On Some Coupled Fixed Points of Generalized T-Contraction Mappings in a b_v(s)-Metric Space and Its Application

Abstract

Common coupled fixed point theorems for generalized T-contractions are proved for a pair of mappings $S:X\times X\to X$ and $g:X\to X$ in a $b_v\left(s\right)$-metric space, which generalize, extend, and improve some recent results on coupled fixed points. As an application, we prove an existence and uniqueness theorem for the solution of a system of nonlinear integral equations under some weaker conditions and given a convergence criteria for the unique solution, which has been properly verified by using suitable example.

1. Introduction

In the last three decades, the definition of a metric space has been altered by many authors to give new and generalized forms of a metric space. In 1989, Bakhtin [1] introduced one such generalization in the form of a b-metric space and in the year 2000 Branciari [2] gave another generalization in the form a rectangular metric space and generalized metric space. Thereafter, using the above two concepts, many generalizations of a metric space appeared in the form of rectangular b-metric space [3], hexagonal b-metric space [4], pentagonal b-metric space [5], etc. The latest such generalization was given by Mitrović and Radenović [6] in which the authors defined a $b_v\left(s\right)$-metric space which is a generalization of all the concepts told above. Some recent fixed point theorems in such generalized metric spaces can be found in [6,7,8,9]. In [10,11,12], one can find some interesting coupled fixed point theorems and their applications proved in some generalized forms of a metric space. In the present note, we have given coupled fixed point results for a pair of generalized T-contraction mappings in a $b_v\left(s\right)$-metric space. Our results are new and it extends, generalize, and improve some of the coupled fixed point theorems recently dealt with in [10,11,12].

In recent years, fixed point theory has been successfully applied in establishing the existence of solution of nonlinear integral equations (see [11,12,13,14,15] ). We have applied one of our results to prove the existence and convergence of a unique solution of a system of nonlinear integral equations using some weaker conditions as compared to those existing in literature.

2. Preliminaries

Definition 1. 

[6] Let X be a nonempty set. Assume that, for all$x,y,\in X$and distinct$u_1,\cdots ,u_v\in X-\{x,y\}$, $d_v:X\times X\to R$satisfies:

1. 

$d_v(x,y)\ge 0$and$d_v(x,y)=0$if and only if$x=y$,

2. 

$d_v(x,y)=d_v(y,x),$

3. 

$d_v(x,y)\le s[d_v(x,u_1)+d_v(u_1,u_2)+\cdots +d_v(u_{v-1},u_v)+d_v(u_v,y)]$, for some$s\ge 1$.

Then,$(X,d_v)$is a$b_v\left(s\right)$-metric space.

Definition 2. 

[6] In the$b_v\left(s\right)$-metric space$(X,d_v)$, the sequence$<u_n>$

(a) 

converges to$u\in X$if$d_v(u_n,u)\to 0$as$n\to \infty$;

(b) 

is a Cauchy sequence if$d_v(u_n,u_m)\to 0$as$n,m\to +\infty$.

Clearly, $b_1\left(1\right)$-metric space is the usual metric space, whereas $b_1\left(s\right)$, $b_2\left(1\right)$, $b_2\left(s\right)$, and $b_v\left(1\right)$-metric spaces are, respectively, the b-metric space ([1]), rectangular metric space ([2]), rectangular b-metric space ([3]), and v-generalized metric space ([2]).

Lemma 1. 

[6] If$(X,d_v)$is a$b_v\left(s\right)$-metric space, then$(X,d_v)$is a$b_{2v}\left(s^2\right)$-metric space.

Definition 3. 

An element$(u,v)\in X\times X$is called a coupled coincidence point of$S:X\times X\to X$and$g:X\to X$if$g\left(u\right)=S(u,v)$and$g\left(v\right)=S(v,u)$. In this case, we also say that$\left(g\right(u),g(v\left)\right)$is the point of coupled coincidence of S and g. If$u=g\left(u\right)=S(u,v)$and$v=g\left(v\right)=S(v,u)$, then we say that$(u,v)$is a common coupled fixed point of S and g.

We will denote by $COCP\{S,g\}$ and $CCOFP\{S,g\}$ respectively the set of all coupled coincidence points and the set of all common coupled fixed points of S and g.

Definition 4. 

$S:X\times X\to X$and$g:X\to X$are said to be weakly compatible if and only if$S\left(g\right(u),g(v\left)\right)=g\left(S\right(u,v\left)\right)$for all$(u,v)\in COCP\{S,g\}$.

3. Main Results

We will start this section by proving the following lemma which is an extension of Lemma 1.12 of [6] to two sequences:

Lemma 2. 

Let$(X,d_v)$be a$b_v\left(s\right)$-metric space and let$<u_n>$and$<v_n>$be two sequences in X such that$u_n\ne u_{n+1},v_n\ne v_{n+1}$$(n\ge 0)$. Suppose that$\lambda \in [0,1)$and$c_1,c_2$are real nonnegative numbers such that

$$K_{m,n}\le \lambda K_{m-1,n-1}+c_1{\lambda }^m+c_2{\lambda }^n,forallm,n\in \mathbb{N},$$

(1)

where$K_{m,n}=max\{d_v(u_m,u_n),d_v(v_m,v_n)\}$or$K_{m,n}=d_v(u_m,u_n)+d_v(v_m,v_n)$. Then,$<u_n>$and$<v_n>$are Cauchy sequences.

Proof. 

From (1), we have

$$\begin{array}{ccc}{K}_{n,n+1}\hfill & \le \hfill & \lambda K_{n-1,n}+c_1{\lambda }^n+c_2{\lambda }^{n+1}\hfill \\ \hfill & \hfill & \le \cdots \hfill \\ \hfill & \hfill & \le {\lambda }^n{K}_{0,1}+c_{1}n{\lambda }^n+c_{2}n{\lambda }^{n+1}\hfill \\ \hfill & \hfill & \le {\lambda }^n{K}_{0,1}+C_{0}n{\lambda }^n.\hfill \end{array}$$

(2)

For $m,n,k\in N$, by (1), we have

$$\begin{array}{ccc}{K}_{m+k,n+k}\hfill & \le \hfill & \lambda max\{K_{m+k-1,n+k-1},c_1{\lambda }^{m+k-1}+c_2{\lambda }^{n+k-1})\}\hfill \\ \hfill & \hfill & \le \lambda K_{m+k-1,n+k-1}+c_1{\lambda }^{m+k}+c_2{\lambda }^{n+k})\hfill \\ \hfill & \hfill & \cdots \hfill \\ \hfill & \hfill & \le {\lambda }^k{K}_{m,n}+k{C}_1{\lambda }^k({\lambda }^m+{\lambda }^n).\hfill \end{array}$$

(3)

