Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

Abstract

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

1. Introduction and Preliminaries

Fixed-point theory is a basic tool in functional analysis. Banach [1] has shown significant result for contraction mappings. Due to its significance, a large number of authors have proved newsworthy of this result (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]). In the sequel George et al. [2] furnished the notion of rectangular b-metric space (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. Further recent results on rectangular b-metric spaces can be seen in [10,11]. In this paper, we achieved fixed-point results for a pair of $\alpha$-dominated mappings fulfilling a generalized rational F-dominated contractive condition in complete rectangular b-metric spaces. Therefore, here, we investigate our results in a better framework of rectangular b-metric space. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results. First, we give the precise definitions that we will use.

Definition 1.

([2]). Let Z be a nonempty set and let $d_l:Z\times Z\to [0,\infty )$ be a function, called a rectangular b-metric (or simply $d_l$-metric), if there exists $b\ge 1$ such that the following conditions hold:

(i) $d_l(g,p)=0,$ if and only if $g=p;$

(ii) $d_l(g,p)=d_l(p,g);$

(iii) $d_l(g,p)\le b[d_l(g,q)+d_l(q,h)+d_l(h,p)]$ for all $g,p\in Z$ and all distinct points $q,h\in Z\setminus \{g,p\}.$ The pair $(Z,d_l)$ is said a rectangular b-metric space (in short $R.B.M.S$) with coefficient b.

Definition 2.

([2]). Let $(Z,d_l)$ be a $R.B.M.S$.

(i) A sequence $\left\{g_n\right\}$ in $(Z,d_l)$ said to be Cauchy sequence if for each $\epsilon >0$, there corresponds $n_0\in N$ such that for all $n,m\ge$ $n_0$ we have $d_l(g_m,g_n)$ $<\epsilon$ or $\underset{n,m\to \infty }{\lim }{d}_l(g_n,g_m)=0.$

(ii) A sequence $\left\{g_n\right\}$ rectangular b-converges (for short $d_l$ -converges) to g if $\underset{n\to \infty }{\lim }{d}_l(g_n,g)=0.$ In this case, g is called a $d_l$-limit of $\left\{g_n\right\}.$

(iii) $(Z,d_l)$ is complete if every Cauchy sequence in Z converges to a point $g\in Z$ for which $d_l(g,g)=0$.

Example 1.

([2]). Let Z $=N$ define $d:Z\times Z\to Z$ such that $d(u,v)=d(v,u)$ for all $u,v\in Z$ and

$$d(u,v)=\left\{\begin{array}{c}0,ifu=v;\\ 10\alpha ,ifu=1,v=2;\\ \alpha ,ifu\in \{1,2\}andv\in \left\{3\right\};\\ 2\alpha ,ifu\in \{1,2,3\}andv\in \left\{4\right\};\\ 3\alpha ,ifu\mathrm{or}v\notin \{1,2,3,4\}andu\ne v;\end{array}\right.$$

where $\alpha >0$ is a constant. Then $(Z,d)$ is a $R.B.M.S$ with coefficient $b=2>1,$ but $(Z,d)$ does not be a rectangular metric, since

$$d(1,2)=10\alpha >5\alpha =d(1,3)+d(3,4)+d(4,2).$$

Definition 3.

([26]). Let $(Z,d_l)$ be a metric space, $S:Z\to P\left(Z\right)$ be a multivalued mapping and $\alpha :Z\times Z\to [0,+\infty )$. Let $A\subseteq Z,$ the mapping S is said semi ${\alpha }_{\ast }$-admissible on $A,$ if $\alpha (x,y)\ge 1$ implies ${\alpha }_{\star }(Sx,Sy)\ge 1,$ for all $x\in A,$ where ${\alpha }_{\ast }(Sx,Sy)=\inf \{\alpha (a,b):a\in Sx,b\in Sy\}.$ When $A=Z$, we say that the S is ${\alpha }_{\ast }$-admissible on $Z.$ In the case in which S is a single valued mapping, the previous definition becomes.

Definition 4.

Let $(Z,d_l)$ be a $R.B.M.S$. Let $S:Z\to Z$ be a mapping and $\alpha :Z\times Z\to [0,+\infty )$. If $A\subseteq Z,$ we say that the S is α-dominated on $A,$ whenever $\alpha (i,Si)\ge 1$ for all $i\in A.$ If $A=Z$, we say that S is α-dominated.

