk [11, 12, 13]. He showed that every nonexpansive (singlevalued) mapping defned on a bounded closed convex subset of a complete CAT(0) space always has a fxed point. Since then, the fxed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2000, Moudaf's [14] introduce viscosity approximation methods as following:

Theorem 1.1 Let $C$ be a nonempty closed convex subset of the real CAT(0) space $X$. Let $T$ be a nonexpansive mapping of $C$ into itself such that $Fix(T)$ is nonempty. Let $f$ be a contraction of $C$ into itself with coefficient $\theta\in [0,1)$. Pick any $x_{0}\in [0,1)$, let $\{x_{n}\}$ be a sequence generated by $$x_{n+1}=\frac{\gamma_{n}}{1+\gamma_{n}}f(x_{n})+\frac{1}{1+\gamma_{n}}T(x_{n}),\;\;\;n\geq 0.$$ Where $\{\gamma_{n}\}$ is a sequence in $(0,1)$ satisfying the following conditions:

$\lim\limits_{n\rightarrow \infty}\gamma_{n}=0, $
$\sum\limits_{n=0}^{\infty}\gamma_{n}=\infty, $
$\sum_{n=0}^{\infty}|\frac{1}{\gamma_{n+1}}-\frac{1}{\gamma_{n}}|=0. $

Then $\{x_{n}\}$ converges strongly to a fixed point $x^\ast$ of the mapping $T$, which is also the unique solution of the variational inequality $$\langle x-f(x), x-y\rangle\geq 0, \;\;\; \forall \;y\in \textrm{Fix}(T),$$ in other words, $x^{\ast}$ is the unique fixed point of the contraction $P_{Fix(T)}f$, that is $P_{Fix(T)}f(x^\ast)=x^\ast$.

Shi and Chen [15] studied the convergence theorems of the following Moudaf's viscosity iterations for a nonexpansive mapping in CAT(0) spaces.

\begin{equation}\label{fc0} x_{n+1}=tf(x_{n})\oplus (1-t)T(x_{n}) \end{equation}

(1)

\begin{equation}\label{fc1} x_{n+1}=\alpha_{n}f(x_{n})\oplus (1-\alpha_{n})T(x_{n}). \end{equation}

(2)

They proved that $\{x_{n}\}$ defned by (1) and $\{x_{n}\}$ defned by (2) converged strongly to a fxed point of $T$ in the framework of CAT(0) space. In 2017, Zhao et al. [16] applied viscosity approximation methods for the implicit midpoint rule for non-expansive mappings $$x_{n+1}=\alpha_nf(x_n)\oplus(1-\alpha_n)T\left(\frac{x_n\oplus x_{n+1}}{2}\right),\forall n\geq 0.$$ C.Y. Jung [17], proposed two generalized viscosity implicit rules:

\begin{equation} x_{n+1}=\alpha_nf(x_n)\oplus(1-\alpha_n)T\left(s_nx_n\oplus(1-s_n)x_{n+1}\right), \end{equation}

(3)

\begin{equation} x_{n+1}=\alpha_nx_n\oplus\beta f(x_n)+\gamma_nT(s_nx_n\oplus(1-s_n)x_{n+1}). \end{equation}

(4)

Motivated and inspired by the idea of C.Y. Jung [17], in this paper, we extend and study the implicit viscosity rules of nonexpansive mappings in the framework of CAT(0) spaces $$\left\{ \begin{array}{ll} x_{n+1}=T(y_{n}),\\ y_{n}=\alpha_{n}(w_{n})\oplus\beta_{n}f(w_{n})\oplus\gamma_{n}T(w_{n}), \\ w_{n}=\frac{x_{n}\oplus x_{n+1}}{2}. \end{array} \right.$$

2. Preliminaries

Let $(X,d)$ be a metric space. A geodesic path joining $x\in X$ to $y\in X$ (or, more briefly, a geodesic from $x$ to $y$) is a map $c$ from a closed interval $[0,l]\subset R$ to $X$ such that $c(0)=x$, $c(l)=y$, and $d(c(t),c(t'))=|t-t'|$ for all $t,t'\in [0, l]$. In particular, $c$ is an isometry and $d(x,y)=l$. The image $\alpha$ of $c$ is called a geodesic (or metric) segment joining $x$ and $y$. When it is unique, this geodesic segment is denoted by $[x, y]$. The space $(X,d)$ is said to be a geodesic space if every two points of $X$ are joined by a geodesic, and $X$ is said to be uniquely geodesic if there is exactly one geodesic joining $x$ and $y$ for each $x,y \in X$. A subset $Y \subset X$ is said to be convex if $Y$ includes every geodesic segment joining any two of its points. A geodesic triangle $\triangle(x_{1}, x_{2}, x_{3})$ in a geodesic metric space $(X,d)$ consists of three points $x_{1},x_{2}$,and $x_{3}$ in $X$ (the vertices of $\triangle$) and a geodesic segment between each pair of vertices (the edges of $\triangle$). A comparison triangle for the geodesic triangle $\triangle (x_{1},x_{2},x_{3}$ in $(X,d)$ is a triangle $\overline{\triangle}(x_{1},x_{2},x_{3}) :=\triangle (\overline{x_{1}},\overline{ x_{2}}, \overline{x_{3}})$ in the Euclidean plane $\mathbb{E}^2$ such that $d_{\mathbb{E}^2}d(x_{i},x_{j})=d(x_{i},x_{j})$ for $i,j=1,2,3.$\\ A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. CAT(0): let $\triangle$ be a geodesic triangle in $X$, and let $\overline{\triangle}$ be a comparison triangle for $\triangle$ . Then, is said to satisfy the CAT(0) inequality if for all $x, y\in \triangle$ and all comparison points $\overline{x},\overline{y}\in \overline{\triangle}$,

