Graphic Contraction Mappings via Graphical b-Metric Spaces with Applications

Abstract

In this paper, we introduce graphic contraction mappings in b-metric spaces and establish some fixed point theorems in the aforesaid space. Appropriate graphs are presented to visualize related definitions and concepts. On the other hand, some examples are presented to verify the authenticity and effectiveness of the established results. Our results generalize and extend some pioneer results existing in the literature. Moreover, applications to existence of solution for ordinary differential equations and for integral equations are also established to show the adequacy of our results.

1 Introduction

It is well known that Banach’s contraction principle theorem is one of the pivotal results in the fixed point theory and it has been improved in different directions at different spaces by mathematicians over the years. Czerwik introduced the concept of b-metric spaces in [9]. Following this initial paper of Czerwik [9], a number of researchers investigate the topology of the paper and proved several fixed point results in the frame of a complete b-metric space (see, e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and the related references therein).

In recent investigations, the study of fixed point theory endowed with a graph occupies a prominent place in many aspects. Echenique [12] in 2005, deliberated fixed point theory by using graph. In 2007, Jachymski [16] introduced a new approach in metric fixed point theory by replacing the order structure with a graph structure on a metric space. After that, there are various extensions for constructing the fixed point theorems in metric space, generalized metric space and also Banach space, we refer to [1, 19, 24] and the references therein. Olatinwo [20] established a fixed point theorem for multi-valued operators in a complete b-metric space using the concept on multi-valued weak contractions for the Picard iteration in a metric space. In 2016, Debnath, Neog [11] defined the new concept of start point in a directed graph. They gave characterizations for a directed graph to have a start point. The notion of self path set valued map has also been defined and its relation with start point is established in the setting of a metric space with a graph.

On the other hand, in 2017, Chuensupantharat at el. [8] gave a new proof of the Brouwer’s fixed point theorem without using the Tietze extension theorem by using graphical which is simplifying the proof in [11]. In 2018, Choudhury et al. [7] established some common fixed point results for a pair of mappings in a metric space having the additional structure of a directed graph. The mappings are assumed to satisfy certain almost G-contractions without and with rational terms in metric spaces endowed with the structure of a graph. The approach here is a blending of analytic and graph theoretic methodologies

Recently, Samreen et al. given some contractive mappings in a b-metric space including a graph G and consequently proved some related fixed point theorems for such type of contractions in [22] and furthermore, we refer to [4, 14, 15] and the references therein. Most recently, the notion of graphical metric space has been introduced by Shukla et al. [23], in which the authors replaced the triangular inequality by weaker one. Notably, the triangular inequality is fulfilled by only those points positioned on some path involved in graphical structure related with the space.

Inspired by the idea given in [23], we introduced a new notion called a graphical b-metric space and for the same we present some fixed point results. From now onward, we assume that the graphs under consultation are directed, including nonempty sets of vertices and edges. Finally some applications to existence of solution for ordinary differential equations and for integral equations are also obtained.

2 Basic Facts and Definitions

In this section we list some basic definitions and useful results that are fruitful tools in subsequent analysis. Furthermore, Let X be a nonempty set and $\varDelta$ denotes the diagonal of $X\times X$. Let G be a directed graph, which has no parallel edges, such that the set V(G) of its vertices coincides with X and $E(G)\subseteq X\times X$ contains all loops (i.e., $\varDelta \subseteq E(G)$). Hence, G is identified by the pair (V(G), E(G)).

Denote by $G^{-1}$ the graph obtained from G by reversing the direction of its edges. That is

$$\begin{aligned} E(G^{-1})=\{(x,y)\in X\times X: (y,x)\in E(G)\}. \end{aligned}$$

It is more adaptable to treat $\widetilde{G}$ as a directed graph for which the set of its edges is symmetric and under this convention, we have that

$$\begin{aligned} E(\widetilde{G})=E(G)\cup E(G^{-1}) \end{aligned}$$

If x, y are vertices of the directed graph G, then a path in G is a sequence $\{x_i\}_{i=0}^{n}$ of $(n+1)$ vertices such that $x_0=x$,  $x_n=y$ and $(x_{i-1},x_i) \in E(G)$ for $i=1,2,\ldots , n$. A graph G is connected if there is at least one path between every pair of vertices of G. A directed graph G is said to be strongly connected if there is at least one directed path from every vertex to every other vertex. A directed graph G is called weakly connected if its corresponding undirected graph is connected but G is not strongly connected, i.e., G is weakly connected if $\widetilde{G}$ is connected. A subgraph H of a graph G is a graph whose set of vertices and set of edges are all subsets of G. (Since every set is a subset of itself, every graph is a subgraph of itself.) All the edges and vertices of G might not be present in H; but if a vertex is present in H, it has a corresponding vertex in G and any edge that connects two vertices in H will also connect the corresponding vertices in G.

As a generalization and unification of metric space involving a graphical structure, Shukla et al. [23] introduced the concept of graphical metric space as follows:

Definition 2.1

[23] Let X be a nonempty set endowed with a graph G and $d_{G}: X \times X \rightarrow \mathbb {R}$ be a function satisfying the following conditions:

• (GM1) $d_{G}(x,y)\ge 0$ for all $x,y\in X$;

• (GM2) $d_{G}(x,y)= 0$ if and only if $x=y$;

• (GM3) $d_{G}(x,y)= d_{G}(y,x)$ for all $x,y\in X$;

• (GM4) $(xPy)_{G}, z\in (xPy)_{G}$ implies $d_{G}(x,y)\le d_{G}(x,z)+d_{G}(z,y)$ for all $x,y,z\in X$.