Since $0<\lambda <1$, we can find a positive integer $q_k$ such that $0<{\lambda }^{q_k}<\frac{1}{s}$. Now, suppose $v\ge 2$. Then, by using condition $3.$ of a $b_v\left(s\right)$-metric and inequalities (2) and (3), we have

$$\begin{array}{ccc}\hfill K_{m,n}& \le & s[K_{m,m+1}+K_{m+1,m+2}+\cdots +K_{m+v-3,m+v-2}+K_{m+v-2,m+q_k}+K_{m+q_k,n+q_k}+K_{n+q_k,n}]\hfill \\ \hfill & & \le s[{\lambda }^m+{\lambda }^{m+1}+\cdots +{\lambda }^{m+v-3}]K_0+s{C}_0[m{\lambda }^m+(m+1){\lambda }^{m+1}+\cdots +(m+v-3){\lambda }^{m+v-2}]\hfill \\ \hfill & & +s[{\lambda }^m{K}_{v-2,q_k}+m{\lambda }^m({\lambda }^{v-2}+{\lambda }^{q_k})K_0]\hfill \\ \hfill & & +s[{\lambda }^{q_k}{K}_{m,n}+q_k{\lambda }^{q_k}({\lambda }^m+{\lambda }^n)K_0]+s[{\lambda }^n{K}_{q_k,0}+n{\lambda }^n({\lambda }^{q_k}+1)K_0].\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill K_{m,n}& \le & \frac{s{\lambda }^m}{(1-s{\lambda }^{q_k})(1-\lambda )}{K}_{0,1}+\frac{s(m+v-3){\lambda }^m}{(1-\lambda )(1-s{\lambda }^{q_k})}\hfill \\ \hfill & & +\frac{s}{1-s{\lambda }^{q_k}}[{\lambda }^m{K}_{v-2,q_k}+m{\lambda }^m({\lambda }^{v-2}+{\lambda }^{q_k})K_{0,1}]\hfill \\ \hfill & & +\frac{s}{1-s{\lambda }^{q_k}}\left[q_k{\lambda }^{q_k}({\lambda }^m+{\lambda }^n)K_{0,1}\right]+\frac{s}{1-s{\lambda }^{q_k}}[{\lambda }^n{K}_{q_k,0}+n{\lambda }^n({\lambda }^{q_k}+1)K_{0,1}].\hfill \end{array}$$

Thus, from the definition of $K_{m,n}$, we see that, as $m,n\to +\infty$, $d_v(u_m,u_n)\to 0$ and $d_v(v_m,v_n)\to 0$ and thus $<u_n>$ and $<v_n>$ are Cauchy sequences. □

3.1. Coupled Fixed Point Theorems

We now present our main theorems as follows:

Theorem 1. 

Let$(X,d_v)$be a$b_v\left(s\right)$-metric space,$T:X\to X$be a one to one mapping,$S:X\times X\to X$and$g:X\to X$be mappings such that$S(X\times X)\subset g\left(X\right)$, $Tg\left(X\right)$is complete. If there exist real numbers$\lambda ,\mu ,\nu$with$0\le \lambda <1$, $0\le \mu ,\nu \le 1$, $min\{\lambda \mu ,\lambda \nu \}<\frac{1}{s}$such that, for all$u,v,w,z\in X$

$$\begin{array}{ccc}{d}_v(TS(u,v),TS(w,z))& \le & \lambda max\{d_v(Tgu,Tgw),d_v(Tgv,Tgz),\mu d_v(Tgu,TS(u,v)),\mu d_v(Tgv,TS(v,u),\hfill \\ & & \nu d_v(Tgw,TS(w,z)),\nu d_v(Tgz,TS(z,w))\}\hfill \end{array}$$

(4)

then the following holds:

1. 

There exist$w_{x_0},w_{y_0}$in X, such that sequences$<Tg{u}_n>$and$<Tg{v}_n>$converge to$Tg{w}_{x_0}$and$Tg{w}_{y_0}$respectively, where the iterative sequences$<g{u}_n>$and$<g{v}_n>$are defined by$g{u}_n=S(u_{n-1},v_{n-1})$and$g{v}_n=S(v_{n-1},u_{n-1})$for some arbitrary$(u_0,v_0)\in X\times X$.

2. 

$(w_{x_0},w_{y_0})\in COCP\{S,g\}$.

3. 

If S and g are weakly compatible, then S and g have a unique common coupled fixed point.

Proof. 

1. We shall start the proof by showing that the sequences $<Tg{u}_n>$ and $<Tg{v}_n>$ are Cauchy sequences, where $<g{u}_n>$ and $<g{v}_n>$ are as mentioned in the hypothesis.

By (4), we have

$$\begin{array}{ccc}{d}_v(Tg{u}_n,Tg{u}_{n+1})& =& d_v(TS(u_{n-1},v_{n-1}),TS(u_n,v_n))\hfill \\ & \le & \lambda max\{d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_{n-1},Tg{v}_n),\mu d_v(Tg{u}_{n-1},TS(u_{n-1},v_{n-1})),\hfill \\ & & \mu d_v(Tg{v}_{n-1},TS(v_{n-1},u_{n-1})),\nu d_v(Tg{u}_n,TS(u_n,v_n)),\nu d_v(Tg{v}_n,TS(v_n,u_n))\}\hfill \\ & \le & \lambda max\{d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n),\hfill \\ & & d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_n,Tg{u}_{n+1}),d_v(Tg{v}_n,Tg{v}_{n+1})\}.\hfill \end{array}$$

(5)

Similarly, we get

$$\begin{array}{ccc}\hfill d_v(Tg{v}_n,Tg{v}_{n+1})& \le & \lambda max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_{n-1},Tg{v}_n),\hfill \\ \hfill & & d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_n,Tg{v}_{n+1}),d_v(Tg{u}_n,Tg{u}_{n+1})\}.\hfill \end{array}$$

(6)

Let $K_n=max\{d_v(Tg{u}_n,Tg{u}_{n+1}),d_v(Tg{v}_n,Tg{v}_{n+1})\}.$ By (5) and (6), we get

$$K_n\le \lambda max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_n,Tg{v}_{n+1}),d_v(Tg{u}_n,Tg{u}_{n+1})\}.$$

(7)

If

$$\begin{array}{c}\hfill max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_n,Tg{v}_{n+1}),d_v(Tg{u}_n,Tg{u}_{n+1})\}\\ \hfill =d_v(Tg{v}_n,Tg{v}_{n+1})\mathrm{or}{d}_v(Tg{u}_n,Tg{u}_{n+1}),\end{array}$$

then (7) will yield a contradiction. Thus, we have

$$\begin{array}{c}\hfill max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_n,Tg{v}_{n+1}),d_v(Tg{u}_n,Tg{u}_{n+1})\}\\ \hfill =max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n)\},\end{array}$$

and then (7) gives

$$K_n\le \lambda max\{d_v(Tg{v}_{n-1},Tg{v}_n),d_v(Tg{u}_{n-1},Tg{u}_n)\}=\lambda K_{n-1}⪯{\lambda }^2{K}_{n-2}⪯\cdots ⪯{\lambda }^n{K}_0.$$

(8)