\begin{equation} d(x,y)=d_{\mathbb{E}^2}(\overline{x},\overline{y}). \end{equation}

(5)

Let $x,y\in X$ and by the Lemma $2.1(iv)$ of [18] for each $t\in [0,1]$, there exist a unique point $z\in [x,y]$ such that

\begin{equation} d(x,z)=td(x,y),\;\;\;\; d(y,z)=(1-t)d(x,y). \end{equation}

(6)

From now on, we will use the notation $(1-t)x\oplus ty$ for the unique fixed point $z$ satisfying the above equation. We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 2.1. Let $X$ be a CAT(0) spaces.

For any $x,y,z\in X$ and $t\in [0,1]$,

\begin{equation} d((1-t)x\oplus ty,z)\leq (1-t)d(x,z)+td(y,z) \end{equation}

(7)
For any $x,y,z\in X$ and $t\in [0,1]$,

\begin{equation} d^2((1-t)x\oplus ty,z)\leq (1-t)^2d(x,z)+td^2(y,z)-t(1-t)d^2(x,y). \end{equation}

(8)

Complete CAT(0) spaces are often called Hadamard spaces (see[1]). If $x,y_{1},y_{2}$ are points of a CAT(0) spaces and $y_{0}$ is the midpoint of the segment $[y_{1},y_{2}]$, which we will denoted by $\frac{(y_{1}\oplus y_{2})}{2}$, then the CAT(0) inequality implies

\begin{equation} d^{2}\left(x, \frac{y_{1}\oplus y_{2}}{2}\right)\leq \frac{1}{2}d^2(x,y_{1})+\frac{1}{2}d^2(x,y_{2})-\frac{1}{4}d^2(y_{1},y_{2}). \end{equation}

(9)

This inequality is the (CN) inequality of Bruhat and Tits [19]. In fact, a geodesic space is a CAT(0) space if and only if it satisfes the (CN) inequality (cf. [1], page 163).

Definition 2.2 Let $X$ be a CAT(0) space and $T: X\rightarrow X$ be a mapping. Then $T$ is called nonexpensive if $$d(T(x), T(y))\leq d(x,y), \;\;\; x,y\in C$$

Definition 2.3 Let $X$ be a CAT(0) space and $T: X\rightarrow X$ be a mapping. Then $T$ is called contraction if $$d(T(x), T(y))\leq \theta d(x,y), \;\;\; x,y\in C\;\; \theta \in [0,1)$$

Berg and Nikolaev [4] introduce the concept of quasilinearization as follow. Let us denote the pair $(a,b)\in X\times X$ by the $\overrightarrow{ab}$ and call it a vector. Then, quasilinearization is defined as a map $$\langle .,.\rangle: (x\times X)\times (X\times X) \longrightarrow\mathbb{R}$$ defined as

\begin{equation} \langle \overrightarrow{ab},\overrightarrow{cd}\rangle=\frac{1}{2}(d^{2}(a,d)+d^{2}(b,c)-d^{2}(a,c)-d^{2}(b,d)) \end{equation}

(10)

it is easy to see that $\langle \overrightarrow{ab},\overrightarrow{cd}\rangle=\langle \overrightarrow{cd},\overrightarrow{ab}\rangle$, $\langle \overrightarrow{ab},\overrightarrow{cd}\rangle=-\langle \overrightarrow{ba},\overrightarrow{cd}\rangle$ and $\langle \overrightarrow{ax},\overrightarrow{cd}\rangle+\langle \overrightarrow{xb},\overrightarrow{cd}\rangle=\langle \overrightarrow{ab},\overrightarrow{cd}\rangle$ for all $a,b,c,d\in X$. We say that $X$ satisfies the Cauchy-Schwarz inequality if $$\langle \overrightarrow{ab},\overrightarrow{cd}\rangle\leq d(a,b)d(a,c)$$ for all $a,b,c,d\in X$. It is well-known [4] that a geodesically connected metric space is a CAT(0) space of and only if it satisfy the Cauchy-Schwarz inequality. Let $C$ be a non-empty closed convex subset of a complete CAT(0) space $X$. The metric projection $P_{c}: X\rightarrow C$ is defined by $$u=P_{c}(x)\Longleftrightarrow \inf\{d(y,x):y\in C\},\;\;\; \forall x\in X$$

Definition 2.4. Let $P_{c}: X\rightarrow C$ is called the metric projection if for every $x\in X$ there exist a unique nearest point in $C$, denoted by $P_{c}x$, such that $$d(x, P_{c}x)\leq d(x,y), \;\;\; y\in C.$$

The following theorem gives you the conditions for a projection mapping to be non-expensive.