Then, the mapping $d_{G}$ is called a graphical metric on X and the pair $(X,d_{G})$ is called a graphical metric space. For a relation P that defined on X, $(xPy)_{G}$ and $[x]_{G}^{l}$ can be defined by

• (i) $(xPy)_{G}$ if and only if there is a directed path from x to y in G and $z\in (xPy)_{G}$ if z is contained in some directed path from x to y in G,

• (ii) $[x]_{G}^{l}$ stands for $[x]_{G}^{l}$=$\{y\in X:$ there is a directed path from x to y of length $l\}$ see in [23].

Moreover, a sequence $\{x_n\}\in X$ is said to be G-termwise connected if $(x_nPx_{n+1})_{G}$ for all $n\in \mathbb {N}$.

Example 2.2

[23] Every metric space (X, d) is graphical metric space with graph G, where $V(G)=X$ and $E(G)=X\times X$.

In the following definition, we will define the concept of graphical b-metric space by combining graphical metric space and b-metric space.

Definition 2.3

Let X be a nonempty set endowed with a graph G and $d_{G_b}:X\times X\rightarrow [0,\infty )$ be a function such that for all $x,y,z\in X$ and some $s\ge 1$, we have

($G_bM1$):

$d_{G_b}(x,y)= 0$ if and only if $x=y$;

($G_bM2$):

$d_{G_b}(x,y)= d_{G_b}(y,x)$ for all $x,y\in X$;

($G_bM3$):

$(xPy)_{G}, z\in (xPy)_{G}$$\Longrightarrow$$d_{G_b}(x,y)\le s[d_{G_b}(x,z)+d_{G_b}(z,y)]$.

Then, the mapping $d_{G_b}$ is called a graphical b-metric on X and the pair $(X,d_{G_b},s)$ is called a graphical b-metric space.

Remark 1

It should be noted that the notion of graphical b-metric is a real generalization of graphical metric space since a graphical b-metric space is a graphical metric space when $s=1$.

Example 2.4

Every b-metric space (X, d, s) is a graphical b-metric space including the graph G in which $V(G)=X$ and $E(G)=X\times X$.

For instance, let $X=\{a,b,c\}$. Define a b-metric on X as follows:

$d(a,a)=d(b,b)=d(c,c)=0$; $d(a,c)=d(c,a)=2$; $d(c,b)=d(b,c)=2$ and $d(a,b)=d(b,a)=5$.

Then (X, d, s) is a b-metric space with $s=5/4$. Consider the graph G such that $X=V(G)$ and $E(G)=\varDelta \cup \{(a,b),(a,c),(c,b)\}$, then it is easy to conclude that the pair (X, d, s) is a graphical b-metric space along with the graph G (see Fig. 1).

Fig. 1

Graph associated with the graphical b-metric space

Next, we present an example which shows that a graphical b-metric on X need not to be a graphical metric on X.

Example 2.5

Let $X=\{1,2,3,4\}$ and $d_{G_b}:X\times X\rightarrow [0,\infty )$ is defined by

$$\begin{aligned} d_{G_b}(x,y)= \left\{ \begin{array}{ll} 0, &{} ~~ \text {if} \,\, x=y;\\ 3a, &{} ~~ \text {if} \,\,x,y\in \{1,2\}\,\, \text {and} \,\,x\ne y;\\ a, &{} ~~ \text {if} \,\,x\,\, \text {or} \,y\,\notin \{1,2\} \,\, \text {and} \,\,x\ne y, \end{array} \right. \end{aligned}$$

where $a>0$ is a constant including the graph $G=(V(G),E(G))$ where $V(G)=X$ and $E(G)=X\times X$ (see Fig. 2). Then $(X,d_{G_b},s)$ is a graphical b-metric space with coefficient $s=3/2>1$, but, it is not a graphical metric space as

$$\begin{aligned} d_{G_b}(1,2)=3a>2a=d_{G_b}(1,3)+d_{G_b}(3,2). \end{aligned}$$

Fig. 2

Graph associated with the graphical b-metric space

Remark 2

It should be noted that, it is always possible to find a graphical b-metric from an ordered b-metric space.

Let $(X,d_{\precsim },s)$ be an ordered b-metric space. Let $G=(V(G),E(G))$ be a graph in which $V(G)=X$ and $E(G)=\{(x,y)\in X\times X: x\precsim y\}$. Then $(X,d_{G_b},s)$ is a graphical b-metric space. Along these lines we conclude that every ordered b-metric space is a graphical b-metric space including the graph $G_{b}$ which is known as included graph. It need mentioning that this graphical b-metric space need not to be a b-metric space.

Here we present an example for supporting the above statement.

Example 2.6

Let $X=\{0,1,2,3\}$ be equipped with the following partial order $\precsim$:

$$\begin{aligned} \precsim := \{(0,0),(1,1),(2,2),(3,3),(0,1),(0,3),(2,1),(1,3),(2,3)\} \end{aligned}$$

Define a mapping $d_{\precsim }:X\times X\rightarrow [0,\infty )$ by

$$\begin{aligned} d_{\precsim }(x,y)= \left\{ \begin{array}{ll} 0,&{} \text {if} \,\, x=y;\\ |x-y|^{2},&{} \text {if} \,\, x\ne y.\\ \end{array}\right. \end{aligned}$$

Then $(X, d_{\precsim },s)$ is an ordered b-metric space with $s=9/5$. Let G be the graph induced by the partial order $\precsim$. Thus $(X, d_{G_{\precsim }},s)$ is a graphical b-metric on X but not a b-metric on X. Indeed for $x=0, y=2, z=1$, we get $d_{\precsim }(0,2)=4>s(1+1)=s(d_{\precsim }(0,1)+d_{\precsim }(1,2))$ (see Fig.  3).

Fig. 3

Graph associated with the graphical b-metric space

Remark 3

We reformulate the remark given in [23] with the adjoining graph which will give batter insight to the Remark 2.6 [23].