For any $m,n\in N$, we have

$$\begin{array}{ccc}\hfill d_v(Tg{u}_m,Tg{u}_n)& =& d_v(TS(u_{m-1},v_{m-1}),TS(u_{n-1},v_{n-1})\hfill \\ \hfill & \le & \lambda max\{d_v(Tg{u}_{m-1},Tg{u}_{n-1}),d_v(Tg{v}_{m-1},Tg{v}_{n-1}),\hfill \\ \hfill & & \mu d_v(Tg{u}_{m-1},TS(u_{m-1},v_{m-1})),\mu d_v(Tg{v}_{m-1},TS(v_{m-1},u_{m-1})),\hfill \\ \hfill & & \nu d_v(Tg{u}_{n-1},TS(u_{n-1},v_{n-1})),\nu d_v(Tg{v}_{n-1},TS(v_{n-1},u_{n-1}))\}\hfill \\ \hfill & \le & \lambda max\{d_v(Tg{u}_{m-1},Tg{u}_{n-1}),d_v(Tg{v}_{m-1},Tg{v}_{n-1}),d_v(Tg{u}_{m-1},Tg{u}_m),\hfill \\ \hfill & & d_v(Tg{v}_{m-1},Tg{v}_m),d_v(Tg{u}_{n-1},Tg{u}_n),d_v(Tg{v}_{n-1},Tg{v}_n)\}.\hfill \end{array}$$

Then, by using (8), we get

$$\begin{array}{ccc}\hfill d_v(Tg{u}_m,Tg{u}_n)& \le & \lambda max\{d_v(Tg{u}_{m-1},Tg{u}_{n-1}),d_v(Tg{v}_{m-1},Tg{v}_{n-1})\}\hfill \\ & & +({\lambda }^m+{\lambda }^n)K_0\}.\hfill \end{array}$$

(9)

Similarly, we have

$$\begin{array}{ccc}\hfill d_v(Tg{v}_m,Tg{v}_n)& \le & \lambda max\{d_v(Tg{u}_{m-1},Tg{u}_{n-1}),d_v(Tg{v}_{m-1},Tg{v}_{n-1})\}\hfill \\ & & +({\lambda }^m+{\lambda }^n)K_0\}.\hfill \end{array}$$

(10)

Let $K_{m,n}=max\{d_v(Tg{u}_m,Tg{u}_n),d_v(Tg{v}_m,Tg{v}_n)\}.$ By (9) and (10), we get

$$\begin{array}{ccc}\hfill K_{m,n}& \le & \lambda K_{m-1,n-1}+({\lambda }^m+{\lambda }^n)K_0.\hfill \end{array}$$

Thus, we see that inequality (1) is satisfied with $c_1=c_2=K_0$. Hence, by Lemma 2, $<Tg{u}_n>$ and $<Tg{v}_n>$ are Cauchy sequences. For $v=1$, the same follows from Lemma 1.

Since $\left(Tg\right(X),d)$ is complete, we can find $w_{x_0},w_{y_0}\in X$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}Tg{u}_n=Tg{w}_{x_0}and\underset{n\to \infty }{lim}Tg{v}_n=Tg{w}_{y_0}.\end{array}$$

2. Now,

$$\begin{array}{ccc}\hfill & \hfill & d_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0})\le s\left[d_v\right(TS(w_{x_0},w_{y_0}),TS(u_n,v_n)+d_v(TS(u_n,v_n),TS(u_{n+1},v_{n+1}))\hfill \\ \hfill & \hfill & +\cdots +d_v(TS(u_{n+v-2},v_{n+v-2}),TS(u_{n+v-1},v_{n+v-1})+d_v(TS(u_{n+v-1},v_{n+v-1}),Tg{w}_{x_0})\hfill \\ \hfill & \le \hfill & s\left[\lambda max\right\{d_v(Tg{w}_{x_0},Tg{u}_n),d_v(Tg{w}_{y_0},Tg{v}_n),\mu d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),\hfill \\ \hfill & \hfill & \mu d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}),\nu d_v(Tg{u}_n,TS(u_n,v_n)),\nu d_v(Tg{v}_n,TS(v_n,u_n))\}\hfill \\ \hfill & \hfill & +d_v(Tg{u}_{n+1},Tg{u}_{n+2})+\cdots +d_v(Tg{u}_{n+v-1},Tg{u}_{n+v})+d_v(Tg{u}_{n+v},Tg{w}_{x_0})\hfill \\ \hfill & \le \hfill & s\left[\lambda max\right\{d_v(Tg{w}_{x_0},Tg{u}_n),d_v(Tg{w}_{y_0},Tg{v}_n),\mu d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),\hfill \\ \hfill & \hfill & \mu d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}),\nu d_v(Tg{u}_n,Tg{u}_{n+1}),\nu d_v(Tg{v}_n,Tg{v}_{n+1})\}\hfill \\ \hfill & \hfill & +d_v(Tg{u}_{n+1},Tg{u}_{n+2})+\cdots +d_v(Tg{u}_{n+v-1},Tg{u}_{n+v}+d_v(Tg{u}_{n+v},Tg{w}_{x_0}).\hfill \end{array}$$

(11)

Note that, since $<Tg{u}_n>$ and $<Tg{v}_n>$ are Cauchy sequences, by definition, $d_v(Tg{u}_n,Tg{u}_{n+1})\to 0$, $d_v(Tg{v}_n,Tg{v}_{n+1})\to 0$ as $n\to \infty$. Thus, from (11), as $n\to \infty$, we get

$$\begin{array}{ccc}\hfill d_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0})& \le & s\lambda max\{\mu d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),\mu d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))\}.\hfill \end{array}$$

Similarly, we get

$$\begin{array}{ccc}\hfill d_v(TS(w_{y_0},w_{x_0}),Tg{w}_{y_0})& \le & s\lambda max\{\mu d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),\mu d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})\}.\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill & & max\{d_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0}),d_v(TS(w_{y_0},w_{x_0}),Tg{w}_{y_0})\}\hfill \\ \hfill & \le & s\lambda \mu max\{d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})\}.\hfill \end{array}$$

(12)

Proceeding along the same lines as above, we also have

$$\begin{array}{ccc}\hfill & & max\{d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))\}\hfill \\ \hfill & \le & s\lambda \nu max\{d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})\}.\hfill \end{array}$$

(13)

Using (12) and (13) along with the condition $min\{\lambda \mu ,\lambda \nu \}<\frac{1}{s}$, we get $TS(w_{x_0},w_{y_0})=Tg{w}_{x_0}$ and $TS(w_{y_0},w_{x_0})=Tg{w}_{y_0}$. As T is one to one, we have $S(w_{x_0},w_{y_0})=g{w}_{x_0}$ and $S(w_{y_0},w_{x_0})=g{w}_{y_0}.$ Therefore, $(w_{x_0},w_{y_0})\in COCP\{S,g\}$.

