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Urysohn integral equations approach by common fixed points in complexvalued metric spaces
Advances in Difference Equations volume 2013, Article number: 49 (2013)
Abstract
Recently, the complexvalued metric spaces which are more general than the metric spaces were first introduced by Azam et al. (Numer. Funct. Anal. Optim. 32:243253, 2011). They also established the existence of fixed point theorems under the contraction condition in these spaces. The aim of this paper is to introduce the concepts of a CCauchy sequence and Ccomplete in complexvalued metric spaces and establish the existence of common fixed point theorems in Ccomplete complexvalued metric spaces. Furthermore, we apply our result to obtain the existence theorem for a common solution of the Urysohn integral equations
where $t\in [a,b]\subseteq \mathbb{R}$, $x,g,h\in C([a,b],{\mathbb{R}}^{n})$ and ${K}_{1},{K}_{2}:[a,b]\times [a,b]\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$.
MSC:47H09, 47H10.
1 Introduction
The study of metric spaces has played a vital role in many branches of pure and applied sciences. We can find useful applications of metric spaces in mathematics, biology, medicine, physics and computer science (see [1–3]). Several mathematicians improved, generalized and extended the concept of metric spaces to vectorvalued metric spaces of Perov [4], Gmetric spaces of Mustafa and Sims [5], cone metric spaces of Huang and Zhang [6], modular metric spaces of Chistyakov [7], partial metric spaces of Matthews [8] and others. Since Banach [9] introduced his contraction principle in complete metric spaces in 1922, this field of fixed point theory has been rapidly growing. It has been very useful in many fields such as optimization problems, control theory, differential equations, economics and many others. A number of papers in this field have been dedicated to the improvement and generalization of Banach’s contraction mapping principle in many spaces and ways (see [10–13]).
Recently, Azam et al. [14] introduced a new space, the socalled complexvalued metric space, and established a fixed point theorem for some type of contraction mappings as follows.
Theorem 1.1 (Azam et al. [14])
Let $(X,d)$ be a complete complexvalued metric space and $S,T:X\to X$ be two mappings. If S and T satisfy
for all $x,y\in X$, where λ, μ are nonnegative reals with $\lambda +\mu <1$, then S and T have a unique common fixed point in X.
Theorem 1.1 of Azam et al. in [14] is an essential tool in the complexvalued metric space to claim the existence of a common fixed point for some mappings. However, it is most interesting to find another new auxiliary tool to claim the existence of a common fixed point. Some other works related to the results in a complexvalued metric space are [15, 16].
In this paper, we introduce the concept of a CCauchy sequence and Ccomplete in complexvalued metric spaces and also prove some common fixed point theorems for new generalized contraction mappings in Ccomplete complexvalued metric spaces.
On the other hand, integral equations arise naturally from many applications in describing numerous real world problems. These equations have been studied by many authors. Existence theorems for the Urysohn integral equations can be obtained applying various fixed point principles.
As applications, we show the existence of a common solution of the following system of Urysohn integral equations by using our common fixed point results:
where $t\in [a,b]\subseteq \mathbb{R}$, $x,g,h\in C([a,b],{\mathbb{R}}^{n})$ and ${K}_{1},{K}_{2}:[a,b]\times [a,b]\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$.
2 Preliminaries
In this section, we discuss some background of the complexvalued metric spaces of Azam et al. in [14] and give some notions for our results. Also, some essential lemmas which are useful for our results are given.
Let ℂ be the set of complex numbers. For ${z}_{1},{z}_{2}\in \mathbb{C}$, we will define a partial order ≾ on ℂ as follows:
We note that ${z}_{1}\precsim {z}_{2}$ if one of the following holds:
(C1) $Re({z}_{1})=Re({z}_{2})$ and $Im({z}_{1})=Im({z}_{2})$;
(C2) $Re({z}_{1})<Re({z}_{2})$ and $Im({z}_{1})=Im({z}_{2})$;
(C3) $Re({z}_{1})=Re({z}_{2})$ and $Im({z}_{1})<Im({z}_{2})$;
(C4) $Re({z}_{1})<Re({z}_{2})$ and $Im({z}_{1})<Im({z}_{2})$.
It obvious that if $a,b\in \mathbb{R}$ such that $a\le b$, then $az\precsim bz$ for all $z\in \mathbb{C}$.
In particular, we write ${z}_{1}\u22e8{z}_{2}$ if ${z}_{1}\ne {z}_{2}$ and one of (C2), (C3) and (C4) is satisfied, and we write ${z}_{1}\prec {z}_{2}$ if only (C4) is satisfied. The following are well known:
Definition 2.1 [14]
Let X be a nonempty set. Suppose that the mapping $d:X\times X\to \mathbb{C}$ satisfies the following conditions:
(d1) $0\precsim d(x,y)$ for all $x,y\in X$ and $d(x,y)=0$ if and only if $x=y$;
(d2) $d(x,y)=d(y,x)$ for all $x,y\in X$;
(d3) $d(x,y)\precsim d(x,z)+d(z,y)$ for all $x,y,z\in X$.
Then d is called a complexvalued metric on X and $(X,d)$ is called a complexvalued metric space.
Example 2.2 Let $X=\mathbb{C}$. Define the mapping $d:X\times X\to \mathbb{C}$ by
where ${z}_{1}={x}_{1}+i{y}_{1}$ and ${z}_{2}={x}_{2}+i{y}_{2}$. Then $(X,d)$ is a complexvalued metric space.
Example 2.3 Let $X={X}_{1}\cup {X}_{2}$, where
and
Define the mapping $d:X\times X\to \mathbb{C}$ by
where ${z}_{1}={x}_{1}+i{y}_{1}$ and ${z}_{2}={x}_{2}+i{y}_{2}$. Then $(X,d)$ is a complexvalued metric space.
Example 2.4 Let $X={X}_{1}\cup {X}_{2}$, where
and
Define the mapping $d:X\times X\to \mathbb{C}$ by
where ${z}_{1}={x}_{1}+i{y}_{1}$ and ${z}_{2}={x}_{2}+i{y}_{2}$. Then $(X,d)$ is a complexvalued metric space.
Definition 2.5 [14]
Let $(X,d)$ be a complexvalued metric space.