Example 2.7

Let $G=(V(G),E(G))$ be an undirected disconnected graph, in which $V(G)=\{v_1,v_2,v_3,v_4,v_5,v_6, v_7\}$ and $E(G)=\varDelta \cup \{e_i=(v_i,v_{i+1}),i\in \{1,2,3,4,6\}\} \cup \{e_5=(v_1,v_3)\}$. Let $G_1=(V(G_1),E(G_1))$ and $G_2=(V(G_2),E(G_2))$ are two connected components of such that $V(G)=V(G_1)\cup V(G_2)$ and $E(G)=\varDelta \cup E(G_1)\cup E(G_2)$, where $(V(G_1),E(G_1))=(\{v_1,v_2,\ldots , v_5\}, \{e_1,e_2,\ldots , e_5\})$ and $(V(G_2),E(G_2))=(\{v_6,v_7\}, \{e_6\})$. Figure 4 represents the undirected graph G including two connected components $G_1$ and $G_2$.

Fig. 4

Disconnected graph G with two components $G_1$ and $G_2$

We define $d_{G_{b}}:X \times X\rightarrow [0,\infty )$ by

$$\begin{aligned} d_{G_{b}}(x,y)= \left\{ \begin{array}{ll} l_{xy}, &{} ~~ \text {if} \,\, x,y\in V(G_{i}) \,\, \text {for} \,\,\, i=1,2;\\ 1,&{} ~~ \text {if} \,\, x\in V(G_{i}), y\in V(G_{j}) \,\,\text {for} \,\,\,i,j\in \{1,2\} \,\,\text {and}\,\, i\ne j.\\ \end{array}\right. \end{aligned}$$

where $l_{xy}$ is the length of the shortest path from x to y with $l_{xx}=0$, which yields that

$$\begin{aligned}&d_{G_{b}}(v_i,v_i)=0 \,\, for\,\,i=\{1,2,\ldots ,7\};\\&d_{G_{b}}(v_1,v_2)=1;\,\, d_{G_{b}}(v_1,v_3)=1;\,\,d_{G_{b}}(v_1,v_4)=2;\,\,d_{G_{b}}(v_1,v_5)=3;\\&d_{G_{b}}(v_1,v_6)=d_{G_{b}}(v_1,v_7)=1;\\&d_{G_{b}}(v_2,v_3){=}1;\,\,d_{G_{b}}(v_2,v_4){=}2;\,\,d_{G_{b}}(v_2,v_5){=}3;\,\,d_{G_{b}}(v_2,v_6){=}d_{G_{b}}(v_2,v_7)=1;\\&d_{G_{b}}(v_3,v_4)=1;\,\,d_{G_{b}}(v_3,v_5)=2;\,\,d_{G_{b}}(v_3,v_6)=d_{G_{b}}(v_3,v_7)=1;\\&d_{G_{b}}(v_4,v_5)=1;\,\,d_{G_{b}}(v_4,v_6)=d_{G_{b}}(v_4,v_7)=1;\\&d_{G_{b}}(v_5,v_6)=d_{G_{b}}(v_5,v_7)=d_{G_{b}}(v_6,v_7)=1. \end{aligned}$$

Thus $(X,d_{G_{b}})$ is a graphical b-metric space for $s=1$. Notice that $(X,d_{G_{b}})$ is not a metric space since $d_{G_{b}}(v_1,v_5)=3> (1+1)=d_{G_{b}}(v_1,v_6)+d_{G_{b}}(v_6,v_5)$.

Definition 2.8

Let $(X,d_{G_{b}})$ be a graphical b-metric space. Then for $x\in X$ and $\epsilon >0$, the $d_{G_{b}}$-open ball with center x and radius $\epsilon$ is

$$\begin{aligned} B_{d_{G_{b}}}(x,\epsilon )=\{y\in X| (xPy)_{G_{b}}, d_{G_{b}}(x,y)<\epsilon \}. \end{aligned}$$

As $\varDelta \subseteq E(G)$, which implies that $x\in B_{d_{G_{b}}}(x,\epsilon )$ and so $B_{d_{G_{b}}}(x,\epsilon )\ne \phi$ for all $x\in X$ and $\epsilon >0$. The set $\mathbb {B}=\{B_{d_{G_{b}}}(x,\epsilon )| x\in X, \epsilon >0\}$ constructs a neighborhood system for the topology $\tau _{G_{b}}$ on X inclosed by the graphical b-metric $d_{G_{b}}$. Finally, a subset H of X is called open if for every $x\in H$ there exists an $\epsilon >0$ such that $B_{d_{G_{b}}}(x,\epsilon )\subset H$. Obviously a subset S of X is said to be closed if its complement is open.

Proposition 2.9

Let $(X,d_{G_{b}})$ be a graphical b-metric space, $x\in X$ and $\epsilon >0$. If $y\in B_{d_{G_{b}}}(x,\epsilon )$ then there exists $\delta >0$ such that $B_{d_{G_{b}}}(y,\delta )\subseteq B_{d_{G_{b}}}(x,\epsilon )$.