Definition 5.

([28]). Let $(Z,d)$ be a metric space. A mapping $H:Z\to Z$ is said to be an A−contraction if there exists $\tau >0$ such that

$$\forall j,k\in Z,d(Hj,Hk)>0\Rightarrow \tau +A\left(d(Hj,Hk)\right)\le A\left(d(j,k)\right)$$

with $A:{\mathbb{R}}_{+}\to \mathbb{R}$ real function which satisfies three assumptions:

(F1) A is strictly increasing

(F2) For any sequence ${\left\{{\alpha }_n\right\}}_{n=1}^{\infty }$ of positive real numbers, ${\lim }_{n\to \infty }$ ${\alpha }_n$ $=0$ is equivalent to ${\lim }_{n\to \infty }A\left({\alpha }_n\right)=-\infty$;

(F3) There is $k\in (0,1)$ for which $\lim \alpha \to 0^{+}{\alpha }^{k}A\left(\alpha \right)=0$.

Example 2.

([19]). Let $Z=\mathbb{R}.$ Define the mapping $\alpha :Z\times Z\to [0,\infty )$ by

$$\alpha (x,y)=\left\{\begin{array}{c}1ifx>y\\ \frac{1}{2}otherwise\end{array}\right\}.$$

Define the self-mappings $S,T:Z\to Z$ by $Sx=\frac{x}{4},$ and $Ty=\frac{y}{2},$ where $x,y\in Z.$ Suppose $x=3$ and $y=2.$ As $3>2,$ then $\alpha (3,2)\ge 1.$ Now, $\alpha (S3,T2)=\frac{1}{2}\ngeqq 1,$ this means the pair $(S,T)$ is not α-admissible. Also, $\alpha (S3,S2)\ngeqq 1$ and $\alpha (T3,T2)\ngeqq 1.$ This implies S and T are not α-admissible individually. Now, $\alpha (x,Sx)\ge 1,$ for all $x\in Z.$ Hence S is α-dominated mapping. Similarly it is clear that $\alpha (y,Ty)\ge 1$ for all $x\in Z.$ Hence it is clear that S and T are α-dominated but not α-admissible.

2. Main Result

Theorem 1.

Let $(Z,d_l)$ be a complete $R.B.M.S$ with coefficient $b\ge 1$. Let $\alpha :Z\times Z\to [0,\infty )$ be a function and $S,T:Z\to Z$ be the α-dominated mappings on $Z.$ Suppose that the following condition is satisfied:

There exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(d_l(Se,Ty)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(e,y)+{\eta }_2{d}_l(e,Se)\\ +{\eta }_3{d}_l(e,Ty)+{\eta }_4\frac{d_l^2(e,Se).d_l(y,Ty)}{1+d_l^2(e,y)}\end{array}\right),$$

(1)

whenever $e,y\in \left\{g_n\right\},$ $\alpha (e,y)\ge 1$ and $d_l(Se,Ty)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S{\left(TS\right)}^n{g}_0$ and $g_{2n}={\left(TS\right)}^{n+1}{g}_0"$. Then $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$ and $\left\{g_n\right\}\to u\in Z.$ Also, if the inequality (1) holds for u and either $\alpha (g_n,u)\ge 1$ or $\alpha (u,g_n)\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have a common fixed point u in Z.

Proof. 

Chose a point $g_0$ in Z such that $g_1=S{g}_0$ and $g_2=T{g}_1.$ Continuing this process we construct a sequence $\left\{g_n\right\}$ of points in Z such that $g_{2n+1}=S{g}_{2n}$ and $g_{2n+2}=T{g}_{2n+1}$ for all for all $n\in \mathbb{N}\cup \left\{0\right\}.$ Let $g_1,\cdots ,g_j\in Z$ for some $j\in \mathbb{N}$. If j is odd, then $j=2\stackrel{`}{ı}+1$ for some $\stackrel{`}{ı}\in \mathbb{N}$. Since $S,T:Z\to Z$ be the $\alpha$-dominated mappings on Z, so $\alpha (g_{2\stackrel{`}{ı}},S{g}_{2\stackrel{`}{ı}})\ge 1$ and $\alpha (g_{2\stackrel{`}{ı}+1},T{g}_{2\stackrel{`}{ı}+1})\ge 1.$ As $\alpha (g_{2\stackrel{`}{ı}},S{g}_{2\stackrel{`}{ı}})\ge 1,$ this implies $\alpha (g_{2\stackrel{`}{ı}},S{g}_{2\stackrel{`}{ı}})=\alpha (g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1})\ge 1$ where $g_{2\stackrel{`}{ı}+1}=S{g}_{2\stackrel{`}{ı}}.$ Now, by using inequality (1),