(1)
A point $x\in X$ is called an interior point of a set $A\subseteq X$ whenever there exists $0\prec r\in \mathbb{C}$ such that
$$B(x,r)=\{y\in X:d(x,y)\prec r\}\subseteq A.$$ 
(2)
A point $x\in X$ is called a limit point of A whenever, for all $0\prec r\in \mathbb{C}$,
$$B(x,r)\cap (AX)\ne \mathrm{\varnothing}.$$ 
(3)
A set $A\subseteq X$ is called open whenever each element of A is an interior point of A.

(4)
A set $A\subseteq X$ is called closed whenever each limit point of A belongs to A.

(5)
A subbasis for a Hausdorff topology τ on X is the family
$$F=\{B(x,r):x\in X\text{and}0\prec r\}.$$
Definition 2.6 [14]
Let $(X,d)$ be a complexvalued metric space, $\{{x}_{n}\}$ be a sequence in X and let $x\in X$.

(1)
If, for any $c\in \mathbb{C}$ with $0\prec c$, there exists $N\in \mathbb{N}$ such that, for all $n>N$, $d({x}_{n},x)\prec c$, then $\{{x}_{n}\}$ is said to be convergent to a point $x\in X$ or $\{{x}_{n}\}$ converges to a point $x\in X$ and x is the limit point of $\{{x}_{n}\}$. We denote this by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ or ${x}_{n}\to x$ as $n\to \mathrm{\infty}$.