Theorem 2.5 Let $C$ be a non-empty closed convex subset of a real CAT(0) space $X$ and $P_{c}: X\rightarrow X$ a metric projection. Then

$d(P_{c}x, P_{c}y)\leq \langle \overrightarrow{xy}, \overrightarrow{P_{c}xP_{c}y}\rangle$ for all $x,y\in X$,
$P_{c}$ is non-expensive mapping , that is, $d(x,p_{c}x)\leq d(x,y)$ for all $y\in C$,
$\langle \overrightarrow{xP_{c}x}, \overrightarrow{yP_{c}y}\rangle\leq 0$ for all $x\in X$ and $y\in C$ .

Further if, in addition, $C$ is bounded, then $F(T)$ is nonempty. The following Lemmas are very useful for proving our main results:

Lemma 2.6 (The demiclosedness principle) Let $C$ be a nonempty closed convex subset of the real CAT(0) space $X$ and $T:C\rightarrow C$ such that $$x_n\rightharpoonup x^\ast \in C\,\, \mbox{and}\,\, (I-T)x_n \rightarrow 0.$$ Then $x^\ast=Tx^\ast$. (Here $\rightarrow$ (respectively ⇀) denotes strong (respectively weak) convergence.) Moreover, the following result gives the conditions for the convergence of a nonnegative real sequences.

Lemma 2.7. Assume that $\{a_n\}$ is a sequence of nonnegative real numbers such that $a_{n+1}\leq(1-\beta_n)a_n+\delta_n, \forall n\geq0$, where $\{\beta_n\}$ is a sequence in $(0,1)$ and $\{\delta_n\}$ is a sequence with

$\sum_{n=0}^\infty\beta_n=\infty$
$\lim_{n\rightarrow\infty}\sup\frac{\delta_n}{\beta_n}\leq0$ or $\sum_{n=0}^{\infty}|\delta_n|<\infty$.

Then $\lim\limits_{n\rightarrow \infty} a_n\rightarrow0$.

3. The Main Result

Theorem 3.1. Let $C$ be a non-empty closed convex subset of a complete CAT(0) space $X$ and $T:C\longrightarrow C$ be a non-expensive mapping with $\textrm{Fix}(T)\neq\emptyset$. Let $f:C\longrightarrow C$ be a contraction with coefficient $\theta\in [0,1)$ and for arbitrary initial point $x_{0}\in C$. Let $\{x_{n}\}$ be a sequence generated by $$\left\{ \begin{array}{ll} x_{n+1}=T(y_{n}),\\ y_{n}=\alpha_{n}(w_{n})\oplus\beta_{n}f(w_{n})\oplus\gamma_{n}T(w_{n}), \\ w_{n}=\frac{x_{n}\oplus x_{n+1}}{2}. \end{array} \right.$$ Where $\{\alpha_{n}\},\{\beta_{n}\}$ and $\{\gamma_{n}\}$ are the sequence in $(0,1)$ satisfying the following conditions:

$\alpha_{n}+\beta_{n}+\gamma_{n}=1$
$\lim_{n\longrightarrow \infty}\alpha_{n}=0=\lim_{n\longrightarrow \infty}\beta_{n}$ and $\lim_{n\longrightarrow \infty}\gamma_{n}=1$,
$\sum_{n=0}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty$,
$\sum_{n=0}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty$,
$\lim_{n\longrightarrow \infty}d(x_{n},T(x_{n}))=0 $.

Then $\{x_{n}\}$= converges strongly to a fixed point $x^\ast$ of the mapping $T$, which is also the unique solution of the variational inequality $$\langle \overrightarrow{xf(x)}, \overrightarrow{xy}\rangle\geq 0, \;\;\; \forall \;y\in \textrm{Fix}(T),$$ in other words, $x^{\ast}$ is the unique fixed point of the contraction $P_{Fix(T)}f$, that is $P_{Fix(T)}f(x^\ast)=x^\ast$.