Proof

Let $y\in B_{d_{G_{b}}}(x,\frac{\epsilon }{s})$ and if $y=x$, then we choose $\delta =\frac{\epsilon }{s}$. Suppose that $y\ne x$, then we get $d_{G_{b}}(x,y)\ne 0$. We choose $\delta = \frac{\epsilon }{s}-d_{G_{b}}(x,y)>0$ and let $z\in B_{d_{G_{b}}}(y,\delta )$. By the definition we get $(xPy)_{{G_{b}}}$ and $(yPz)_{{G_{b}}}$ and so $(xPz)_{{G_{b}}}$. Now it follows from the property $(G_bM3)$ that

$$\begin{aligned} \begin{aligned} d_{G_{b}}(z,x)&\le s[d_{G_{b}}(z,y)+d_{G_{b}}(y,x)]\\&<s[\delta + d_{G_{b}}(y,x)]\\&=s[\frac{\epsilon }{s}-d_{G_{b}}(y,x)+d_{G_{b}}(y,x)]=\epsilon .\\ \text {That is} \,\,\,\,d_{G_{b}}(z,x)&<\epsilon .\\ \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} B_{d_{G_{b}}}(y,\delta )\subseteq B_{d_{G_{b}}}(x,\epsilon ). \end{aligned}$$

Thus we conclude that every open ball in X is an open set. Note that the topological space $(X,d_{G_{b}})$ is $T_1$ but need not to $T_2$. $\square$

Definition 2.10

Let $(X, d_{G_{b}})$ be a graphical b-metric space. Then a sequence $\{x_n\}$ in X is called:

• (i) convergent sequence if and only if there exists $x\in X$ such that $d_{G_{b}}(x_n,x)\rightarrow 0$ as $n\rightarrow \infty$;

• (ii) Cauchy if and only if $d_{G_{b}}(x_n,x_m)\rightarrow 0$ as $n,m\rightarrow \infty$.

3 Main Results

We begin this section by introducing the following definition that generalized by the concept of graphical contraction in metric spaces.

Definition 3.1

Let $(X,d_{G_b})$ be a graphical b-metric space. Let $f:X \rightarrow X$ be a self-mapping on X and $G^*$ be a subgraph of G such that $\varDelta \subseteq E(G^{*})$. We say that f is a graphical $(G,G^{*})$-contraction on a graphical b-metric space X, if

($G_{b}C1$):

f preserves edges of $G^{*}$; that is for each $(x,y)\in E(G^*)$, we have $(fx,fy) \in E(G^*)$;

($G_{b}C2$):

there exists $\alpha \in [0,1)$ such that for all $x,y\in X$ with $(x,y)\in E(G^*)$, we have

$$\begin{aligned} d_{G_b}(fx,fy)\le \frac{\alpha }{s^2}d_{G_b}(x,y). \end{aligned}$$

(3.1)

Notice that, the graph $G^*$ may be considered as a weighted graph by referring to each edge the graphical distance between its vertices. A sequence $\{x_n\}$ with initial value $x_0\in X$ is said to be a f-Picard sequence (or Picard sequence generated by f) if $x_n=fx_{n-1}$ for all $n\in \mathbb {N}$. For further discussion, hypothesize that $G^*$ is a subgraph of G such that $\varDelta \subseteq E(G^{*})$. Furthermore, we will use the following property:

A graph $G^*$ is said to satisfy the property $(P^{*})$, if a $G^*$-termwise connected f-Picard sequence $\{x_n\}$ converges in X implies that there is a limit $z\in X$ of $\{x_n\}$ and $n_0\in \mathbb {N}$ such that $(x_n,z)\in E(G^*)$ or $(z,x_n)\in E(G^*)$ for all $n>n_0$.

Theorem 3.2

Let $(X,d_{G_{b}})$ be a $G^*$-complete graphical b-metric space and $f:X\rightarrow X$ be a graphical $(G,G^{*})$-contraction on X. Assume that the following assertions are satisfied:

• (i) there exists $x_0\in X$ such that $fx_0\in [x_0]_{G^*}^{l}$ for some $l\in \mathbb {N}$;

• (ii) the graph $G^*$ satisfies the property $(P^{*})$.

Then there exists $x^*\in X$ such that the f-Picard sequence $\{x_n\}$ with initial value $x_0\in X$ is $G^*$-termwise connected and converges to both $x^*$ and $fx^*$.

Proof

Starting from $x_0\in X$ such that $fx_0\in [x_0]_{G^*}^{l}$, for some $l\in \mathbb {N}$ and using the assumption that $\{x_n\}$ is a f-Picard sequence with initial value $x_0$, then there exists a path $\{y_i\}_{i=0}^{l}$ such that $x_0=y_0$, $fx_0=y_l$ and $(y_{i-1},y_i)\in E(G^*)$ for $i=1,2,\ldots ,l$. As f preserves edges in $G^*$ so that from $(G_{b}C1)$, we have $(fy_{i-1},fy_i)\in E(G^*)$ for $i=1,2,\ldots ,l$. Thus $\{fy_i\}_{i=0}^{l}$ is a path of length l from $fy_0=fx_0=x_1$ to $fy_l=f^{2}x_0=x_2$, which gives, $x_2\in [x_1]_{G^*}^{l}$. By repeating this process we get that $\{f^{n}y_i\}_{i=0}^{l}$ is a path from $f^{n}y_0=f^{n}x_0=x_n$ to $f^{n}y_l=f^{n}fx_0=x_{n+1}$ of length l and then $x_{n+1}\in [x_n]_{G^*}^{l}$, for all $n\in \mathbb {N}$.

Hence $\{x_n\}$ is a $G^*$-termwise connected sequence. Since $(f^{n}y_{i-1},f^{n}y_i)\in E(G^*)$ for $i=1,2,\ldots ,l$ and $n\in \mathbb {N}$, by using the graphical $(G,G^*)$-contractive condition, we get

$$\begin{aligned} d_{G_b}(f^{n}y_{i-1},f^{n}y_{i})\le \frac{\alpha }{s^2}d_{G_b}(f^{n-1}y_{i-1},f^{n-1}y_{i}) \end{aligned}$$

(3.2)

By the repeated use of (3.2), we arrive at

$$\begin{aligned} d_{G_b}(f^{n}y_{i-1},f^{n}y_{i})\le & {} \frac{\alpha }{s^2}d_{G_b}(f^{n-1}y_{i-1},f^{n-1}y_{i})\\\le & {} \frac{\alpha ^2}{s^4}d_{G_b}(f^{n-2}y_{i-1},f^{n-2}y_{i})\\&\vdots \\\le & {} \frac{\alpha ^n}{s^{2n}}d_{G_b}(y_{i-1},y_{i})\\ \end{aligned}$$