(2)
If, for any $c\in \mathbb{C}$ with $0\prec c$, there exists $N\in \mathbb{N}$ such that, for all $n>N$, $d({x}_{n},{x}_{n+m})\prec c$, where $m\in \mathbb{N}$, then $\{{x}_{n}\}$ is called a Cauchy sequence in X.

(3)
If every Cauchy sequence in X is convergent, then $(X,d)$ is said to be a complete complexvalued metric space.
Next, we give some lemmas which are an essential tool in the proof of main results.
Lemma 2.7 [[14], see Definition 2.5]
Let $(X,d)$ be a complexvalued metric space and $\{{x}_{n}\}$ be a sequence in X. Then $\{{x}_{n}\}$ converges to a point $x\in X$ if and only if $d({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$.
Lemma 2.8 [14]
Let $(X,d)$ be a complexvalued metric space and $\{{x}_{n}\}$ be a sequence in X. Then $\{{x}_{n}\}$ is a Cauchy sequence if and only if $d({x}_{n},{x}_{n+m})\to 0$ as $n\to \mathrm{\infty}$, where $m\in \mathbb{N}$.
Definition 2.9 Let S and T be selfmappings of a nonempty set X.

(1)
A point $x\in X$ is called a fixed point of T if $Tx=x$.

(2)
A point $x\in X$ is called a coincidence point of S and T if $Sx=Tx$ and the point $w\in X$ such that $w=Sx=Tx$ is called a point of coincidence of S and T.

(3)
A point $x\in X$ is called a common fixed point of S and T if $x=Sx=Tx$.
Lemma 2.10 [17]
Let X be a nonempty set and $T:X\to X$ be a function. Then there exists a subset $E\subseteq X$ such that $T(E)=T(X)$ and $T:E\to X$ is onetoone.
3 Common fixed points (I)
Throughout this paper, ℝ denotes a set of real numbers, ${\mathbb{C}}_{+}$ denotes a set $\{c\in \mathbb{C}:0\precsim c\}$ and Γ denotes the class of all functions $\gamma :{\mathbb{C}}_{+}\to [0,1)$ which satisfies the condition: for any sequences $\{{x}_{n}\}$ in ${\mathbb{C}}_{+}$,
The following are examples of the function in Γ:

(1)
${\gamma}_{1}(x)=k$, where $k\in [0,1)$;

(2)
${\gamma}_{2}(x)=\frac{1}{1+kx}$, where $k\in (0,\mathrm{\infty})$.
Now, we introduce the concepts of a CCauchy sequence and Ccomplete in complexvalued metric spaces.
Definition 3.1 Let $(X,d)$ be a complexvalued metric space and $\{{x}_{n}\}$ be a sequence in X.

(1)
If, for any $c\in \mathbb{C}$ with $0\prec c$, there exists $N\in \mathbb{N}$ such that, for all $m,n>N$, $d({x}_{n},{x}_{m})\prec c$, then $\{{x}_{n}\}$ is called a CCauchy sequence in X.

(2)
If every CCauchy sequence in X is convergent, then $(X,d)$ is said to be a Ccomplete complexvalued metric space.
Next, we prove our main results.
Theorem 3.2 Let $(X,d)$ be a Ccomplete complexvalued metric space and $S,T:X\to X$ be mappings. If there exist two mappings $\alpha ,\beta :{\mathbb{C}}_{+}\to [0,1)$ such that, for all $x,y\in X$,

(a)
$\alpha (x)+\beta (x)<1$;

(b)
the mapping $\gamma :{\mathbb{C}}_{+}\to [0,1)$ defined by $\gamma (x):=\frac{\alpha (x)}{1\beta (x)}$ belongs to Γ;