Proof. We divide the proof into four steps:

Step 1: Firstly we show that the sequence $\{x_{n}\}$ is bounded. Indeed take $p\in \textrm{Fix}(T)$ arbitrary, we have \begin{eqnarray*} d(x_{n+1},p)&=& d(T(y_{n}),p)\\ &=& d(T(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}(w_{n})), p)\\ &\leq& d(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}T(w_{n}), p)\\ &=& d(\alpha_{n}(w_{n})-\alpha_{n}p+\beta_{n}(w_{n})-\beta_{n}p+\gamma_{n}T(w_{n})+\alpha_{n}p+\beta_{n}p, p)\\ &\leq & \alpha_{n}d((w_{n}),p)+\beta_{n}d((w_{n}),p)+\gamma_{n}d(T(w_{n}), p)\\ &\leq &\frac{\alpha_{n}}{2}d((x_{n}),p)+\frac{\alpha_{n}}{2}d((x_{n+1}),p)+\beta_{n}d((w_{n}),f(p))+\beta_{n}d(f(p),p)+\gamma_{n}d(T(w_{n}), p)\\ &=& \frac{\alpha_{n}}{2}d((x_{n}),p)+\frac{\alpha_{n}}{2}d((x_{n+1}),p)+\theta\beta d((w_{n}),p)+\beta d(f(p),p)\\ &+&\gamma_{n}\left(\frac{1}{2}d(x_{n}, p)+\frac{1}{2}d(x_{n+1}, p)\right)\\ &=& \left(\frac{\alpha_{n}+\gamma_{n}+\theta\beta_{n}}{2}\right)d(x_{n},p)+\left(\frac{\alpha_{n}+\gamma_{n}+\theta\beta_{n}}{2}\right)d(x_{n+1},p)\\ &+& \frac{\gamma_{n}}{2}d(x_{n+1}),p)+\beta_{n}d(f(p),p)\\ &=& \left(\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n},p)+\left(\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n+1},p)\\ &+& \frac{\gamma_{n}}{2}d(x_{n+1}),p)+\beta_{n}d(f(p),p). \end{eqnarray*} It follows that \begin{eqnarray*} \left(1-\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n+1},p)&=&\left(\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n},p)\\ &&+\beta_{n}d(f(p),p). \end{eqnarray*} implies that

\begin{equation}\label{a1} (1+\beta_{n}(1-\theta))d(x_{n+1},p)\leq (1-\beta_{n}(1-\theta))d(x_{n},p)+2\beta_{n}d(f(p),p). \end{equation}

(11)

Since $\beta_{n}, \theta\in (0,1), 1-\beta_{n}(1-\theta)\geq 0$. Moreover, by (11) and $\alpha_{n}+\beta_{n}+\gamma_{n}=1$, we get \begin{eqnarray*} d(x_{n+1},p)&=&\frac{1-\beta_{n}(1-\theta)}{1+\beta_{n}(1-\theta)}d(x_{n},p)+\frac{2\beta_{n}}{1+\beta_{n}(1-\theta)}d(f(p),p)\\ &\leq &\left[1-\frac{2\beta_{n}(1-\theta)}{1+\beta_{n}(1-\theta)}\right]d(x_{n},p)+\left[\frac{2\beta_{n}(1-\theta)}{1+\beta_{n}(1-\theta)}\right]\left(\frac{1}{1-\theta}d(f(p),p)\right). \end{eqnarray*} Thus we have $$d(x_{n+1},p)\leq \max\left\{d(x_{n},p),\frac{1}{1-\theta} d(f(p),p)\right\}.$$ By applying induction, we obtain $$d(x_{n+1},p)\leq \max\left\{d(x_{0},p),\frac{1}{1-\theta} d(f(p),p)\right\}.$$ Hence, we conclude that $\{x_{n}\}$ is bounded. Consequently, we deduce immediately from it that $\{f(w_{n})\}$ and $\{T(w_{n})\}$ are bounded.