3. Suppose S and g are weakly compatible. First, we will show that, if $(w_{x_0}^{\ast },w_{y_0}^{\ast })\in COCP\{S,g\}$, then $g{w}_{x_0}^{\ast }=g{w}_{x_0}$ and $g{w}_{y_0}^{\ast }=g{w}_{y_0}$, or in other words the point of coupled coincidence of S and g is unique. By (5), we have

$$\begin{array}{ccc}\hfill d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0})& =& d_v(TS(w_{x_0}^{\ast },w_{y_0}^{\ast }),TS(w_{x_0},w_{y_0}))\hfill \\ & & \le \lambda max\{d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0}),d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0}),\mu d_v(Tg{w}_{x_0}^{\ast },TS(w_{x_0}^{\ast },w_{y_0}^{\ast })),\hfill \\ & & \mu d_v(Tg{w}_{y_0}^{\ast },TS(w_{y_0}^{\ast },w_{x_0}^{\ast }),\nu d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0})),\nu d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))\}\hfill \\ & & \le \lambda max\{d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0}),d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})\}.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})& \le & \lambda max\{d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0}),d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})\}.\hfill \end{array}$$

Thus, from the above two inequalities, we get

$$\begin{array}{ccc}\hfill max\{d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0}),d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})& \le & \lambda max\{d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0}),d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})\}\hfill \end{array}$$

which implies that $Tg{w}_{x_0}^{\ast }=Tg{w}_{x_0}$ and $Tg{w}_{y_0}^{\ast }=Tg{w}_{y_0}$. Since T is one to one, we get $g{w}_{x_0}^{\ast }=g{w}_{x_0}$ and $g{w}_{y_0}^{\ast }=g{w}_{y_0}$, which is the point of coupled coincidence of S and g is unique. Since S and g are weakly compatible and, since $(w_{x_0},w_{y_0})\in COCP\{S,g\}$, we have

$$gg{w}_{x_0}=gS(w_{x_0},w_{y_0})=S(g{w}_{x_0},g{w}_{y_0})$$

and

$$gg{w}_{y_0}=gS(w_{y_0},w_{x_0})=S(g{w}_{y_0},g{w}_{x_0})$$

which shows that $(g{w}_{x_0},g{w}_{y_0})\in COCP\{S,g\}$. By the uniqueness of the point of coupled coincidence, we get $gg{w}_{x_0}=g{w}_{x_0}$ and $gg{w}_{y_0}=g{w}_{y_0}$ and thus $(g{w}_{x_0},g{w}_{y_0})\in CCOFP\{S,g\}$. Uniqueness of the coupled fixed point follows easily from (4). □

Our next result is a generalized version of Theorem 2.1 of Gu [10].

Theorem 2. 

Let$(X,d_v)$, T, S and g be as in Theorem 1 and suppose there exist${\beta }_1,{\beta }_2,{\beta }_3$in the interval [0,1), such that${\beta }_1+{\beta }_2+{\beta }_3<1$, minimum$\{{\beta }_2,{\beta }_3\}<\frac{1}{s}$and for all$u,v,w,z\in X$

$$\begin{array}{c}\hfill d_v(TS(u,v),TS(w,z)+d_v(TS(v,u),TS(z,w)\le {\beta }_1(d_v(Tgu,Tgw)+d_v(Tgv,Tgz))+\\ \hfill {\beta }_2(d_v(Tgu,TS(u,v))+d_v(Tgv,TS(v,u))+{\beta }_3(d_v(Tgw,TS(w,z))+d_v(Tgz,TS(z,w))).\end{array}$$

(14)

Then, conclusions 1, 2, and 3 of Theorem 1 are true.

Proof. 

Let $K_n^{{}^{\prime }}=d_v(Tg{u}_n,Tg{u}_{n+1})+d_v(Tg{v}_n,Tg{v}_{n+1})$ and $K_{m,n}^{{}^{\prime }}=d_v(Tg{u}_m,Tg{u}_n)+d_v(Tg{v}_m,Tg{v}_n)$. From condition (14), we obtain

$$\begin{array}{ccc}\hfill & & d_v(Tg{u}_n,Tg{u}_{n+1})+d_v(Tg{v}_n,Tg{v}_{n+1})=d_v(TS(u_{n-1},v_{n-1}),TS(u_n,v_n))+\hfill \\ \hfill & & d_v(TS(v_{n-1},u_{n-1}),TS(v_n,u_n))\hfill \\ \hfill & \le & {\beta }_1[d_v(Tg{u}_{n-1},Tg{u}_n)+d_v(Tg{v}_{n-1},Tg{v}_n)]+{\beta }_2[d_v(Tg{u}_{n-1},TS(u_{n-1},v_{n-1}))\hfill \\ \hfill & & +d_v(Tg{v}_{n-1},TS(v_{n-1},u_{n-1}))]+{\beta }_3[d_v(Tg{u}_n,TS(u_n,v_n))+d_v(Tg{v}_n,TS(v_n,u_n))]\hfill \\ \hfill & \le & ({\beta }_1+{\beta }_2)[d_v(Tg{u}_{n-1},Tg{u}_n)+d_v(Tg{v}_{n-1},Tg{v}_n)]\hfill \\ \hfill & & +{\beta }_3[d_v(Tg{u}_n,Tg{u}_{n+1})+d_v(Tg{v}_n,Tg{v}_{n+1})].\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill d_v(Tg{u}_n,Tg{u}_{n+1})+d_v(Tg{v}_n,Tg{v}_{n+1})\le {\lambda }^{{}^{\prime }}[d_v(Tg{u}_{n-1},Tg{u}_n)+d_v(Tg{v}_{n-1},Tg{v}_n)],\end{array}$$

where ${\lambda }^{{}^{\prime }}={\displaystyle \frac{{\beta }_1+{\beta }_2}{1-{\beta }_3}}<1$. Thus, we get

$$\begin{array}{c}\hfill K_n^{{}^{\prime }}\le {\lambda }^{{}^{\prime }}{K}_{n-1}^{{}^{\prime }}\le \cdots \le {{\lambda }^{{}^{\prime }}}^n{K}_0^{{}^{\prime }}.\end{array}$$

(15)

For any $m,n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill d_v(Tg{u}_m,Tg{u}_n)& +& d_v(Tg{v}_m,Tg{v}_n)=d_v(TS(u_{m-1},v_{m-1}),TS(u_{n-1},v_{n-1})+\hfill \\ \hfill & & d_v(TS(v_{m-1},u_{m-1}),TS(v_{n-1},u_{n-1})\hfill \\ \hfill & \le & {\beta }_1[d_v(Tg{u}_{m-1},Tg{u}_{n-1})+d_v(Tg{v}_{m-1},Tg{v}_{n-1})]\hfill \\ \hfill & & +{\beta }_2[d_v(Tg{u}_{m-1},TS(u_{m-1},v_{m-1}))+d_v(Tg{v}_{m-1},TS(v_{m-1},u_{m-1}))]\hfill \\ \hfill & & +{\beta }_3[d_v(Tg{u}_{n-1},TS(u_{n-1},v_{n-1}))+d_v(Tg{v}_{n-1},TS(v_{n-1},u_{n-1}))]\hfill \\ \hfill & \le & {\beta }_{[}{d}_v(Tg{u}_{m-1},Tg{u}_{n-1})+d_v(Tg{v}_{m-1},Tg{v}_{n-1})]+{\beta }_2[d_v(Tg{u}_{m-1},Tg{u}_m)\hfill \\ \hfill & & +d_v(Tg{v}_{m-1},Tg{v}_m)]+{\beta }_3[d_v(Tg{u}_{n-1},Tg{u}_n)+d_v(Tg{v}_{n-1},Tg{v}_n)].\hfill \end{array}$$

Then, by using (15), we get

$$\begin{array}{ccc}\hfill d_v(Tg{u}_m,Tg{u}_n)+d_v(Tg{v}_m,Tg{v}_n)& \le & {\beta }_1[d_v(Tg{u}_{m-1},Tg{u}_{n-1})+d_v(Tg{v}_{m-1},Tg{v}_{n-1})]\hfill \\ & & +({\beta }_2{{\lambda }^{{}^{\prime }}}^m+{\beta }_3{{\lambda }^{{}^{\prime }}}^n)K_0^{{}^{\prime }}\}.\hfill \end{array}$$

That is,

$$\begin{array}{ccc}\hfill K_{m,n}^{{}^{\prime }}& \le & \lambda K_{m-1,n-1}^{{}^{\prime }}+({\lambda }^m+{\lambda }^n)K_0^{{}^{\prime }}\hfill \end{array}$$

where ${\lambda }^{{}^{\prime }}={\beta }_1+{\beta }_2+{\beta }_3<1$. Now for $m,n,r\in N$. Thus, we see that inequality (1) is satisfied with $c_1=c_2=K_0$. Hence, by Lemma 2, $<Tg{u}_n>$ and $<Tg{v}_n>$ are Cauchy sequences. For $v=1$, the same follows from Lemma 1.