(c)
$d(Sx,Ty)\precsim \alpha (d(x,y))d(x,y)+\frac{\beta (d(x,y))d(x,Sx)d(y,Ty)}{1+d(x,y)}$.
Then S and T have a unique common fixed point in X.
Proof Let ${x}_{0}$ be an arbitrary point in X. We construct the sequence $\{{x}_{n}\}$ in X such that
for all $n\ge 0$. For all $n\ge 0$, we get
which implies that
Similarly, for all $n\ge 0$, we get
which implies that
From (3.2) and (3.3), we have
for all $n\in \mathbb{N}$. Therefore, we get
for all $n\in \mathbb{N}$. This implies the sequence ${\{d({x}_{n1},{x}_{n})\}}_{n\in \mathbb{N}}$ is monotone nonincreasing and bounded below. Therefore, $d({x}_{n1},{x}_{n})\to d$ for some $d\ge 0$.
Next, we claim that $d=0$. Assume to the contrary that $d>0$. In (3.4), taking $n\to \mathrm{\infty}$, we have
Since $\gamma \in \mathrm{\Gamma}$, we get $d({x}_{n1},{x}_{n})\to 0$, which is a contradiction. Therefore, we have $d=0$, that is,
Next, we prove that $\{{x}_{n}\}$ is a CCauchy sequence. According to (3.5), it is sufficient to show that the subsequence $\{{x}_{2n}\}$ is a CCauchy sequence. On the contrary, assume that $\{{x}_{2n}\}$ is not a CCauchy sequence. By Definition 3.1(1), there is $c\in \mathbb{C}$ with $0\prec c$ for which, for all $k\in \mathbb{N}$, there exists ${m}_{k}>{n}_{k}\ge k$ such that
Further, corresponding to ${n}_{k}$, we can choose ${m}_{k}$ in such a way that it is the smallest integer with ${m}_{k}>{n}_{k}\ge k$ satisfying (3.6). Then we have
and
By (3.7), (3.8) and the notion of a complexvalued metric, we have
This implies
On taking limit as $k\to \mathrm{\infty}$, we have
Further, we have
and then
Passing to the limit when $k\to \mathrm{\infty}$ and using (3.5) and (3.9), we get
Now, from the triangle inequality for a complexvalued metric d, we obtain that
which implies that
Taking $k\to \mathrm{\infty}$, we have
that is,
Since $\gamma \in \mathrm{\Gamma}$, we get $d({x}_{2{n}_{k}},{x}_{2{m}_{k}+1})\to 0$, which contradicts $0\prec c$. Therefore, we can conclude that $\{{x}_{2n}\}$ is a CCauchy sequence and hence $\{{x}_{n}\}$ is a CCauchy sequence. By the completeness of X, there exists a point $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$.
Next, we claim that $Sz=z$. If $Sz\ne z$, then $d(z,Sz)>0$. By the notion of a complexvalued metric d, we have
which implies that
Taking $n\to \mathrm{\infty}$, we have $d(z,Sz)=0$, which is a contradiction. Thus, we get $Sz=z$. It follows similarly that $Tz=z$. Therefore, $z=Sz=Tz$, that is, z is a common fixed point of S and T.
Finally, we show that z is a unique common fixed point of S and T. Assume that there exists another point $\stackrel{\u02c6}{z}$ such that $\stackrel{\u02c6}{z}=S\stackrel{\u02c6}{z}=T\stackrel{\u02c6}{z}$. Now, we have
Hence $d(z,\stackrel{\u02c6}{z})\le \alpha (d(z,\stackrel{\u02c6}{z}))d(z,\stackrel{\u02c6}{z})$. Since $0\le \alpha (d(z,\stackrel{\u02c6}{z}))<1$, we get $d(z,\stackrel{\u02c6}{z})=0$ and then $z=\stackrel{\u02c6}{z}$. Therefore, z is a unique common fixed point of S and T. This completes the proof. □
Corollary 3.3 Let $(X,d)$ be a Ccomplete complexvalued metric space and $S,T:X\to X$ be mappings. If S and T satisfy
for all $x,y\in X$, where λ, μ are nonnegative reals with $\lambda +\mu <1$, then S and T have a unique common fixed point in X.
Proof We can prove this result by applying Theorem 3.2 by setting $\alpha (x)=\lambda $ and $\beta (x)=\mu $. □
Corollary 3.4 Let $(X,d)$ be a Ccomplete complexvalued metric space and $T:X\to X$ be a mapping. If there exist two mappings $\alpha ,\beta :{\mathbb{C}}_{+}\to [0,1)$ such that, for all $x,y\in X$,