Step 2: Now, we prove that $\lim\limits_{n\rightarrow \infty}d(x_{n+1},x_{n})=0$ \begin{eqnarray*} d(x_{n+1},x_{n})&=& d(T(y_{n}),T(y_{n-1}))\\ &=& d(T(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}(w_{n})), T(\alpha_{n-1}(w_{n-1})\oplus\beta_{n-1}(w_{n-1})\oplus\gamma_{n-1}(w_{n-1})))\\ &\leq& d(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}T(w_{n}), [\alpha_{n-1}(w_{n-1})\oplus\beta_{n-1}(w_{n-1})\oplus\gamma_{n-1}T(w_{n-1})])\\ &\leq &\frac{\alpha_{n}}{2}d(x_{n+1},x_{n})+\frac{\alpha_{n}}{2}d(x_{n},x_{n-1})+\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|d\left((x_{n-1}+x_{n}),2T(w_{n-1})\right)\\ &+&\beta_{n}d(f(w_{n}),f(w_{n-1}))+|\beta_{n}-\beta_{n-1}|d(f(w_{n-1}),T(w_{n-1}))+\gamma_{n}d(T(w_{n}),T(w_{n-1}))\\ &=& \frac{\alpha_{n}}{2}d(x_{n+1},x_{n})+\frac{\alpha_{n}}{2}d(x_{n},x_{n-1})+\left(\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|\right)M\\ &+&\theta\beta_{n}d(w_{n},w_{n-1})+\gamma_{n}(w_{n},w_{n-1})\\ &=& \frac{\alpha_{n}}{2}d(x_{n+1},x_{n})+\frac{\alpha_{n}}{2}d(x_{n},x_{n-1})+\left(\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|\right)M\\ &+&\frac{\theta\beta_{n}}{2}d(x_{n+1},x_{n})+\frac{\theta\beta_{n}}{2}d(x_{n},x_{n-1})+ \frac{\gamma_{n}}{2}d(x_{n+1},x_{n})+\frac{\gamma_{n}}{2}d(x_{n},x_{n-1})\\ &=& \left(\frac{\alpha_{n}+\gamma_{n}+\theta\beta_{n}}{2}\right)d(x_{n+1},x_{n})+\left(\frac{\alpha_{n}+\gamma_{n}+\theta\beta_{n}}{2}\right)d(x_{n},x_{n-1})\\ &+& \left(\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|\right)M \end{eqnarray*} Where $M>0$ is constant such that $$M\geq \max\left\{\sup_{n\geq 0}d((x_{n}+x_{n+1},2T(w_{n-1})),\sup_{n\geq 0}d(f(w_{n-1}),T(w_{n-1}))\right\}$$ It gives \begin{eqnarray*} \left(1-\frac{\alpha_{n}+\theta\beta_{n}+\gamma_{n}}{2}\right)d(x_{n+1},x_{n})&=&\left(\frac{\alpha_{n}+\theta\beta_{n}+\gamma_{n}}{2}\right)d(x_{n},x_{n-1})\\ &+& \left(\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|\right)M \end{eqnarray*} implies that \begin{eqnarray*} \left(1-\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n+1},x_{n})&=&\left(\frac{1-\beta_{n}+\theta\beta_{n}}{2}\right)d(x_{n},x_{n-1})\\ &+& \left(\frac{1}{2}|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|\right)M \end{eqnarray*} implies \begin{eqnarray*} (1+\beta_{n}(1-\theta))d(x_{n+1},x_{n})&\leq &(1-\beta_{n}(1-\theta))d(x_{n},x_{n-11})\\ &+& \left(|\alpha_{n}-\alpha_{n-1}|+2|\beta_{n}-\beta_{n-1}|\right)M. \end{eqnarray*} Thus, we have \begin{eqnarray*} d(x_{n+1},x_{n})&\leq&\left(\frac{1-\beta_{n}(1-\theta)}{1+\beta_{n}(1-\theta)}\right)d(x_{n},x_{n-1})\\ &+&\frac{M}{(1+\beta_{n}(1-\theta))} \left(|\alpha_{n}-\alpha_{n-1}|+2|\beta_{n}-\beta_{n-1}|\right). \end{eqnarray*} Since $\beta_{n}, \theta\in (0,1), 1+\beta_{n}(1-\theta)\geq 1$ and $\left(\frac{1-\beta_{n}(1-\theta)}{1+\beta_{n}(1-\theta)}\right)\leq 1-\beta_{n}(1-\theta)$ Thus \begin{eqnarray*} d(x_{n+1},x_{n})&\leq&[1-\beta_{n}(1-\theta)]d(x_{n},x_{n-1})\\ &+&\frac{M}{(1+\beta_{n}(1-\theta))} \left(|\alpha_{n}-\alpha_{n-1}|+2|\beta_{n}-\beta_{n-1}|\right). \end{eqnarray*} Since $\sum_{n=0}^{\infty}\beta_{n}=\infty$, $\sum_{n=0}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty$,and $\sum_{n=0}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty$, by the Lemma (2.7) we have $\lim\limits_{n\rightarrow \infty}d(x_{n+1},x_{n})=0$.

Step 3: In this step, we claim that $$\limsup\limits_{x\rightarrow\infty}\langle\overrightarrow{x^{\ast}f(x^\ast)},\overrightarrow{x^{\ast}x_{n}}\rangle\leq 0,$$ where $x^{\ast}=P_{Fix(T)}f(x^{\ast})$. Indeed, we take a subsequence $\{x_{n_{i}}\}$ of $\{x_{n}\}$ which converges weakly to a fixed point $p$ of $T$. Without loss of generality, we may assume that $\{x_{n_{i}}\}\rightharpoonup p$. From $\lim\limits_{n\rightarrow \infty}d(x_{n},T(x_{n})=0$ and the Lemma (2.6) we have $p=Tp$. This together, with the properity of metric projection implies that \begin{eqnarray*} \limsup\limits_{x\rightarrow\infty}\langle\overrightarrow{x^{\ast}f(x^\ast)},\overrightarrow{x^{\ast}x_{n}}\rangle &=&\limsup\limits_{x\rightarrow\infty}\langle\overrightarrow{x^{\ast}f(x^\ast)},\overrightarrow{x^{\ast}x_{n_{i}}}\rangle\\ &=&\limsup\limits_{x\rightarrow\infty}\langle\overrightarrow{x^{\ast}f(x^\ast)},\overrightarrow{x^{\ast}p}\rangle\\ &\leq & 0. \end{eqnarray*}