Since $\left(Tg\right(X),d)$ is complete, we can find $w_{x_0},w_{y_0}\in X$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}Tg{u}_n=Tg{w}_{x_0}and\underset{n\to \infty }{lim}Tg{v}_n=Tg{w}_{y_0}.\end{array}$$

Again, from condition 3 in Definition 1, we have

$$\begin{array}{ccc}{d}_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0}))& \le & s[d_v(TS(w_{x_0},w_{y_0}),TS(u_n,v_n))+d_v(TS(u_n,v_n),TS(u_{n+1},v_{n+1}))+\cdots +\hfill \\ & & +d_v(TS(u_{n+v-2},v_{n+v-2}),TS(u_{n+v-1},v_{n+v-1}))+\hfill \\ & & d_v(TS(u_{n+v-1},v_{n+v-1}),Tg{w}_{x_0})\left)\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}{d}_v(TS(w_{y_0},w_{x_0}),Tg{w}_{y_0}))& \le & s[d_v(TS(w_{y_0},w_{x_0}),TS(v_n,u_n))+d_v(TS(v_n,u_n),TS(v_{n+1},u_{n+1}))+\cdots +\hfill \\ & & d_v(TS(v_{n+v-2},u_{n+v-2}),TS(v_{n+v-1},u_{n+v-1}))+\hfill \\ & & d_v(TS(v_{n+v-1},u_{n+v-1}),Tg{w}_{x_0})\left)\right].\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & d_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0})+d_v(TS(w_{y_0},w_{x_0}),Tg{w}_{y_0})\le s\left[d_v\right(TS(w_{x_0},w_{y_0}),TS(u_n,v_n)\hfill \\ & & +d_v(TS(w_{y_0},w_{x_0}),TS(v_n,u_n)\hfill \\ & & +d_v(TS(u_n,v_n),TS(u_{n+1},v_{n+1}))+\cdots +d_v(TS(u_{n+v-2},v_{n+v-2}),TS(u_{n+v-1},v_{n+v-1}))\hfill \\ & & +d_v(TS(v_n,u_n),TS(v_{n+1},u_{n+1}))+\cdots +d_v(TS(v_{n+v-2},u_{n+v-2}),TS(v_{n+v-1},u_{n+v-1}))\hfill \\ & & +d_v(TS(u_{n+v-1},v_{n+v-1}),Tg{w}_{x_0})+d_v(TS(v_{n+v-1},u_{n+v-1}),Tg{w}_{y_0})]\hfill \\ & & \le s[{\beta }_1(d_v(Tg{w}_{x_0},Tg{u}_n)+d_v(Tg{w}_{y_0},Tg{v}_n))+{\beta }_2(d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+\hfill \\ & & d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))+{\beta }_3(d_v(Tg{u}_n,TS(u_n,v_n))+d_v(Tg{v}_n,TS(v_n,u_n)))\}\hfill \\ & & +d_v(Tg{u}_n,Tg{u}_{n+1})+\cdots +d_v(Tg{u}_{n-1},Tg{u}_n)++d_v(Tg{v}_n,Tg{v}_{n+1})+\cdots +d_v(Tg{v}_{n-1},Tg{v}_n)\hfill \\ & & +d_v(Tg{u}_{n+v-1},Tg{w}_{x_0})+d_v(Tg{v}_{n+v-1},Tg{w}_{y_0})].\hfill \end{array}$$

As $n\to \infty$, we get

$$\begin{array}{c}{d}_v(TS(w_{x_0},w_{y_0}),Tg{w}_{x_0})+d_v(TS(w_{y_0},w_{x_0}),Tg{w}_{y_0})\\ \le s{\beta }_2\left[d_v\right(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})\left)\right].\end{array}$$

(16)

Similarly, we can show that

$$\begin{array}{c}{d}_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))\\ \le s{\beta }_3\left[d_v\right(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})].\end{array}$$

(17)

Using (16) and (17) along with the condition $min\{{\beta }_2,{\beta }_3\}<\frac{1}{s}$, we get $d_v(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0}))=0$, i.e., $TS(w_{x_0},w_{y_0})=Tg{w}_{x_0}$ and $TS(w_{y_0},w_{x_0})=Tg{w}_{y_0}$. As T is one to one, we have $S(w_{x_0},w_{y_0})=g{w}_{x_0}$ and $S(w_{y_0},w_{x_0})=g{w}_{y_0}.$ Therefore, $(w_{x_0},w_{y_0})\in COCP\{S,g\}$.

If $(w_{x_0}^{\ast },w_{y_0}^{\ast })\in COCP\{S,g\}$, then, by (14), we have

$$\begin{array}{cc}{d}_v(Tg{w}_{x_0}^{\ast },w_{x_0})\hspace{1em}+& d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})=d_v\left(TS\right(w_{x_0}^{\ast },w_{y_0}^{\ast }),TS(w_{x_0},w_{y_0}\left)\right)+d_v\left(TS\right(w_{y_0}^{\ast },w_{x_0}^{\ast }),TS(w_{y_0},w_{x_0}\left)\right)\\ & \le {\beta }_1\left[d_v\right(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0})+d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0}\left)\right]+{\beta }_2\left[d_v\right(Tg{w}_{x_0}^{\ast },TS(w_{x_0}^{\ast },w_{y_0}^{\ast }))\hfill \\ & +d_v(Tg{w}_{y_0}^{\ast },TS(w_{y_0}^{\ast },w_{x_0}^{\ast }\left)\right]+{\beta }_3\left[d_v\right(Tg{w}_{x_0},TS(w_{x_0},w_{y_0}))+d_v(Tg{w}_{y_0},TS(w_{y_0},w_{x_0})\left)\right]\hfill \\ & \le {\beta }_1\left[d_v\right(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0})+d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0}\left)\right].\hfill \end{array}$$

Thus, $d_v(Tg{w}_{x_0}^{\ast },Tg{w}_{x_0})+d_v(Tg{w}_{y_0}^{\ast },Tg{w}_{y_0})=0$, which implies that $Tg{w}_{x_0}^{\ast }=Tg{w}_{x_0}$ and $Tg{w}_{y_0}^{\ast }=Tg{w}_{y_0}$. Since T is one to one, we get $g{w}_{x_0}^{\ast }=g{w}_{x_0}$ and $g{w}_{y_0}^{\ast }=g{w}_{y_0}$, which is the point of coupled coincidence of S, and g is unique. The remaining part of the proof is the same as in the proof of Theorem 1. □

The next results can be proved as in Theorems 1 and 2 and so we will not give the proof.