As $G^*$ is a subgraph of G and the sequence $\{x_n\}$ is a $G^*$-termwise connected sequence for any $n\in \mathbb {N}$ then we can apply $(G_{b}M3)$ and (3.2) which yield that

$$\begin{aligned} d_{G_b}(x_n,x_{n+1})= & {} d_{G_b}(f^{n}x_0,f^{n+1}x_{0})=d_{G_b}(f^{n}y_0,f^{n}y_{l})\\\le & {} s d_{G_b}(f^{n}y_0,f^{n}y_{1})+s d_{G_b}(f^{n}y_1,f^{n}y_{l})\\\le & {} s d_{G_b}(f^{n}y_0,f^{n}y_{1})+s^2 d_{G_b}(f^{n}y_1,f^{n}y_{2})+ s^2 d_{G_b}(f^{n}y_2,f^{n}y_{l})\\\le & {} s d_{G_b}(f^{n}y_0,f^{n}y_{1})+s^2 d_{G_b}(f^{n}y_1,f^{n}y_{2})\\&\quad +\cdots + s^l d_{G_b}(f^{n}y_{l-1},f^{n}y_{l})\\\le & {} s \frac{\alpha ^n}{s^{2n}} d_{G_b}(y_{0},y_{1})+ s^2\frac{\alpha ^n}{s^{2n}} d_{G_b}(y_{1},y_{2})+\cdots + s^l \frac{\alpha ^n}{s^{2n}} d_{G_b}(y_{l-1},y_{l})\\\le & {} \frac{\alpha ^n}{s^{2n-1}}[d_{G_b}(y_{0},y_{1})+s d_{G_b}(y_{1},y_{2})+ \cdots + s^{l-1}d_{G_b}(y_{l-1},y_{l})]\\\le & {} \frac{\alpha ^n}{s^{2n-1}} \varSigma _{i=1}^{l} s^{i-1} d_{G_b}(y_{i-1},y_{i}). \end{aligned}$$

If we put $D_{l_{b}}=\varSigma _{i=1}^{l} s^{i-1} d_{G_b}(y_{i-1},y_{i})$ then the above inequality turns into the following

$$\begin{aligned} \begin{aligned} d_{G_b}(x_n,x_{n+1}) \le \frac{\alpha ^n}{s^{2n-1}}D_{l_{b}}. \end{aligned} \end{aligned}$$

Using the fact that $G^*$-termwise connected sequence, for $n,m\in \mathbb {N},\,\,m>n$, we get

$$\begin{aligned} d_{G_b}(x_n,x_{m})&\le s d_{G_b}(x_n,x_{n+1}) + s d_{G_b}(x_{n+1},x_{m}) \\&\le s d_{G_b}(x_n,x_{n+1}) + s [ s d_{G_b}(x_{n+1},x_{n+2}) + s d_{G_b}(x_{n+2},x_{m})] \\&\le s d_{G_b}(x_n,x_{n+1}) + s^2 d_{G_b}(x_{n+1},x_{n+2}) + s^2 [ s d_{G_b}(x_{n+2},x_{n+3}) \\&~~~ + s d_{G_b}(x_{n+3},x_{m})] \\&\le s d_{G_b}(x_n,x_{n+1}) + s^2 d_{G_b}(x_{n+1},x_{n+2}) + s^3 d_{G_b}(x_{n+2},x_{n+3}) \\&~~~ + \cdots + s^{m-n}d_{G_b}(x_{m-1},x_{m})\\&\le s \left( \frac{\alpha ^n}{s^{2n-1}}D_{l_{b}}\right) + s^2 \left( \frac{\alpha ^{n+1}}{s^{2(n+1)-1}}D_{l_{b}} \right) + s^3 \left( \frac{\alpha ^{n+2}}{s^{2(n+2)-1}}D_{l_{b}} \right) \\&~~~ + \cdots + s^{m-n}\left( \frac{\alpha ^{m-1}}{s^{2(m-1)-1}}D_{l_{b}} \right) \\&\le \frac{\alpha ^n}{s^{2n-2}}\left( 1 + \frac{\alpha }{s} + \frac{\alpha ^2}{s^2} + \cdots + \frac{\alpha ^{m-n-1}}{s^{m-n-1}}\right) D_{l_{b}}\\&\le \left( \frac{\alpha ^n}{s^{2n-2}}\right) D_{l_{b}} \varSigma _{i=1}^{\infty } \left( \frac{\alpha }{s}\right) ^{i-1}\\&\le \left( \frac{\alpha ^n}{s^{2n-2}}\right) \left( \frac{s}{s-\alpha }\right) D_{l_{b}} \end{aligned}$$

and hence $d_{G_b}(x_n,x_{m}) \le \left( \frac{\alpha ^n}{s^{2n-2}}\right) \left( \frac{s}{s-\alpha }\right) D_{l_{b}} \longrightarrow 0$, as $m,n\rightarrow \infty$, which means that $\{x_n\}$ is a Cauchy sequence. Since X is $G^*$-complete, it implies that the sequence $\{x_n\}$ converges in X and from condition (ii), there exists $x^*\in X \text {and} \,n_0\in \mathbb {N}$ such that $(x_n,x^*)\in E(G^*)$ or $(x^*,x_n)\in E(G^*)$ for all $n>n_0$ and

$$\begin{aligned} \mathop {\lim }\limits _{n\rightarrow \infty } d_{G_b}(x_{n}, x^*)=0, \end{aligned}$$

which show that $\{x_n\}$ converges to $x^*$. Utilizing the condition $(G_{b}C2)$ and (3.1) when $(x_n,x^*)\in E(G^*)$, then we have

$$\begin{aligned} d_{G_b}(x_{n+1}, fx^*)=d_{G_b}(fx_{n}, fx^*)\le \frac{\alpha }{s^2}d_{G_b}(x_{n}, x^*) \end{aligned}$$

for all $n>n_0$. That is

$$\begin{aligned} \mathop {\lim }\limits _{n\rightarrow \infty } d_{G_b}(x_{n+1}, fx^*)=0, \end{aligned}$$

which yields that the sequence $\{x_n\}$ converges to both $x^*$ and $fx^*$. $\square$