Step 4: Finally, we show that $x_{n}\rightarrow x^{\ast}$ as $n\rightarrow \infty$. Now, we prove that $\lim\limits_{n\rightarrow \infty}d(x_{n+1},x_{n})=0$. Now, we again take $x^{\ast}\in \textrm{Fix}(T)$ is the unique fixed point of the contraction $P_{\textrm{Fix}(T)}f$. Consider \begin{eqnarray*} d^2(x_{n+1},x_{n})&=& d^2(T(y_{n}),x^{\ast})\\ &=& d^2(T(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}(w_{n})), x^{\ast})\\ &\leq& d^{2}(\alpha_{n}(w_{n})\oplus\beta_{n}(w_{n})\oplus\gamma_{n}T(w_{n}),x^{\ast} )\\ &=& d^{2}(\alpha_{n}(w_{n})-\alpha_{n}x^{\ast}+\beta_{n}(w_{n})-\beta_{n}x^{\ast}+\gamma_{n}T(w_{n})+\alpha_{n}x^{\ast}+\beta_{n}x^{\ast}, x^{\ast})\\ &=& \alpha^{2}_{n}d^2((w_{n}),x^{\ast})+\beta^{2}_{n}d^2((w_{n}),x^{\ast})+\gamma^{2}_{n}d^2((w_{n}),x^{\ast})\\ &+&2\alpha_{n}\beta_{n}\langle\overrightarrow{x^{\ast}w_{n}},\overrightarrow{x^{\ast}f(w_{n})}\rangle+2\alpha_{n}\gamma_{n}\langle\overrightarrow{x^{\ast}w_{n}},\overrightarrow{x^{\ast}T(w_{n})}\rangle\\ &+& 2\beta_{n}\gamma_{n}\langle\overrightarrow{x^{\ast}f(w_{n})},\overrightarrow{x^{\ast}T(w_{n})}\rangle\\ &=& \alpha^{2}_{n}d^2((w_{n}),x^{\ast})+\beta^{2}_{n}d^2((w_{n}),x^{\ast})+\gamma^{2}_{n}d^2((w_{n}),x^{\ast})\\ &+&2\alpha_{n}\beta_{n}\langle\overrightarrow{x^{\ast}w_{n}},\overrightarrow{x^{\ast}f(w_{n})}\rangle+2\alpha_{n}\gamma_{n}d(x_{n},x^{\ast})d(T(w_{n}),x^{\ast})\\ &+& 2\beta_{n}\gamma_{n}\langle\overrightarrow{x^{\ast}f(w_{n})},\overrightarrow{x^{\ast}T(w_{n})}\rangle\\ &\leq& (\alpha^{2}_{n}+\gamma^{2}_{n})d^{2}(w_{n},x^{\ast})+2\alpha_{n}\gamma_{n}d^{2}(w_{n},x^{\ast})+2\beta_{n}\gamma_{n}d^{2}(f(w_{n}),f(x^{\ast}))d^{2}(w_{n},x^{\ast})+K_{n}\\ &\leq& (\alpha^{2}_{n}+\gamma^{2}_{n})d^{2}(w_{n},x^{\ast})+2\theta\beta_{n}\gamma_{n}d^{2}(w_{n},x^{\ast})+K_{n}\\ &\leq&(\alpha^{2}_{n}+\gamma^{2}_{n}+2\theta\beta_{n}\gamma_{n})d^{2}(w_{n},x^{\ast})+K_{n}\\ &\leq&((1-\beta^{2}_{n})^2+2\theta\beta_{n}\gamma_{n})d^{2}(w_{n},x^{\ast})+K_{n} \end{eqnarray*} where \begin{eqnarray*} K_{n}&=&\beta^{2}_{n}d^2(f(w_{n}),x^{\ast})+2\alpha_{n}\beta_{n}\langle\overrightarrow{x^{\ast}w_{n}},\overrightarrow{x^{\ast}f(w_{n})}\rangle\\ &+&2\beta_{n}\gamma_{n}\langle\overrightarrow{f(w_{n})}x^{\ast},\overrightarrow{T(w_{n})x^{\ast}}\rangle \end{eqnarray*} it become $$[(1-\beta)^2+2\theta\beta_{n}\gamma_{n})]d^{2}(w_{n},x^{\ast})\geq d^{2}(x_{n+1},x_{n})-K_{n}$$ implies $$\sqrt{(1-\beta)^2+2\theta\beta_{n}\gamma_{n}}d(w_{n},x^{\ast})\geq \sqrt{d^{2}(x_{n+1},x_{n})-K_{n}}$$ implies \begin{align*} \frac{1}{2}\sqrt{(1-\beta)^2+2\theta\beta_{n}\gamma_{n}}d(w_{n},x^{\ast})(d(x_{n+1},x^{\ast})+d(x_{n},x^{\ast}))\\ \geq \sqrt{d^{2}(x_{n+1},x_{n})-K_{n}} \end{align*} implies \begin{eqnarray*} \frac{1}{4}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})(d^{2}(x_{n+1},x^{\ast})+d^{2}(x_{n},x^{\ast}))&+&2(d(x_{n+1},x^{\ast})+d(x_{n},x^{\ast}))\\ &\geq& d^{2}(x_{n+1},x_{n})-K_{n} \end{eqnarray*} implies \begin{eqnarray*} \frac{1}{4}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})(d^{2}(x_{n+1},x^{\ast})+d^{2}(x_{n},x^{\ast}))&+&(d^{2}(x_{n+1},x^{\ast})+d^{2}(x_{n},x^{\ast}))\\ &\geq& d^{2}(x_{n+1},x_{n})-K_{n} \end{eqnarray*} implies \begin{align*} \left[1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})\right]d^{2}(x_{n+1},x^{\ast})\\ \leq \left[\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})\right]d^{2}(x_{n+1},x^{\ast})+K_{n}. \end{align*} Thus, we have \begin{eqnarray*} d^2(x_{n+1},x_{n})&\leq& \frac{\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\\ &=&\frac{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})-1+((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\\ &=&\left[1-\frac{1-(1-\beta)^2+2\theta\beta_{n}\gamma_{n})}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\right]d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}. \end{eqnarray*} Note that $$0<1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})<1$$ implies $$\frac{1-(1-\beta)^2+2\theta\beta_{n}\gamma_{n})}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\geq 1-((1-\beta)^2+2\theta\beta_{n}\gamma_{n}). $$ Thus, we have \begin{eqnarray*} d^2(x_{n+1},x_{n})&\leq&1-((1-\beta)^2+2\theta\beta_{n}\gamma_{n})d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\\ &=&[((1-\beta)^2+2\theta\beta_{n}\gamma_{n})]d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}\\ &=&(1-\beta)^2d^{2}(x_{n+1},x^{\ast})\\ &+&\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})}. \end{eqnarray*} Since $0< 1-\beta_{n}<1$, this gives $(1-\beta_{n})^2<(1-\beta_{n})$ and