Theorem 3. 

Theorem 1 holds if we replace condition (4) with the following condition:

There exist${\beta }_i\in [0,1),i\in \{1,\dots ,6\}$such that${\sum }_{i=1}^6{\beta }_i<1$, $min\{{\beta }_3+{\beta }_4,{\beta }_5+{\beta }_6\}<\frac{1}{s}$and for all$u,v,w,z\in X$,

$$\begin{array}{c}{d}_v\left(TS\right(u,v),TS(w,z\left)\right)\le {\beta }_1{d}_v(Tgu,Tgw)+{\beta }_2{d}_v(Tgv,Tgz)+{\beta }_3{d}_v(Tgu,TS(u,v\left)\right)\\ +{\beta }_4{d}_v(Tgv,TS(v,u)+{\beta }_5{d}_v(Tgw,TS(w,z))+{\beta }_6{d}_v(Tgz,TS(z,w)).\end{array}$$

(18)

Taking T to be the identity mapping in Theorems 1–3, we have the following:

Corollary 1. 

Let$(X,d_v)$, S, g,$\lambda ,\mu$and ν be as in Theorem 1 such that, for all$u,v,w,z\in X$, the following holds:

$$\begin{array}{cc}{d}_v\left(S\right(u,v),S(w,z)\hspace{1em}\le & \lambda max\left\{d_v\right(gu,gw),d_v(gv,gz),\mu d_v(gu,S(u,v)),\mu d_v(gv,S(v,u),\\ & v{d}_v(gw,S(w,z\left)\right),v{d}_v(gz,S(z,w\left)\right)\}.\hfill \end{array}$$

(19)

Then,$COCP\{S,g\}\ne \varphi$. Furthermore, if S and g are weakly compatible, then S and g has a unique common coupled fixed point. Moreover, for some arbitrary$(u_0,v_0)\in X\times X$, the iterative sequences$(<g{u}_n>,<g{v}_n>)$defined by$g{u}_n=S(u_{n-1},v_{n-1})$and$g{v}_n=S(v_{n-1},u_{n-1})$converge to the unique common coupled fixed point of S and g.

Corollary 2. 

Corollary 1 holds if the condition (19) is replaced with the following condition:

There exist${\beta }_1,{\beta }_2,{\beta }_3$in the interval [0,1), such that${\beta }_1+{\beta }_2+{\beta }_3<1$, $min\{{\beta }_2,{\beta }_3\}<\frac{1}{s}$and for all$u,v,w,z\in X$

$$\begin{array}{c}{d}_v\left(S\right(u,v),S(w,z)+d_v(S(v,u),S(z,w)\le {\beta }_1\left(d_v\right(gu,gw)+d_v(gv,gz\left)\right)+\\ {\beta }_2\left(d_v\right(gu,S(u,v))+d_v(gv,S(v,u))+{\beta }_3(d_v(gw,S(w,z\left)\right)+d_v(gz,S(z,w\left)\right)).\end{array}$$

(20)

Corollary 3. 

Corollary 1 holds if the condition (19) is replaced with the following condition:

There exist${\beta }_i\in [0,1),i\in \{1,\dots 6\}$ such that ${\sum }_{i=1}^6{\beta }_i<1$, $min\{{\beta }_3+{\beta }_4,{\beta }_5+{\beta }_6\}<\frac{1}{s}$ and, for all $u,v,w,z\in X$,

$$\begin{array}{cc}& d_v\left(S\right(u,v),S(w,z)\le {\beta }_1{d}_v(gu,gw)+{\beta }_2{d}_v(gv,gz)+\\ {\beta }_3{d}_v(gu,S(u,v\left)\right)& +{\beta }_4{d}_v(gv,S(v,u)+{\beta }_5{d}_v(gw,S(w,z))+{\beta }_6{d}_v(gz,S(z,w)).\end{array}$$

(21)

Remark 1. 

Since every b-metric space is a$b_1\left(s\right)$metric space, we note that Theorem 1 is a substantial generalization of Theorem 2.2 of Ramesh and Pitchamani [11]. In fact, we do not require continuity and sub sequential convergence of the function T.

Remark 2. 

Note that condition (2.1) of Gu [10] implies (20) and hence Corollary 2 gives an improved version of Theorem 2.1 of Gu [10].

Remark 3. 

Condition (3.1) of Hussain et al. [12] implies (18) and hence Theorem 3 is an extended and generalized version of Theorem 3.1 of [12].

3.2. Application to a System of Integral Equations

In this section, we give an application of Theorem 1 to study the existence and uniqueness of solution of a system of nonlinear integral equations.

Let $X=C[0,A]$ be the space of all continuous real valued functions defined on $[0,A]$, $A>0$. Our problem is to find $\left(u\right(t),v(t\left)\right)\in X\times X$, $t\in [0,A]$ such that, for $f:[0,A]\times R\times R\to R$ and $G:[0,A]\times [0,A]\to R$ and $K\in C\left(\right[0,A]$, the following holds:

$$\begin{array}{c}u\left(t\right)={\int }_0^{A}G(t,r)f(t,u\left(r\right),v\left(r\right))dr+K\left(t\right)\\ v\left(t\right)={\int }_0^{A}G(t,r)f(t,v\left(r\right),u\left(r\right))dr+K\left(t\right).\end{array}$$

(22)

Now, suppose $F:X\times X\to X$ is given by

$$\begin{array}{c}\hfill F(u\left(t\right),v\left(t\right))={\int }_0^{A}G(t,r)f(t,u\left(r\right),v\left(r\right))dr+K\left(t\right).\end{array}$$

$$\begin{array}{c}\hfill F(v\left(t\right),u\left(t\right))={\int }_0^{A}G(t,r)f(t,v\left(r\right),u\left(r\right))dr+K\left(t\right).\end{array}$$

Then, (22) is equivalent to the coupled fixed point problem $F\left(u\right(t),v(t\left)\right)=u\left(t\right)$, $F\left(v\right(t),u(t\left)\right)=v\left(t\right)$.

Theorem 4. 

The system of Equation (22) has a unique solution provided the following holds:

(i) 

$G:[0,A]\times [0,A]\to R$and$f:[0,A]\times R\times R\to R$are continuous functions.