$$\begin{array}{ccc}\hfill \tau +F\left(d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})\right)& \le & \tau +F\left(d_l(S{g}_{2\stackrel{`}{ı}},T{g}_{2\stackrel{`}{ı}+1})\right)\hfill \\ & \le & F\left[\begin{array}{c}{\eta }_1{d}_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)+{\eta }_2{d}_l\left(g_{2\stackrel{`}{ı}},S{g}_{2\stackrel{`}{ı}}\right)+{\eta }_3{d}_l\left(g_{2\stackrel{`}{ı}},T{g}_{2\stackrel{`}{ı}+1}\right)\\ +{\eta }_4\frac{d_l^2\left(g_{2\stackrel{`}{ı}},S{g}_{2\stackrel{`}{ı}}\right).d_l(g_{2\stackrel{`}{ı}+1},T{g}_{2\stackrel{`}{ı}+1})}{1+d_l^2\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)}\end{array}\right]\hfill \\ & \le & F\left[\begin{array}{c}{\eta }_1{d}_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)+{\eta }_2{d}_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)+b{\eta }_3{d}_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)\\ +b{\eta }_3{d}_l\left(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2}\right)+{\eta }_4\frac{d_l^2\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right).d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})}{1+d_l^2\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)}\end{array}\right]\hfill \\ & \le & F\left[({\eta }_1+{\eta }_2+b{\eta }_3)d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)+(b{\eta }_3+{\eta }_4)d_l\left(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2}\right)\right].\hfill \end{array}$$

This implies

$$F\left(d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})\right)<F\left[\begin{array}{c}({\eta }_1+{\eta }_2+b{\eta }_3)d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)\\ +(b{\eta }_3+{\eta }_4)d_l\left(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2}\right)\end{array}\right]$$

As F is strictly increasing. Therefore, we have

$$d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})<\left[\begin{array}{c}({\eta }_1+{\eta }_2+b{\eta }_3)d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)\\ +(b{\eta }_3+{\eta }_4)d_l\left(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2}\right).\end{array}\right]$$

Which implies

$$\begin{array}{ccc}\hfill (1-b{\eta }_3-{\eta }_4)d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})& <& ({\eta }_1+{\eta }_2+b{\eta }_3)d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)\hfill \\ \hfill d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})& <& \left(\frac{{\eta }_1+{\eta }_2+b{\eta }_3}{1-b{\eta }_3-{\eta }_4}\right)d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right).\hfill \end{array}$$

Now, we note that by assumption of inequality (1) it immediately follows $\lambda =\frac{{\eta }_1+{\eta }_2+b{\eta }_3}{1-b{\eta }_3-{\eta }_4}<1.$ Hence

$$d_l(g_{2\stackrel{`}{ı}+1},g_{2\stackrel{`}{ı}+2})<\lambda d_l\left(g_{2\stackrel{`}{ı}},g_{2\stackrel{`}{ı}+1}\right)<{\lambda }^2{d}_l\left(g_{2\stackrel{`}{ı}-1},g_{2\stackrel{`}{ı}}\right)<\cdots <{\lambda }^{2i+1}{d}_l\left(g_0,g_1\right).$$

Similarly, if j is even, we have

$$d_l(g_{2\stackrel{`}{ı}+2},g_{2\stackrel{`}{ı}+3})<{\lambda }^{2i+2}{d}_l\left(g_0,g_1\right).$$

(2)

Now, we have

$$d_l(g_j,g_{j+1})<{\lambda }^j{d}_l\left(g_0,g_1\right)\mathrm{for}j\in \mathbb{N}.$$