\begin{equation}\label{aab} d^2(x_{n+1},x_{n})\leq (1-\beta)d^{2}(x_{n+1},x^{\ast})+\frac{K_{n}}{1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n})} \end{equation}

(12)

by $\lim\limits_{n\rightarrow \infty}\alpha_{n}=\lim\limits_{ n\rightarrow \infty}\beta_{n}=0$ and $\lim\limits_{ n\rightarrow \infty}\gamma_{n}=1$ we have

\begin{equation}\label{ab} \begin{split} \limsup\limits_{x\rightarrow\infty}\frac{K_{n}}{\beta_{n}(1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n}))}\\ =\limsup\limits_{x\rightarrow\infty}\left(\frac{\beta^{2}_{n}d^2(f(w_{n}),x^{\ast})+2\alpha_{n}\beta_{n}\langle\overrightarrow{x^{\ast}w_{n}},\overrightarrow{x^{\ast}f(w_{n})}\rangle}{(1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n}))}\right.\\ \left.+\frac{2\beta_{n}\gamma_{n}\langle\overrightarrow{x^{\ast}f(w_{n})},\overrightarrow{x^{\ast}T(w_{n})}\rangle}{(1-\frac{1}{2}((1-\beta)^2+2\theta\beta_{n}\gamma_{n}))}\right)\\ \leq 0. \end{split} \end{equation}

(13)

From (12) and (13) and the Lemma 2.6, we have $$\lim\limits_{n\rightarrow \infty}d(x_{n+1},x^{\ast})=0.$$ This implies that $x_{n}\rightarrow x^{\ast}$ as $n\longrightarrow \infty$. This complete the proof.

Competing Interests

The author do not have any competing interests in the manuscript.

References

Bridson, M. R., & Haefliger, A. (2013). Metric spaces of non-positive curvature (Vol. 319). Springer Science & Business Media. [Google Scholar]
Alghamdi, M. A., Alghamdi, M. A., Shahzad, N., & Xu, H. K. (2014). The implicit midpoint rule for nonexpansive mappings. Fixed Point Theory and Applications, 2014(1), 96. [Google Scholar]
Rajagopal, Attouch, H. (1996). Viscosity solutions of minimization problems. SIAM Journal on Optimization, 6(3), 769-806. [Google Scholar]
Berg, I. D., & Nikolaev, I. G. (2008). Quasilinearization and curvature of Aleksandrov spaces. Geometriae Dedicata, 133(1), 195-218. [Google Scholar]
Auzinger, W., & Frank, R. (1989). Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case. Numerische Mathematik, 56(5), 469-499. [Google Scholar]
Petrusel, A., & Yao, J. C. (2008). Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Analysis: Theory, Methods & Applications, 69(4), 1100-1111. [Google Scholar]
Shimoji, K., & Takahashi, W. (2001). Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics,387-404. [Google Scholar]
Wu, D., Chang, S. S., & Yuan, G. X. (2005). Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space. Nonlinear Analysis: Theory, Methods & Applications, 63(5), 987-999. [Google Scholar]
Xu, H. K. (2004). Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications, 298(1), 279-291. [Google Scholar]
Yao, Y., & Shahzad, N. (2011). New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory and Applications, 2011(1), 79. [Google Scholar]
Dhompongsa, S., Kirk, W. A., & Panyanak, B. (2007). Nonexpansive set-valued mappings in metric and Banach spaces. Journal of Nonlinear and Convex Analysis, 8(1), 35. [Google Scholar]
Kirk, W. A. (2003, February). Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003)(Vol. 64, pp. 195-225). [Google Scholar]
Kirk, W. A. (2004). Geodesic geometry and fixed point theory II. Fixed Point Theory and Applications. [Google Scholar]
Moudafi, A. (2000). Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications, 241(1), 46-55. [Google Scholar]
Shi, L. Y., & Chen, R. D. (2012). Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT (0) spaces. Journal of Applied Mathematics, 2012. [Google Scholar]
Zhao, L. C., Chang, S. S., Wang, L., & Wang, G. (2017). Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT (0) Spaces. Journal of Nonlinear Sciences & Applications (JNSA), 10(2). [Google Scholar]
Ke, Y., & Ma, C. (2015). The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications, 2015(1), 190. [Google Scholar]
Dhompongsa, S., & Panyanak, B. (2008). On $\triangle$-convergence theorems in CAT (0) spaces. Computers & Mathematics with Applications, 56(10), 2572-2579. [Google Scholar]
Bruhat, F., & Tits, J. (1972). Groupes réductifs sur un corps local. Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 41(1), 5-251. [Google Scholar]