(ii) 

$K\in C\left(\right[0,A].$

(iii) 

For all$x,y,u,v\in X$and$t\in [0,A]$, we can find a function$g:X\to X$and real numbers$p\ge 1$, $\lambda ,\mu ,\nu$with$0\le \lambda <1$, $0\le \mu ,\nu \le 1$, minimum$\{\lambda \mu ,\lambda \nu \}<\frac{1}{3^{s-1}}$satisfying

$$\begin{array}{ll}(iii-a):\mid f(t,u\left(r\right),v\left(r\right)))-f(t,x\left(r\right),y\left(r\right))){\mid }^p\hspace{1em}\le & {\lambda }^{p}max\{\mid g\left(u\left(r\right)\right)-g\left(x\left(r\right)\right){\mid }^p,\mid g\left(v\left(r\right)\right)-g\left(y\left(r\right)\right){\mid }^p,\\ & \mu \mid g\left(u\left(r\right)\right)-F(u\left(r\right),v\left(r\right)){\mid }^p,\mu \mid g\left(v\left(r\right)\right)-F(v\left(r\right),u\left(r\right)){\mid }^p,\\ & \nu \mid g\left(x\left(r\right)\right)-F(x\left(r\right),y\left(r\right)){\mid }^p,\nu \mid g\left(y\left(r\right)\right)-F(y\left(r\right),x\left(r\right)){\mid }^p\}.\\ \hspace{1em}(iii-b)F\left(g\right(u\left(t\right)),g(v\left(t\right)\left)\right)=g\left(F\right(u\left(t\right),v\left(t\right)\left)\right)& \end{array}$$

(iv) 

${sup}_{t\in [0,A]}{\int }_0^A\mid G(t,r){\mid }^{p}dr\le \frac{1}{{\lambda }^{p-1}}$.

Moreover, for some arbitrary$u_0\left(t\right),v_0\left(t\right)$in X, the sequence$(<g{u}_n\left(t\right)>,<g{v}_n\left(t\right)>)$defined by

$$\begin{array}{c}g{u}_n\left(t\right)={\int }_0^{A}G(t,r)f(t,u_{n-1}\left(r\right),v_{n-1}\left(r\right))dr+K\left(t\right)\\ g{v}_n\left(t\right)={\int }_0^{A}G(t,r)f(t,v_{n-1}\left(r\right),u_{n-1}\left(r\right))dr+K\left(t\right).\end{array}$$

(23)

converges to the unique solution.

Proof. 

Define $d_v:X\times X\to R$ such that for all $u,v\in X$,

$$d_v(u,v)={sup}_{t\in [0,A]}\mid u\left(t\right)-v\left(t\right){\mid }^s.$$

(24)

Clearly, $d_v$ is a $b_v\left({(v+1)}^{s-1}\right)$-metric space.

For some $r\in [0,A]$, we have

$$\begin{array}{ccc}\mid F\left(u\right(t),v(t\left)\right)\hfill & -\hfill & F(x\left(t\right),y\left(t\right)){\mid }^p\hfill \\ \hfill & =\hfill & \mid {\int }_0^{A}G(t,r)f(t,u\left(r\right),v\left(r\right))dr+g\left(t\right)-{\int }_0^{A}G(t,r)f(t,x\left(r\right),y\left(r\right))dr+g\left(t\right){\mid }^p\hfill \\ \hfill & \le \hfill & {\int }_0^A\mid G(t,r){\mid }^p\mid f(t,u\left(r\right),v\left(r\right))-f(t,x\left(r\right),y\left(r\right)){\mid }^{p}dr\hfill \\ \hfill & \le \hfill & ({\int }_0^A\mid G(t,r){\mid }^{p}dr){\lambda }^p\left[max\right\{\mid g\left(u\left(r\right)\right)-g\left(x\left(r\right)\right){\mid }^p,\mid g\left(v\left(r\right)\right)-g\left(y\left(r\right)\right){\mid }^p,\hfill \\ \hfill & \hfill & \mu \mid g\left(u\left(r\right)\right)-F(u\left(r\right),v\left(r\right)){\mid }^p,\mu \mid g\left(v\left(r\right)\right)-F(v\left(r\right),u\left(r\right)){\mid }^p,\hfill \\ \hfill & \hfill & \nu \mid g\left(x\left(r\right)\right)-F(x\left(r\right),y\left(r\right)){\mid }^p,\nu \mid g\left(y\left(r\right)\right)-F(y\left(r\right),x\left(r\right)){\mid }^p\}.\hfill \\ \hfill & \le \hfill & ({\int }_0^A\mid G(t,r){\mid }^{p}dr){\lambda }^p\left[max\right\{d_v(g\left(u\right),g\left(x\right)),d_v(g\left(v\right),g\left(y\right)),\mu d_v(g\left(u\right),F(u,v)),\mu d_v(g\left(v\right),F(v,u)),\hfill \\ \hfill & \hfill & \nu d_v(g\left(x\right),F(x,y)),\nu d_v(g\left(y\right),F(y,x))\}.\hfill \end{array}$$

Thus, using condition (iv), we have

$$\begin{array}{ccc}\hfill d_v(F(u,v),F(x,y))& =& su{p}_{t\in [0,A]}\mid F(u\left(t\right),v\left(t\right))-F(x\left(t\right),y\left(t\right)){\mid }^p\hfill \\ & \le & \lambda \left[max\right\{d_v(g\left(u\right),g\left(x\right)),d_v(g\left(v\right),g\left(y\right)),\mu d_v(g\left(u\right),F(u,v)),\mu d_v(g\left(v\right),F(v,u)),\hfill \\ & & \nu d_v(g\left(x\right),F(x,y)),\nu d_v(g\left(y\right),F(y,x))\}.\hfill \end{array}$$

Thus, all the conditions of Corollary 1 are satisfied and so F has a unique coupled fixed point $(u^{\prime },v^{\prime })\in C([0,A]\times C([0,A]$, which is the unique solution of (22) and the sequence $(<g{u}_n\left(t\right)>,<g{v}_n\left(t\right)>)$ defined by (23) converges to the unique solution of (22). □

Example 1. 

Let$X=C[0,1]$be the space of all continuous real valued functions defined on$[0,1]$and define$d_3:X\times X\to R$such that, for all$u,v\in X$,

$$d_3(u,v)={sup}_{t\in [0,A]}\mid u\left(t\right)-v\left(t\right){\mid }^2.$$

(25)

Clearly,$d_3$is a$b_2\left(3\right)$-metric. Now, consider the functions$f:[0,1]\times R\times R\to R$given by$f(t,u,v)=t^2+\frac{9}{20}u+\frac{8}{20}v$, $G:[0,1]\times [0,1]\to R$given by$G(t,r)=\frac{\sqrt{45}(t+r)}{10}$, $K\in C\left(\right[0,1]$given by$K\left(t\right)=t$. Then, Equation (22) becomes

$$\begin{array}{c}u\left(t\right)=t+{\int }_0^1\frac{\sqrt{45}(t+r)}{10}(t^2+\frac{9}{20}u(r)+\frac{8}{20}v(r))dr\\ v\left(t\right)=t+{\int }_0^1\frac{\sqrt{45}(t+r)}{10}(t^2+\frac{9}{20}v(r)+\frac{8}{20}u(r))dr.\end{array}$$

(26)

Then,

$$\begin{array}{ccc}\hfill \mid f(t,u,v)-f(t,x,y){\mid }^2& =& \mid \frac{9}{20}(u-x)+\frac{8}{20}(v-y){\mid }^2\hfill \\ & \le & \mid Max\{\frac{9}{10}(u-x),\frac{8}{10}(v-y)\}{\mid }^2\hfill \\ & \le & \frac{81}{100}Max\{\mid u-x{\mid }^2,\mid v-y){\mid }^2\}.\hfill \end{array}$$