(3)

Also $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. Now,

$$d_l(g_n,g_{n+1})<{\lambda }^n{d}_l\left(g_0,g_1\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(4)

Now, for any positive integers $m,n$ $(m>n)$, we have

$$\begin{array}{ccc}\hfill d_l(g_n,g_m)& \le & b[d_l(g_n,g_{n+1})+d_l(g_{n+1},g_{n+2})+d_l(g_{n+2},g_m)]\hfill \\ & \le & b[d_l(g_n,g_{n+1})+d_l(g_{n+1},g_{n+2})]+b^2[d_l(g_{n+2},g_{n+3})\hfill \\ & & +d_l(g_{n+3},g_{n+4})+d_l(g_{n+4},g_m)]\hfill \end{array}$$

$$\begin{array}{cc}\le & b[{\lambda }^n+{\lambda }^{n+1}]d_l(g_0,g_1)+b^2[{\lambda }^{n+2}+{\lambda }^{n+3}]d_l(g_0,g_1)\hfill \\ & +b^3[[{\lambda }^{n+4}+{\lambda }^{n+5}]d_l(g_0,g_1)+\cdots \hfill \\ & +b^{2m-1}{\lambda }^{m-n}{d}_l(g_0,g_1),(\mathrm{by}\left(2.4\right))\hfill \\ \le & b{\lambda }^n[1+b{\lambda }^2+b^2{\lambda }^4+\cdots ]d_l(g_0,g_1)\hfill \\ & +b{\lambda }^{n+1}[1+b{\lambda }^2+b^2{\lambda }^4+\cdots ]d_l(g_0,g_1)\hfill \\ \le & \frac{1+\lambda }{1-b{\lambda }^2}b{\lambda }^n{d}_l(g_0,g_1).\hfill \end{array}$$

As ${\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0,$ $b\ge 1$ and $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1,$ so $\left|b{\lambda }^2\right|<1.$ Then, we have

$$d_l(g_n,g_m)<\frac{1+\lambda }{1-b{\lambda }^2}b{\lambda }^n{d}_l(g_0,g_1)\to 0\mathrm{as}n\to \infty .$$

Hence $\left\{g_n\right\}$ is a Cauchy sequence in Z. Since $(Z,d_l)$ is a complete metric space, so there exist $u\in Z$ such that $\left\{g_n\right\}\to u$ as $n\to \infty ,$ then

$$\underset{n\to \infty }{\lim }{d}_l(g_n,u)=0.$$

(5)

By assumption, $\alpha (u,g_n)\ge 1$. Suppose that $d_l(u,Su)>0,$ then there exists positive integer k such that $d_l(T{g}_{2n+1},Su)>0$ for all $n\ge k$. For $n\ge k,$ we have

$$\begin{array}{ccc}\hfill d_l(u,Su)& \le & b[d_l(u,g_n)+d_l(g_n,g_{2n+2})+d_l(g_{2n+2},Su)]\hfill \\ & \le & b[d_l(u,g_n)+d_l(g_n,g_{2n+1})+d_l(T{g}_{2n+1},Su)]\hfill \\ & \le & b[d_l(u,g_n)+d_l(g_n,g_{2n+1})+d_l(Su,T{g}_{2n+1})]\hfill \\ & <& b\left[\begin{array}{c}{d}_l(u,g_n)+d_l(g_n,g_{2n+1})+{\eta }_1{d}_l(u,g_{2n+1})\\ +{\eta }_2{d}_l(u,Su)+{\eta }_3{d}_l(g_{2n+1},T{g}_{2n+1})\\ +{\eta }_4\frac{d_l(u,Su).d_l^2(g_{2n+1},T{g}_{2n+1})}{1+d_l^2(g_{2n+1},u)}.\end{array}\right]\hfill \end{array}$$

Letting $n\to \infty ,$ and by using the inequalities (4) and (5) we get

$$d_l(u,Su)<{\eta }_3{d}_l(u,Su)<d_l(u,Su),$$

which is a contradiction. So, our supposition is wrong. Hence $d_l(u,Su)=0.$ Similarly, by using the above inequlity

$$\begin{array}{ccc}\hfill d_l(u,Tu)& \le & b[d_l(u,g_n)+d_l(g_n,g_{2n+1})+d_l(g_{2n+1},Tu)]\hfill \\ \hfill d_l(u,Tu)& \le & b[d_l(u,g_n)+d_l(g_n,g_{2n+1})+d_l(S{g}_{2n},Tu)]\hfill \end{array}$$

we can get $d_l(u,Tu)=0.$ This shows that u is a common fixed point of S and T. □

Example 3.