Now, we include an example to illustrate Theorem 3.2

Example 3.3

Let $X=\{0\}\cup \{\frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \ldots \}$ and $G^*=G$ be the graph defined by $V(G)=X$ and $E(G)=\{(x,y)\in X\times X : y \le x\}$. Define a mapping $d_{G_{b}}:X\times X\rightarrow [0,\infty )$ by

$$\begin{aligned} d_{G_{b}}(x,y)=\left\{ \begin{array}{ll} 0,&{} \quad \mathrm{if} \,\,\,\,x=y;\\ |x-y|^2,&{} \quad \mathrm{if}\,\,\,\, x\ne y. \end{array}\right. \end{aligned}$$

Clearly, $d_{G_{b}}$ is a graphical b-metric on X with $s=2$. Let $f:X\rightarrow X$ be a mapping given by $fx=\frac{x}{3}$, for all $x\in X$. It is easy to find that there exists $x_0=\frac{1}{3}$ such that $f(\frac{1}{3})\in [\frac{1}{3}]_{G^*}^{1}$ and the contractive condition (3.1) is satisfied for $\alpha =\frac{1}{2}$, thus f is a graphical $(G,G^*)$-contraction for $\alpha =\frac{1}{2}$ on X. It is easy to deduce that all the conditions of Theorem 3.2 are satisfied and 0 is the required fixed point of the mapping f. As the following, Fig. 5 demonstrates the weighted graph for $n=5$, in which the weight of any edge (x, y) is equal to the value of $d_{G_{b}}(x,y)$.

Fig. 5

Weighted graph for $n=5$ where weight of edge $(x,y)= d_{G_{b}}(x,y)$

Theorem 3.4

By determining the hypotheses of Theorem 3.2, if we additionally assume that the quadruple $(X,d_{G_{b}}, G^*,f)$ has the property (P), that is if $G^*$-termwise connected f-Picard sequence $\{x_n\}$ has two limits $x^*$ and $y^*$, where $x^*\in X$ and $y^*\in f(X)$, then $x^*=y^*$.

Then f has a fixed point.

Proof

Theorem 3.2 guarantees that the f-Picard sequence $\{x_n\}$ with initial value $x_0$ converges to both $x^*$ and $fx^*$. Since $x^*\in X$ and $fx^*\in f(X)$ therefore the property (P) assures that $fx^*=x^*$. It is obtained that $x^*$ is a fixed point f. $\square$

The set of all fixed points of the mapping f is denoted by Fix(f) and we use the notation $X_{f}$ defined by:

$$\begin{aligned} X_{f}=\{x \in X: (x,fx)\in E(G^*)\}. \end{aligned}$$

Remark 4

In light of the comments given by S. Shukla along with S. Radenovic and C. Vetro in [23], our results generalize the results of Ran and Reurings [21], Kirk et al. [17] and Edelstein [13] for $s=1$ in the context of metric space.

4 Some Applications

Let X be the set $C([0,T],\mathbb {R})$ of real continuous functions on [0, T], and

$$\begin{aligned} D=\{u\in X: 0<\inf _{0\le t\le T}u(t)\,\,\, \text {and}\,\,\, u(t)\le 1, t\in [0,T] \}. \end{aligned}$$

Define the graph G and $G^*$ by $G=G^*$, $V(G)=X$ and

$$\begin{aligned} E(G)= \{(u,v)\in X\times X: u,v\in D, u(t)\le v(t), \,\,\,\text {for \,\,all}\,\,\,t\in [0,T]\}. \end{aligned}$$

Clearly, the pair $(X,d_G)$ with the graphical metric $d_G:X\times X\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} d_{G}(u,v)=\left\{ \begin{array}{ll} 0,&{}\quad \text {if} \,\,\,\,u=v;\\ \sup _ {0\le t\le T} \Big \{\ln \Big (\frac{1}{u(t)v(t)}\Big )\Big \},&{}\quad \text {if}\,\,\,\,u,v\in D, u\ne v;\\ 1,&{}\quad {\text {otherwise}}, \end{array}\right. \end{aligned}$$

(4.1)

for all $u,v\in X$ is a $G^*$-complete graphical metric space. We consider the graphical b-metric space $d_{G_{b}}:X\times X\rightarrow \mathbb {R}$ given by

$$\begin{aligned} d_{G_{b}}(u,v)=(d_{G}(u,v))^{q}=\sup _{0\le t\le T}|u(t)-v(t)|^q. \end{aligned}$$

(4.2)

Obviously, $(X,d_{G_{b}})$ is complete graphical b-metric space with coefficient $s=2^{q-1}>1$.

4.1 Application to Solution of Ordinary Differential Equations

Consider the following first-order periodic boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} u^{\prime }(t)=k(t,u(t)),\,\,&{}t\in I=[0,T]\\ u(0)=u(T), &{}\\ \end{array}\right. \end{aligned}$$

(4.3)

where $T>0$ and $k:I\times \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function.