(a)
$\alpha (x)+\beta (x)<1$;

(b)
the mapping $\gamma :{\mathbb{C}}_{+}\to [0,1)$ defined by $\gamma (x):=\frac{\alpha (x)}{1\beta (x)}$ belongs to Γ;

(c)
$d(Tx,Ty)\precsim \alpha (d(x,y))d(x,y)+\frac{\beta (d(x,y))d(x,Tx)d(y,Ty)}{1+d(x,y)}$.
Then T has a unique fixed point in X.
Proof We can prove this result by applying Theorem 3.2 with $S=T$. □
Corollary 3.5 Let $(X,d)$ be a Ccomplete complexvalued metric space and $T:X\to X$ be a mapping. If T satisfies
for all $x,y\in X$, where λ, μ are nonnegative reals with $\lambda +\mu <1$, then T has a unique fixed point in X.
Proof We can prove this result by applying Corollary 3.4 with $\alpha (x)=\lambda $ and $\beta (x)=\mu $. □
Theorem 3.6 Let $(X,d)$ be a Ccomplete complexvalued metric space and $T:X\to X$. If there exist two mappings $\alpha ,\beta :{\mathbb{C}}_{+}\to [0,1)$ such that, for all $x,y\in X$,

(a)
$\alpha (x)+\beta (x)<1$;

(b)
the mapping $\gamma :{\mathbb{C}}_{+}\to [0,1)$ defined by $\gamma (x):=\frac{\alpha (x)}{1\beta (x)}$ belongs to Γ;