Let $Z=A\cup B,$ where $A=\{\frac{1}{n}:n\in \{2,3,4,5\}\}$ and $B=[1,2].$ Define $d_l:Z\times Z\to [0,\infty )$ such that defined by $d_l(x,y)=d_l(y,x)$ for $x,y\in Z$ and

$$\left\{\begin{array}{c}{d}_l(\frac{1}{2},\frac{1}{3})=d_l(\frac{1}{4},\frac{1}{5})=0.03\\ d_l(\frac{1}{2},\frac{1}{5})=d_l(\frac{1}{3},\frac{1}{4})=0.02\\ d_l(\frac{1}{2},\frac{1}{4})=d_l(\frac{1}{5},\frac{1}{3})=0.6\\ d_l(x,y)={\left|x-y\right|}^{2}otherwise.\end{array}\right.$$

be the complete $R.B.M.S$ with coefficient b $=4>1$ but $(Z,d_l)$ is neither a metric space nor a rectangular metric space. Take ${\eta }_1=\frac{1}{10},$ ${\eta }_2=\frac{1}{20},$ ${\eta }_3=\frac{1}{60},$ ${\eta }_4=\frac{1}{30},$ $\tau \in (0,\frac{12}{95}]$ then $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1$, $\lambda =\frac{11}{56}$ and $F\left(x\right)=\ln x$. Consider the mapping $\alpha :Z\times Z\to [0,\infty )$ by

$$\alpha (x,y)=\left\{\begin{array}{c}1ifx>y\\ \frac{1}{2}otherwise\end{array}\right\}.$$

Let $S,T:Z\to Z$ be defined as

$$Sx=\left\{\begin{array}{c}\frac{1}{2}ifx\in A\\ \frac{x}{4}ifx\in B.\end{array}\right.Tx=\left\{\begin{array}{c}\frac{1}{3}ifx\in A\\ \frac{x}{4}ifx\in B.\end{array}\right.$$

As $\frac{1}{2},\frac{1}{3}\in Z,$ $\alpha (\frac{1}{2},\frac{1}{3})>1$ taking $F\left(x\right)=\ln x,$ for any $\tau \in (0,\frac{12}{95}].$ Then S and T satisfy the condition of Theorem 1.

If, we take $S=T$ in Theorem 1, then we are left with result.

Corollary 1.

Let $(Z,d_l)$ be a complete $R.B.M.S$ with coefficient $b\ge 1$. Let $\alpha :Z\times Z\to [0,\infty )$ be a function and $S:Z\to Z$ be the α-dominated mapping on $Z.$ Suppose that the following condition is satisfied:

There exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(d_l(Se,Sy)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(e,y)+{\eta }_2{d}_l(e,Se)\\ +{\eta }_3{d}_l(e,Sy)+{\eta }_4\frac{d_l^2(e,Se).d_l(y,Sy)}{1+d_l^2(e,y)}\end{array}\right),$$

(6)

whenever $e,y\in \left\{g_n\right\},$ $\alpha (e,y)\ge 1$ and $d_l(Se,Sy)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S^{2n}{g}_0".$ Then $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$ and $\left\{g_n\right\}\to u\in Z.$ Also, if the inequality (6) holds for u and either $\alpha (g_n,u)\ge 1$ or $\alpha (u,g_n)\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have a common fixed point u in Z.

If, we take ${\eta }_2=0$ in Theorem 1, then we are left with the result.

Corollary 2.