Definition 4.1

An element $\eta \in X$ is called a lower solution for the problem (4.3) if

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \eta ^{\prime }(t)\le k(t,\eta (t)),\,\,&{} t\in I=[0,T]\\ \eta (0)\le \eta (T). &{}\\ \end{array}\right. \end{aligned} \end{aligned}$$

The problem (4.3) is equivalent to the integral equation

$$\begin{aligned} u(t)= \int _{0}^{T}\varGamma (t,s)[k(s,u(s))+ \lambda u(s)]\mathrm{d}s, \end{aligned}$$

(4.4)

in which

$$\begin{aligned} \varGamma (t,s)=\left\{ \begin{array}{ll} \frac{e^{\lambda (T+s-t)}}{e^{\lambda T}-1}\,\,&{}0 \le s\le t\le T,\\ \frac{e^{\lambda (s-t)}}{e^{\lambda T}-1}\,\,\,&{} 0\le t\le s\le T. \end{array}\right. \end{aligned}$$

(4.5)

Let the function $f:X\rightarrow X$ is given by

$$\begin{aligned} fu(t)= \int _{0}^{T}\varGamma (t,s)[k(s,u(s))+ \lambda u(s)]\mathrm{d}s. \end{aligned}$$

(4.6)

It is easy to note that, if $u\in C(I,\mathbb {R})$ is a fixed point of f then $u\in C^{1}(I,\mathbb {R})$ is a solution of the problem (4.3).

Theorem 4.2

Consider the problem (4.3) and assume that

• (1) $k(s,\cdot ):\mathbb {R}\rightarrow \mathbb {R}$ is increasing on (0, 1], for every $s\in [0,T]$. Furthermore

  $$\begin{aligned} \inf _{0\le t\le T}\varGamma (t,s)>0,\,\,\,\,\varGamma (t,s)[k(s,1)+\lambda ]\le T^{-1}; \end{aligned}$$

• (2) there exist $\alpha \in (0,1)$ such that for $x,y\in X$ with $(x,y)\in E(G)$, we have

  $$\begin{aligned} \Big [ k(t_1,x(t_1))+\lambda x(t_1) \Big ] \Big [ k(t_2,x(t_2))+\lambda x(t_2) \Big ]\ge {[x(t_1)y(t_2)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}} \end{aligned}$$

  for every $t_1,t_2\in [0,T]$.

Then the existence of a lower solution for the periodic boundary value problem provides a solution for (4.3).

Proof

It is easy to verify that, f is well defined. In account of conditions (1)–(2), for $x,y\in X$ with $(x,y)\in E(G)$ and $t \in [0,T]$, then we get

$$\begin{aligned}&\Bigg ( \ln \Bigg (\frac{1}{fx(t)fy(t)}\Bigg )\Bigg )^q \\&=\Bigg ( \ln \Bigg (\frac{1}{\int _{0}^{T}\int _{0}^{T}\varGamma (t,t_1)\varGamma (t,t_2)[k(t_1,x(t_1))+\lambda x(t_1)][k(t_2,y(t_2))+\lambda y(t_2)]\,\,dt_1\,dt_2}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\int _{0}^{T}\int _{0}^{T}\varGamma (t,t_1)\varGamma (t,t_2){[x(t_1)y(t_2)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\,\,dt_1\,dt_2}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\inf _{0\le t\le T}\,\,{[x(t)y(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\int _{0}^{T}\int _{0}^{T}\varGamma (t,t_1)\varGamma (t,t_2)\,\,dt_1\,dt_2}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\inf _{0\le t\le T}\,\,{[x(t)y(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\int _{0}^{T}\Big [\int _{0}^{t}\frac{e^{\lambda (T+t_1-t)}}{e^{\lambda T}-1}dt_1+\int _{t}^{T}\frac{e^{\lambda (t_1-t)}}{e^{\lambda T}-1}dt_1 \Big ]\varGamma (t,t_2)\,\,\,dt_2}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\lambda \inf _{0\le t\le T}\,\,{[x(t)y(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\int _{0}^{T}\Big [\int _{0}^{t}\frac{e^{\lambda (T+t_2-t)}}{e^{\lambda T}-1}dt_2+\int _{t}^{T}\frac{e^{\lambda (t_2-t)}}{e^{\lambda T}-1}dt_2 \Big ]\,}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\lambda ^2 \,\,\inf _{0\le t\le T}{[x(t)y(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}}\Bigg )\Bigg )^q\\&\le \frac{\alpha }{s^2}\Bigg ( d_{G}(x(t),y(t))\Bigg )^q\\&= \frac{\alpha }{s^2}d_{G_{b}}(x(t),y(t)), \end{aligned}$$

which yields that

$$\begin{aligned} d_{G_{b}}(fx(t),fy(t))&=(d_{G}(fx(t),fy(t)))^q \\&=\sup _{0\le t\le T}\Bigg ( \ln \Bigg (\frac{1}{fx(t)fy(t)}\Bigg )\Bigg )^q \\&\le \frac{\alpha }{s^2}\,\,d_{G_{b}}(x(t),y(t)). \end{aligned}$$

Thus the contractive condition of Theorem 3.4 is satisfied. Further, for each $x,y\in X$ such that $(x,y)\in E(G)$, we obtain that $x,y\in D$ and $x(t)\le y(t)$ for all $t\in [0,T]$. Also in account of the condition (1) of Theorem 4.2, we conclude that $\inf _{0\le t\le T} f(x)(t)>0$,

$$\begin{aligned} f(x)(t)=\int _{0}^{T}\varGamma (t,s)[k(s,x(s))+\lambda x(s)] \,\mathrm{d}s\le \int _{0}^{T}\varGamma (t,s)[k(s,1)+\lambda ]\,\mathrm{d}s\le 1 \end{aligned}$$

and

$$\begin{aligned} f(x)(t)=\int _{0}^{T}\varGamma (t,s)[k(s,x(s))+\lambda x(s)]\,\mathrm{d}s&\le \int _{0}^{T}\varGamma (t,s)[k(s,y(s))+\lambda y(s)]\,\mathrm{d}s \\&=f(y)(t). \end{aligned}$$