(c)
$d({T}^{n}x,{T}^{n}y)\precsim \alpha (d(x,y))d(x,y)+\frac{\beta (d(x,y))d(x,{T}^{n}x)d(y,{T}^{n}y)}{1+d(x,y)}$ for some $n\in \mathbb{N}$.
Then T has a unique fixed point in X.
Proof From Corollary 3.4, we get ${T}^{n}$ has a unique fixed point z. Since
we know that Tz is a fixed point of ${T}^{n}$. Therefore, $Tz=z$ by the uniqueness of a fixed point of ${T}^{n}$. Therefore, z is also a fixed point of T. Since the fixed point of T is also a fixed point of ${T}^{n}$, the fixed point of T is also unique. □
Corollary 3.7 Let $(X,d)$ be a Ccomplete complexvalued metric space and $S,T:X\to X$ be mappings. If T satisfy
for all $x,y\in X$ for some $n\in \mathbb{N}$, where λ, μ are nonnegative reals with $\lambda +\mu <1$, then T has a unique fixed point in X.
Proof We can prove this result by applying Theorem 3.6 with $\alpha (x)=\lambda $ and $\beta (x)=\mu $. □
Remark 3.8 It is easy to see that Corollaries 3.3, 3.5 and 3.7 hold in complete complexvalued metric spaces. Therefore, Corollaries 3.3, 3.5 and 3.7 become Theorem 4, Corollary 5 and Corollary 6 of Azam et al. [14] in complete complexvalued metric spaces.
4 Common fixed points (II)
In this section, we prove a common fixed point theorem for weakly compatible mappings in Ccomplete complexvalued metric spaces.
Since Banach’s fixed point theorem, many authors have improved, extended and generalized Banach’s fixed point theorem in several ways. Especially, in [18], Jungck generalized Banach’s fixed point theorem by using the concept of commuting mappings as follows.
Theorem J Let $(X,d)$ be a complete metric space. Then a continuous mapping $S:X\to X$ has a fixed point in X if and only if there exists a number $\alpha \in (0,1)$ and a mapping $T:X\to X$ such that $T(X)\subset S(X)$, S and T are commuting (i.e., $TSx=STx$ for all x in X),
for all $x,y\in X$. Further, S and T have a unique common fixed point in X (i.e., there exists a unique point z in X such that $Sz=Tz=z$).
Note that if we put $S={I}_{X}$ (the identity mapping on X) in Theorem J, we have Banach’s fixed point theorem.
Since Theorem J, in 1986, Jungck [18] introduced more generalized commuting mappings in metric spaces, called compatible mappings, which also are more general than weakly commuting mappings (that is, the mappings $S,T:X\to X$ are said to be weakly commuting if $d(STx,TSx)\le d(Sx,Tx)$ for all $x\in X$) introduced by Sessa [19] as follows.
Definition 4.1 Let S and T be mappings from a metric space $(X,d)$ into itself. The mappings S and T are said to be compatible if
whenever $\{{x}_{n}\}$ is a sequence in X such that ${lim}_{n\to \mathrm{\infty}}S{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n}=z$ for some $z\in X$.
In general, commuting mappings are weakly commuting and weakly commuting mappings are compatible, but the converse is not necessarily true; some examples can be found in [18, 20–22].
Also, some authors introduced some kind of generalizations of compatible mappings in metric spaces and other spaces (see [21–24]) and they proved common fixed point theorems using these kinds of compatible mappings in metric spaces and other spaces.
In [25], Jungck and Rhoades introduced the concept of weakly compatible mappings in symmetric spaces $(X,d)$ and proved some common fixed point theorems for these mappings in symmetric spaces as follows.
Definition 4.2 Let S and T be mappings from a metric space $(X,d)$ into itself. The mappings S and T are said to be weakly compatible if they commute at coincidence points of S and T.
In Djoudi and Nisse [26], we can find an example to show that there exist weakly compatible mappings which are not compatible mappings in metric spaces.
Now, we give the main result in this section.
Theorem 4.3 Let $(X,d)$ be a complexvalued metric space and $S,T:X\to X$ be such that $T(X)\subseteq S(X)$ and $S(X)$ is Ccomplete. If there exist two mappings $\alpha ,\beta :{\mathbb{C}}_{+}\to [0,1)$ such that, for all $x,y\in X$,

(a)
$\alpha (x)+\beta (x)<1$;

(b)
the mapping $\gamma :{\mathbb{C}}_{+}\to [0,1)$ defined by $\gamma (x):=\frac{\alpha (x)}{1\beta (x)}$ belongs to Γ;