Let $(Z,d_l)$ be a complete $R.B.M.S$ with constant $b\ge 1$. Let $\alpha :Z\times Z\to [0,\infty )$ be a function and $S,T:Z\to Z$ be the α-dominated mappings on $Z.$ Suppose that the following condition is satisfied:

There exist $\tau ,{\eta }_1,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+(1+b)b{\eta }_3+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(d_l(Se,Ty)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(e,y)+{\eta }_3{d}_l(e,Ty)\\ +{\eta }_4\frac{d_l^2(e,Se).d_l(y,Ty)}{1+d_l^2(e,y)}\end{array}\right),$$

(7)

whenever $e,y\in \left\{g_n\right\},$ $\alpha (e,y)\ge 1$ and $d_l(Se,Ty)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S{\left(TS\right)}^n{g}_0$ and $g_{2n}={\left(TS\right)}^{n+1}{g}_0"$. Then $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$ and $\left\{g_n\right\}\to u\in Z.$ Also, if the inequality (7) holds for u and either $\alpha (g_n,u)\ge 1$ or $\alpha (u,g_n)\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have common fixed point u in Z.

If, we take ${\eta }_3=0$ in Theorem 1, then we are left with the result.

Corollary 3.

Let $(Z,d_l)$ be a complete $R.B.M.S$ with constant $b\ge 1$. Let $\alpha :Z\times Z\to [0,\infty )$ be a function and $S,T:Z\to Z$ be the α-dominated mappings on $Z.$ Suppose that the following condition is satisfied: There exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(d_l(Se,Ty)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(e,y)+{\eta }_2{d}_l(e,Se)\\ +{\eta }_4\frac{d_l^2(e,Se).d_l(y,Ty)}{1+d_l^2(e,y)}\end{array}\right),$$

(8)

whenever $e,y\in \left\{g_n\right\},$ $\alpha (e,y)\ge 1$ and $d_l(Se,Ty)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S{\left(TS\right)}^n{g}_0$ and $g_{2n}={\left(TS\right)}^{n+1}{g}_0”.$ Then $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$ and $\left\{g_n\right\}\to u\in Z.$ Also, if the inequality (8) holds for u and either $\alpha (g_n,u)\ge 1$ or $\alpha (u,g_n)\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have common fixed point u in Z.

If, we take ${\eta }_4=0$ in Theorem 1, then we are left with the result.

Corollary 4.

Let $(Z,d_l)$ be a complete $R.B.M.S$ with coefficient $b\ge 1$. Let $\alpha :Z\times Z\to [0,\infty )$ be a function and $S,T:Z\to Z$ be the α-dominated mappings on $Z.$ Suppose that the following condition is satisfied:

There exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(d_l(Se,Ty)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(e,y)+{\eta }_2{d}_l(e,Se)\\ +{\eta }_3{d}_l(e,Ty)\end{array}\right),$$

(9)

whenever $e,y\in \left\{g_n\right\},$ $\alpha (e,y)\ge 1$ and $d_l(Se,Ty)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S{\left(TS\right)}^n{g}_0$ and $g_{2n}={\left(TS\right)}^{n+1}{g}_0"$, Then $\alpha (g_n,g_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$ and $\left\{g_n\right\}\to u\in Z.$ Also, if the inequality (9) holds for u and either $\alpha (g_n,u)\ge 1$ or $\alpha (u,g_n)\ge 1$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have a common fixed point u in Z.

3. Fixed Points for Graphic Contractions

Lastly, we give a realization of Theorem 1 in graph theory. Jachymski, [14], shown the particular case for contraction mappings on metric space with a graph. Hussain et al. [12], introduced the concept of graphic contractions and obtained a point fixed result. Further results on graphic contraction can be seen in [8,21,27]. Shang [25], discussed briefly basic notions of graph limit theory and fix some necessary notations and presented many interesting applications.

Definition 6.

Let Z be a nonempty set and $Q=\left(V\right(Q),W(Q\left)\right)$ be a graph such that $V\left(Q\right)=Z$, $A\subseteq Z$. A mapping $S:Z\to Z$ is said to be a graph dominated on A if $(p,q)\in W\left(Q\right),$ for all $q\in Sp$ and $q\in A$.

Theorem 2.