On the other hand, existence of lower solution of the problem (4.3) guaranteed that there is a path from $\eta$ to $f(\eta )$ of length 1, i.e., $f(\eta )\in [\eta ]_{G^*}^{1}$, so that the condition (i) of Theorem 3.4 is satisfied. Next, it is easy to verify that the condition (ii) of Theorem 3.4 and the property (P) hold. Therefore, Theorem 3.4 yields that f has a fixed point and hence the problem (4.7) has a solution in X. $\square$

4.2 Application to Existence of Solution of Integral Equations

Motivated by Theorem 3.4, we are going to study the existence of solution of the following integral equation for an unknown function u:

$$\begin{aligned} u(t)=\int _{0}^{T}G(t,z)f(z,u(z))\,\mathrm{d}z, \end{aligned}$$

(4.7)

where $T>0, f:[0,T]\times \mathbb {R}\rightarrow \mathbb {R}$ and   $G:[0,T]\times [0,T]\rightarrow [0,\infty ]$ are continuous functions. Let the mapping $F:X\rightarrow X$ is defined by

$$\begin{aligned} F u(t)=\int _{0}^{T}G(t,z)f(z,u(z))\,\mathrm{d}z, \end{aligned}$$

then u is a solution of integral equation (4.7) if and only if it is a fixed point of F. A function $\beta \in X$ with $(X=C([0,T],\mathbb {R}))$ is called a lower solution of (4.7) if

$$\begin{aligned} \beta (t)\le \int _{0}^{T}G(t,z)f(z,\beta (z))\,\mathrm{d}z, t\in [0,T]. \end{aligned}$$

Theorem 4.3

Consider the problem (4.7) and assume that the following assumptions hold:

• (1) $f(h,\cdot ):\mathbb {R}\rightarrow \mathbb {R}$ is increasing on (0,1], for every $h\in [0,T]$. Further

  $$\begin{aligned} \inf _{0\le t\le T}G(t,h)>0,\,\,\,\,G(t,h)f(h,1)\le T^{-1}, \end{aligned}$$

• (2) there exist $\alpha \in (0,1)$ and $\gamma \in [1,\infty )$ such that for $u,v\in X$ with $(u,v)\in E(G)$, we have

  $$\begin{aligned} f(h,u(h))f(k,v(k))\ge {[u(h)v(k)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}} \end{aligned}$$

  and

  $$\begin{aligned} \int _{0}^{T}\int _{0}^{T}G(t,h)G(t,k)\,\mathrm{d}h\,\mathrm{d}k\ge \gamma ,\,\,\, t\in [0,T] \end{aligned}$$

  for every $h,k\in [0,T]$.

Proof

By virtue of our assumption, F is well defined. Using the conditions (1)–(2), for $u,v\in X$ with $(u,v)\in E(G)$, we have that

$$\begin{aligned}&\Bigg ( \ln \Bigg (\frac{1}{Fu(t)Fv(t)}\Bigg )\Bigg )^q \\&=\Bigg ( \ln \Bigg (\frac{1}{\int _{0}^{T}\int _{0}^{T}G(t,h)G(t,k)f(h,u(h))f(k,v(k))\,\,\mathrm{d}h\,\mathrm{d}k}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\int _{0}^{T}\int _{0}^{T}G(t,h)G(t,k){[u(h)v(k)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\,\,\mathrm{d}h\,\mathrm{d}k}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\inf _{0\le t\le T}\,\,{[u(t)v(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}\int _{0}^{T}\int _{0}^{T}G(t,h)G(t,k)\,\,\mathrm{d}h\,\mathrm{d}k}\Bigg )\Bigg )^q\\&\le \Bigg ( \ln \Bigg (\frac{1}{\gamma \,\,\inf _{0\le t\le T}{[u(t)v(t)]^{\Big (\frac{\alpha }{s^2}\Big )}}^{\frac{1}{q}}}\Bigg )\Bigg )^q\\&\le \frac{\alpha }{s^2}\Bigg ( d_{G}(u(t),v(t))\Bigg )^q\\&= \frac{\alpha }{s^2}d_{G_{b}}(u(t),v(t)), \end{aligned}$$

which implies that

$$\begin{aligned} d_{G_{b}}(Fu(t),Fv(t))&= (d_{G}(Fu(t),Fv(t)))^q \\&= \sup _{0\le t\le T}\Bigg ( \ln \Bigg (\frac{1}{Fu(t)Fv(t)}\Bigg )\Bigg )^q \\&\le \frac{\alpha }{s^2}\,\,d_{G_{b}}(u(t),v(t)). \end{aligned}$$

Hence the contractive condition of Theorem 3.4 is fulfilled. Moreover, for each $u,v\in X$ with $(u,v)\in E(G)$, we have $u,v\in D$ and $u(t)\le v(t)$ for all $t\in [0,T]$. Also by using the condition (1) of Theorem 4.3, we obtain that $\inf _{0\le t\le T} F(u)(t)>0$ and

$$\begin{aligned} F(u)(t)=\int _{0}^{T}G(t,h)f(h,u(h))\,\mathrm{d}h\le \int _{0}^{T}G(t,h)f(h,1)\,\mathrm{d}h\le 1 \end{aligned}$$

and

$$\begin{aligned} F(u)(t)=\int _{0}^{T}G(t,h)f(h,u(h))\,\mathrm{d}h\le \int _{0}^{T}G(t,h)f(h,v(h))\,\mathrm{d}h=F(v)(t). \end{aligned}$$

The rest of the proof can be completed on the proof lines of Theorem 4.2. Thus, F has a fixed point in X, which is a solution of (4.3). $\square$