(c)
$d(Tx,Ty)\precsim \alpha (d(Sx,Sy))d(Sx,Sy)+\frac{\beta (d(Sx,Sy))d(Sx,Tx)d(Sy,Ty)}{1+d(Sx,Sy)}$.
Then S and T have a unique point of coincidence in X. Moreover, S and T have a unique common fixed point in X if S and T are weakly compatible.
Proof Consider the mapping $S:X\to X$. By Lemma 2.10, there exists $E\subseteq X$ such that $S(E)=S(X)$ and $S:E\to X$ is onetoone.
Next, we define a mapping $\mathcal{W}:S(E)\to S(E)$ by $\mathcal{W}(Sx)=Tx$ for all $Sx\in S(E)$. Therefore, is well defined since S is onetoone on E. Since $\mathcal{W}\circ S=T$, using (c), we get
for all $Sx,Sy\in S(E)$. Since $S(E)=S(X)$ is Ccomplete and (4.1) holds, we can apply Corollary 3.4 with a mapping . Therefore, there exists a unique fixed point $z\in S(X)$ such that $\mathcal{W}z=z$. It follows from $z\in S(X)$ that $z=S{z}^{\prime}$ for some ${z}^{\prime}\in X$. So, $\mathcal{W}(S{z}^{\prime})=S{z}^{\prime}$, that is, $T{z}^{\prime}=S{z}^{\prime}$. Therefore, T and S have a unique point of coincidence.
Next, we show that S and T have a common fixed point. Now, we have $z=T{z}^{\prime}=S{z}^{\prime}$. Since S and T are weakly compatible, we get
This implies $Sz=Tz$ is a point of coincidence of S and T. But z is a unique point of coincidence of S and T. Therefore, we conclude that $z=Sz=Tz$, which implies that z is a common fixed point of S and T.
Finally, we prove the uniqueness of a common fixed point of S and T. Assume that $\overline{z}$ is another common fixed point of S and T. So, $\overline{z}=S\overline{z}=T\overline{z}$, and then $\overline{z}$ is also a point of coincidence of S and T. However, we know that z is a unique point of coincidence of S and T. Therefore, we get $\overline{z}=z$, that is, z is a unique common fixed point of S and T. This completes the proof. □
5 Urysohn integral equations
In this section, we show that Theorem 3.2 can be applied to the existence of a common solution of the system of the Urysohn integral equations.
Theorem 5.1 Let $X=C([a,b],{\mathbb{R}}^{n})$, $a>0$, and $d:X\times X\to \mathbb{C}$ be defined by
Consider the Urysohn integral equations
where $t\in [a,b]\subseteq \mathbb{R}$, $x,g,h\in X$ and ${K}_{1},{K}_{2}:[a,b]\times [a,b]\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$.
Suppose that ${K}_{1}$, ${K}_{2}$ are such that ${F}_{x},{G}_{x}\in X$ for all $x\in X$, where
for all $t\in [a,b]$.
If there exist two mappings $\alpha ,\beta :{\mathbb{C}}_{+}\to [0,1)$ such that for all $x,y\in X$ the following hold:

(a)
$\alpha (x)+\beta (x)<1$;

(b)
the mapping $\gamma :{\mathbb{C}}_{+}\to [0,1)$ defined by $\gamma (x):=\frac{\alpha (x)}{1\beta (x)}$ belongs to Γ;

(c)
${\parallel {F}_{x}(t){G}_{y}(t)+g(t)h(t)\parallel}_{\mathrm{\infty}}\sqrt{1+{a}^{2}}{e}^{i{tan}^{1}a}\precsim \alpha ({max}_{t\in [a,b]}A(x,y)(t))A(x,y)(t)+\beta ({max}_{t\in [a,b]}A(x,y)(t))B(x,y)(t)$, where
$$\begin{array}{l}A(x,y)(t)={\parallel x(t)y(t)\parallel}_{\mathrm{\infty}}\sqrt{1+{a}^{2}}{e}^{i{tan}^{1}a},\\ B(x,y)(t)=\frac{{\parallel {F}_{x}(t)+g(t)x(t)\parallel}_{\mathrm{\infty}}{\parallel {G}_{y}(t)+h(t)y(t)\parallel}_{\mathrm{\infty}}}{1+d(x,y)}\sqrt{1+{a}^{2}}{e}^{i{tan}^{1}a},\end{array}$$
then the system of integral equations (5.1) and (5.2) has a unique common solution.
Proof Define two mappings $S,T:X\to X$ by $Sx={F}_{x}+g$ and $Tx={G}_{x}+h$. Then we have
and
We can show easily that for all $x,y\in X$,
Now, we can apply Theorem 3.2. Therefore, we get the Urysohn integral equations (5.1) and (5.2) have a unique common solution. □
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Acknowledgements
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 20120008170) and the third author would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support.
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Sintunavarat, W., Cho, Y.J. & Kumam, P. Urysohn integral equations approach by common fixed points in complexvalued metric spaces. Adv Differ Equ 2013, 49 (2013). https://doi.org/10.1186/16871847201349
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Keywords
 complexvalued metric spaces
 Urysohn integral equations
 common fixed points
 weakly compatible