Let $(Z,d_l)$ be a complete $R.B.M.S$ endowed with a graph Q with coefficient $b\ge 1$. Let $S,T:Z\to Z$ be two self mappings. Suppose that the following satisfy:

(i) S and T are graph dominated on $Z.$

(ii) There exist $\tau ,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4>0$ satisfying $b{\eta }_1+b{\eta }_2+(1+b)b{\eta }_3+{\eta }_4<1$ and a continuous and strictly increasing real function F such that

$$\tau +F\left(H_{d_l}(Sp,Tq)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(p,q)+{\eta }_2{d}_l(p,Sp)\\ +{\eta }_3{d}_l(p,Tq)+{\eta }_4\frac{d_l^2(p,Sp).d_l(q,Tq)}{1+d_l^2(p,q)}\end{array}\right),$$

(10)

whenever $p,q\in \left\{g_n\right\}$, $\left(p_{,}q\right)\in W\left(Q\right)$ and $d_l(Sp,Tq)>0$ “where the sequence $g_n$ is defined by $g_0$ arbitrary in Z, $g_{2n+1}=S{\left(TS\right)}^n{g}_0$ and $g_{2n}={\left(TS\right)}^{n+1}{g}_0”.$ Then $(g_n,g_{n+1})\in W\left(Q\right)$ and $\left\{g_n\right\}\to m^{\ast }.$ Also, if the inequality (10) holds for $m^{\ast }$ and $(g_n,m^{\ast })\in W\left(Q\right)$ or $(m^{\ast },g_n)\in W\left(Q\right)$ for all $n\in \mathbb{N}\cup \left\{0\right\}$, then S and T have common fixed point $m^{\ast }$ in Z.

Proof. 

Define, $\alpha :Z\times Z\to [0,\infty )$ by

$$\alpha (p,q)=\left\{\begin{array}{c}1,\mathrm{if}p\in Z,(p,q)\in W\left(Q\right)\\ 0,\mathrm{otherwise}.\end{array}\right.$$

As S and T are graph dominated on $Z,$ then for $p\in Z,$ $(p,q)\in W\left(Q\right)$ for all $q\in Sp$ and $(p,q)\in W\left(Q\right)$ for all $q\in Tp$. Therefore, $\alpha (p,q)=1$ for all $q\in Sp$ and $\alpha (p,q)=1$ for all $q\in Tp.$ Hence ${\alpha }_{\ast }(p,Sp)=1,$ ${\alpha }_{\ast }(p,Tp)=1$ for all $p\in Z.$ Therefore, $S,T:Z\to Z$ are the $\alpha$-dominated mappings on $Z.$ Moreover, inequality (10) can be written as

$$\tau +F\left(H_{d_l}(Sp,Tq)\right)\le F\left(\begin{array}{c}{\eta }_1{d}_l(p,q)+{\eta }_2{d}_l(p,Sp)\\ +{\eta }_3{d}_l(p,Tq)+{\eta }_4\frac{d_l^2(p,Sp).d_l(q,Tq)}{1+d_l^2(p,q)}\end{array}\right)$$

whenever $p,q\in \left\{g_n\right\},$ $\alpha (p,q)\ge 1$ and $d_l(Sp,Tq)>0.$ Also, (ii) holds. Then, by Theorem 1, we have $\left\{g_n\right\}\to s^{\ast }\in Z.$ Now, $g_n,s^{\ast }\in Z$ and either $(g_n,s^{\ast })\in W\left(Q\right)$ or $(s^{\ast },g_n)\in W\left(Q\right)$ implies that either $\alpha (g_n,s^{\ast })\ge 1$ or $\alpha (s^{\ast },g_n)\ge 1.$ Therefore, all the conditions of Theorem 1 are satisfied. Hence, by Theorem 1, S and T have a common fixed point $s^{\ast }$ in Z and $d_l(s^{\ast },s^{\ast })=0.$

4. Conclusions

In the present work, we have achieved fixed-point results for new generalized F-contraction for a more general class of $\alpha$-dominated mappings rather than ${\alpha }_{\ast }$-admissible mappings and for a weaker class of strictly increasing mapping F rather than class of mappings F used by Wordowski [28]. We introduced the concept of a pair of graph-dominated mappings and given a fixed-point existence result of a fixed point for graphic contractions. Our results generalized and extended many recent fixed-point results of Rasham et al. [16,20], Wordowski’s result [28], Ameer et al. [6] and many classical results in the current literature (see [4,7,9,13,17,18,23,24]).