In addition,

$$\begin{array}{c}\hfill su{p}_{t\in [0,1]}{\int }_0^1\mid G(t,r){\mid }^{2}dr={\int }_0^1\frac{45}{100}{(t+r)}^{2}dr=1.05.\end{array}$$

We see that all the conditions of Theorem 4 are satisfied, with$\lambda =\frac{9}{10},\mu =0,\nu =0,p=2$and$g=I_X$(Identity mapping). Hence, Theorem 4 ensures a unique solution of (26). Now, for$u_0\left(t\right)=1$and$v_0\left(t\right)=0$, we construct the sequence$(<u_n\left(t\right)>,<v_n\left(t\right)>\}$given by

$$\begin{array}{c}{u}_n\left(t\right)=t+{\int }_0^1\frac{\sqrt{45}(t+r)}{10}(t^2+\frac{9}{20}{u}_{n-1}(r)+\frac{8}{20}{v}_{n-1}(r))dr\\ v_n\left(t\right)=t+{\int }_0^1\frac{\sqrt{45}(t+r)}{10}(t^2+\frac{9}{20}{v}_{n-1}(r)+\frac{8}{20}{u}_{n-1}(r))dr.\end{array}$$

(27)

Using MATLAB, we see that above sequence converges to$\{0.6708t^3+0.3354t^2+2.2339t+0.7677,0.6708t^3+0.3354t^2+2.2339t+0.7677\}$, and this is the unique solution of the system of nonlinear integral Equation (26). The convergence table is given in Table 1 below.

Table 1. Convergence of sequences $<u_n\left(t\right)>$ and $<v_n\left(t\right)>$.

n	${\mathit{u}}_{\mathit{n}}\left(\mathit{t}\right)=\mathit{t}+{\int }_0^1\frac{\sqrt{45}(\mathit{t}+\mathit{r})}{10}({\mathit{t}}^2+\frac{9}{20}{\mathit{u}}_{\mathit{n}-1}\left(\mathit{r}\right)+\frac{8}{20}{\mathit{v}}_{\mathit{n}-1}\left(\mathit{r}\right))\mathit{d}\mathit{r}$	${\mathit{v}}_{\mathit{n}}\left(\mathit{t}\right)=\mathit{t}+{\int }_0^1\frac{\sqrt{45}(\mathit{t}+\mathit{r})}{10}({\mathit{t}}^2+\frac{9}{20}{\mathit{v}}_{\mathit{n}-1}\left(\mathit{r}\right)+\frac{8}{20}{\mathit{u}}_{\mathit{n}-1}\left(\mathit{r}\right))\mathit{d}\mathit{r}$
1	$u_1\left(t\right)=t+0.0167(2t+1)(20t^2+9))$	$v_1\left(t\right)=t+.0671(2t+1)(5t^2+2))$
2	$u_2\left(t\right)=0.6708t^3+0.3354t^2+1.3t+0.5007$	$v_2\left(t\right)=0.6708t^3+0.3354t^2+1.29t+0.5115$
3	$u_3\left(t\right)=0.6708t^3+0.3354t^2+1.8210t+0.5174$	$v_3\left(t\right)=0.6708t^3+0.3354t^2+1.8208t+0.5171$
4	$u_4\left(t\right)=0.6708t^3+0.3354t^2+1.9734t+0.6179$	$v_4\left(t\right)=0.6708t^3+0.3354t^2+1.9734t+0.6178$
5	$u_5\left(t\right)=0.6708t^3+0.3354t^2+2.0743t+0.6755$	$v_5\left(t\right)=0.6708t^3+0.3354t^2+2.0743t+0.6755$
6	$u_6\left(t\right)=0.6708t^3+0.3354t^2+2.1359t+0.7111$	$v_6\left(t\right)=0.6708t^3+0.3354t^2+2.1359t+0.7111$
7	$u_7\left(t\right)=0.6708t^3+0.3354t^2+2.1737t+0.73298$	$v_7\left(t\right)=0.6708t^3+0.3354t^2+2.1737t+0.73298$
8	$u_8\left(t\right)=0.6708t^3+0.3354t^2+2.19699t+0.7464$	$v_8\left(t\right)=0.6708t^3+0.3354t^2+2.19699t+0.7464$
9	$u_9\left(t\right)=0.6708t^3+0.3354t^2+2.2113t+0.7547$	$v_9\left(t\right)=0.6708t^3+0.3354t^2+2.2113t+0.7547$
10	$u_{10}\left(t\right)=0.6708t^3+0.3354t^2+2.2200t+0.7597$	$v_{10}\left(t\right)=0.6708t^3+0.3354t^2+2.2200t+0.7597$
11	$u_{11}\left(t\right)=0.6708t^3+0.3354t^2+2.2254t+0.7628$	$v_{11}\left(t\right)=0.6708t^3+0.3354t^2+2.2254t+0.7628$
12	$u_{12}\left(t\right)=0.6708t^3+0.3354t^2+2.2287t+0.7647$	$v_{12}\left(t\right)=0.6708t^3+0.3354t^2+2.2287t+0.7647$
13	$u_{13}\left(t\right)=0.6708t^3+0.3354t^2+2.2308t+0.7658$	$v_{13}\left(t\right)=0.6708t^3+0.3354t^2+2.2308t+0.7658$
14	$u_{14}\left(t\right)=0.6708t^3+0.3354t^2+2.23199t+0.7666$	$v_{14}\left(t\right)=0.6708t^3+0.3354t^2+2.23199t+0.7666$
15	$u_{15}\left(t\right)=0.6708t^3+0.3354t^2+2.2328t+0.7671$	$v_{15}\left(t\right)=0.6708t^3+0.3354t^2+2.2328t+0.7671$
16	$u_{16}\left(t\right)=0.6708t^3+0.3354t^2+2.2333t+0.7674$	$v_{16}\left(t\right)=0.6708t^3+0.3354t^2+2.2333t+0.7674$
17	$u_{17}\left(t\right)=0.6708t^3+0.3354t^2+2.2336t+0.7675$	$v_{17}\left(t\right)=0.6708t^3+0.3354t^2+2.2336t+0.7675$
18	$u_{18}\left(t\right)=0.6708t^3+0.3354t^2+2.2338t+0.7676$	$v_{18}\left(t\right)=0.6708t^3+0.3354t^2+2.2338t+0.7676$
19	$u_{19}\left(t\right)=0.6708t^3+0.3354t^2+2.2339t+0.7677$	$v_{19}\left(t\right)=0.6708t^3+0.3354t^2+2.2339t+0.7677$
20	$u_{20}\left(t\right)=0.6708t^3+0.3354t^2+2.2339t+0.7677$	$v_{20}\left(t\right)=0.6708t^3+0.3354t^2+2.2339t+0.7677$

Remark 4. 

Condition (iv) of Theorem 4 above is weaker than the corresponding conditions used in similar theorems of [11,13,14].

Remark 5. 

In example 1 above, we see that$su{p}_{t\in [0,1]}{\int }_0^1\mid G(t,r){\mid }^{2}dr={\int }_0^1\frac{45}{100}{(t+r)}^{2}dr=1.05>1$and thus condition (v) of Theorem 3.1 of [11], condition (30) of Theorem 3.1 of [13] and condition (iii) of Theorem 3.1 of [14] are not satisfied.