\((d_{u}, d_{v})\)	Number of edges
\((2,4)\)	\(8\)
\((3,4)\)	\(4(m+n-4)\)
\((3,6)\)	\(2(m+n-4)\)
\((4,6)\)	\(4(mn-n-m)\)
\((4,4)\)	\(4\)
\((6,6)\)	\(4(mn-2m-2n+4)\)

\((d_{u}, d_{v})\)	Number of edges
\((3,6)\)	\(4\)
\((3,5)\)	\(8\)
(5,6)	\(8\)
\((5,5)\)	\(2(m+n-6)\)
\((6,8)\)	4
\((5,8)\)	\(2(m+n-4)\)
\((5,7)\)	\(4(m+n-6)\)
\((7,7)\)	\(2(m+n-8)\)
\((6,7)\)	\(8\)
\((7,8)\)	\(6(m+n-6)\)
\((8,8)\)	\(8mn-24(m+n)+72\)

\((d_{u}, d_{v})\)	Number of edges
\((2,4)\)	\(2^{r+2}\)
\((4,4)\)	\(2^{r+1}(r-2)\)

\((d_{u}, d_{v})\)	Number of edges
\((2,4)\)	\(2^{r+2}\)
\((4,4)\)	\(2^{r+2}(r-1)\)

\((d_{u}, d_{v})\)	Number of edges	Degree of Edges
\((2,2)\)	\(4n\)	\(2\)
\((2,3)\)	\(4n-4\)	\(3\)
\((2,4)\)	\(28n-16\)	\(4\)
\((3,3)\)	\(9n^{2}-13n+24\)	\(4\)
\((3,4)\)	\(36n^{2}-56n+24\)	\(5\)
\((4,4)\)	\(36n^{2}-56n+20\)	\(6\)

OMA – Vol 1 – Issue 1 (2017) – PISRT

Forgotten polynomial and forgotten index of certain interconnection networks

Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks

Abstract

Keywords:

1. Prelimaniries

2. Mesh derived Networks

Table 1. Edge partition \(MDN1[m, n]\)

3. Mesh derived Networks 2

Table 2. Edge partition MDN2[m, n]

4. Butterfly Network

Table 3. Edge partition \(BF(r)\)

5. Benes Network

Table 4. Edge partition \(BF(r)\)

6. Conclusion

Competing Interests

References

Mapping properties of integral operator involving some special functions

Mapping properties of integral operator involving some special Functions

Abstract

Keywords:

1. Introduction

2. Main Lemmas

3. Sufficient Conditions of Integral Operator Defined by Normalized Struve Function

4. Sufficient Conditions of Integral Operator Defined by Normalized Dini Function

Referance

Integral inequalities for differentiable harmonically \((s,m)\)-preinvex functions

Integral Inequalities for Differentiable Harmonically \((s,m)\)-preinvex Functions

Abstract

Keywords:

1. Introduction

2. Main Results

3. Conclusion

Competing Interests

References

K-Banhatti and K-hyper Banhatti indices of dominating David Derived network

K-Banhatti and K-hyper Banhatti indices of dominating David Derived network

Abstract

Keywords:

Introduction

Edge partition of Dominating David derived network of first type

Edge partition of Dominating David derived network of second type

Edge partition of Dominating David derived network of second type

Conclusion

2. Competing Interests

References

An implicit viscosity technique of nonexpansive mapping in CAT(0) spaces

An implicit viscosity technique of nonexpansive mapping in CAT(0) spaces

Abstract

Keywords:

1. Introduction

2. Preliminaries

3. The Main Result

Competing Interests

References