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Coupled fixed point theorems for singlevalued operators in bmetric spaces
Fixed Point Theory and Applications volume 2015, Article number: 231 (2015)
Abstract
The aim of this paper to present fixed point results for singlevalued operators in bmetric spaces. The case of scalar metric and the case of vectorvalued metric approaches are considered. As an application, a system of integral equations is studied.
Introduction and preliminaries
It is well known that Banach’s contraction principle for singlevalued contractions was extended to several types of generalized metric spaces.
An interesting extension to the case of spaces endowed with vectorvalued metrics was done by Perov [1]. Many other contributions on this topic are known now; see, for example, [2–7].
Another extension of the Banach contraction principle was given for the case of socalled bmetric spaces (also called quasimetric spaces), starting with some results given by Czerwik; see [8]. For previous results on bmetric spaces or extensions of this concept see also Bourbaki [9], Bakhtin [10], Blumenthal [11], among others.
The concept of coupled fixed point and the study of coupled fixed point problems appeared for the first time in some papers of Opoitsev (see [12–14]), while the topic expanded with the work of Guo and Lakshmikantham (see [15]), where the monotone iterations technique is exploited.
If \((X,d)\) is a metric space and \(T:X\times X\to X\) is an operator, then, by definition, a coupled fixed point for T is a pair \((x^{*},y^{*})\in X\times X\) satisfying
Several years later, the theory of coupled fixed points in the setting of an ordered metric space and under some contractive type conditions on the operator T was reconsidered by Gnana Bhaskar and Lakshmikantham in [16] (see also Lakshmikantham and Ćirić in [17]). For other results on coupled fixed point theory see [4, 16–21], among others.
The aim of this paper is to present some fixed point theorems for singlevalued operators in bmetric spaces with applications to a system of integral equations.
We denote by \(M_{mm} (\mathbb{R}_{+} ) \) the set of all \(m\times m\) matrices with positive elements, by I the identity \(m\times m\) matrix and by \(O_{m}\) the null \(m\times m\) matrix. If \(x,y\in\mathbb{R}^{m}\), \(x=(x_{1},\ldots,x_{m})\) and \(y=(y_{1},\ldots,y_{m})\), then, by definition,
Throughout this paper we will make an identification between row and column vectors in \(\mathbb{R}^{m}\).
Let us recall first some important preliminary concepts and results. Let X be a nonempty set. A mapping \(d:X\times X\rightarrow \mathbb{R}^{m}\) is called a vectorvalued metric on X if the following properties are satisfied:

(a)
\(d(x,y)\geq O\) for all \(x,y\in X\); if \(d(x,y)=O\), then \(x=y\) (where \(O:=\underbrace{(0,0,\ldots, 0)}_{m\text{times}}\));

(b)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);

(c)
\(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z\in X\).
A nonempty set X endowed with a vectorvalued metric d is called a generalized metric space in the sense of Perov (in short, a generalized metric space) and it will be denoted by \((X,d)\). The usual notions of analysis (such as convergent sequence, Cauchy sequence, completeness, open subset, closed set, open and closed ball, etc.) are defined similarly to the case of metric spaces.
Notice that the generalized metric space in the sense of Perov is a particular case of the socalled cone metric spaces (or Kmetric space); see [22].
Definition 1.1
A square matrix of real numbers is said to be convergent to zero if and only if all the eigenvalues of A are in the open unit disc (see, for example, [23]).
A classical result in matrix analysis is the following theorem (see, for example, [23, 24]).
Theorem 1.2
Let \(A\in M_{mm} (\mathbb{R}_{+} ) \). The following assertions are equivalent:

(i)
A is convergent toward zero;

(ii)
the spectral radius \(\rho(A)\) is strictly less than 1;

(iii)
\(A^{n}\rightarrow O_{m}\) as \(n\rightarrow\infty\);

(iv)
the matrix \(( IA ) \) is nonsingular and
$$ ( IA ) ^{1}=I+A+\cdots+A^{n}+\cdots; $$(2) 
(v)
the matrix \(( IA ) \) is nonsingular and \(( IA ) ^{1}\) has nonnegative elements;

(vi)
\(A^{n}q\) and \(qA^{n}\) are convergent toward zero as \(n\rightarrow\infty\), for each \(q\in\mathbb{R}^{m}\).
Remark 1.3
Notice also that if \(A,B\in M_{mm} (\mathbb{R}_{+} )\) with \(A\le B\) (in the componentwise meaning), then \(\rho(B)<1\) implies \(\rho(A)<1\).
We will recall now the definition of a bmetric space.
Definition 1.4
Let X be a set and let \(s\geq1\) be a given real number. A functional \(d:X\times X\to\mathbb{R_{+}}\) is said to be a bmetric if the following conditions are satisfied:

1.
if \(x,y\in X\), then \(d(x,y)=0\) if and only if \(x=y\);

2.
\(d(x,y)=d(y,x)\), for all \(x,y\in X\);

3.
\(d(x,z)\leq s[d(x,y)+d(y,z)]\), for all \(x,y,z\in X\).
A pair \((X,d)\) is called a bmetric space.
Some examples of bmetric spaces are given in [8, 20, 25], among others.
If X is a nonempty set and \(f:X\to X\) is a singlevalued operator, then we denote by
the fixed point set for f and by
the set of all invariant subsets of X with respect to f.
Coupled fixed points for mixed monotone singlevalued operators
In this section, we will prove some coupled fixed point theorem for mixed monotone operators in complete bmetric spaces. The approach is based on the iterative construction of a Cauchy successive approximations sequence.
Definition 2.1
Let \((X,\leq)\) a partially ordered set and \(T:X\times X\to X\). We say that T has the mixed monotone property if \(T(\cdot,y)\) is monotone increasing for any \(y\in X\) and \(T(x,\cdot)\) is monotone decreasing for any \(y\in X\).
Let \((X,\leq)\) be a partially ordered set. Even though the notation is the same, we make distinction between the partial order on X and the partial order relation on \(\mathbb{R}\). Notice also that we can endow the product space \(X\times X\) with the following partial order:
Our first main result is the following.
Theorem 2.2
Let \((X,d)\) be a complete bmetric space with \(s\ge1\) and \(T:X\times X\to X\) a continuous mapping with the mixed monotone property on \(X\times X\). Assume that the following conditions are satisfied:

(i)
there exists \(k\in[0,\frac{1}{s})\) such that
$$d\bigl(T(x,y),T(u,v)\bigr)\leq\frac{k}{2}\bigl[d(x,u)+d(y,v)\bigr], \quad \forall x\geq u, y\leq v; $$ 
(ii)
there exists \(x_{0},y_{0}\in X\) such that \(x_{0}\leq T(x_{0},y_{0})\) and \(y_{0}\geq T(y_{0},x_{0})\).
Then there exist \(x,y\in X\) such that \(x=T(x,y)\) and \(y=T(y,x)\).
If, in addition, the bmetric d is continuous, then we have
where \(T^{n+1}(x,y):=T(T^{n}(x,y),T^{n}(y,x))\), for \((x,y)\in X\times X\) and \(n\in\mathbb{N}^{*}\).
Proof
Since \(x_{0}\leq T(x_{0},y_{0}):=x_{1}\) and \(y_{0}\geq T(y_{0},x_{0}):=y_{1}\) we have \((x_{0},y_{0})\leq_{P} (x_{1},y_{1})\).
If we define \(x_{2}:=T(x_{1},y_{1})\) and \(y_{2}:=T(y_{1},x_{1})\), then we get
and
With these notations, due to the mixed monotone property of T, we can prove that
and
Indeed, for \(T(x_{1},y_{1})\geq T(x_{0},y_{0})\), let us notice that, from \((x_{0},y_{0})\leq_{P} (x_{1},y_{1})\), using the mixed monotone property, we have \(T(x_{0},y)\leq T(x_{1},y)\), for any \(y\in X\) and \(T(x,y_{0})\leq T(x,y_{1})\), for any \(x\in X\). Thus, for \(y:=y_{0}\) and \(x:=x_{1}\) and using the transitivity we obtain \(T(x_{0},y_{0})\leq T(x_{1},y_{1})\).
In a similar way one can prove the inequality \(T(y_{1},x_{1})\leq T(y_{0},x_{0})\). Indeed, from \((x_{0},y_{0})\leq_{P} (x_{1},y_{1})\), using the mixed monotone property for T, we have \(T(y_{1},x)\leq T(y_{0},x)\), for any \(x\in X\) and \(T(y,x_{1})\leq T(y,x_{0})\), for any \(y\in X\). Choosing \(y:=y_{0}\) and \(x:=x_{1}\) and using the transitivity, we obtain \(T(y_{1},x_{1})\leq T(y_{0},x_{0})\).
We can easily verify that
and
where \(T^{n+1}(x,y):=T(T^{n}(x,y),T^{n}(y,x))\).
Now we claim that, for \(n\in\mathbb{N}\), one has the relations
We obtain the following inequalities:
and
Indeed, for \(n=1\), since \(T(x_{0},y_{0})\geq x_{0}\) and \(T(y_{0},x_{0})\leq y_{0}\) we get
In a similar way, we have
Now we assume that (3) and (4) hold. Using the inequalities
and
we get the induction step \(P(n+1)\). Indeed, we have
Similarly, we have
This implies that \((T^{n}(x_{0},y_{0}))\) and \((T^{n}(y_{0},x_{0}))\) are Cauchy sequences in X.
Indeed, we have
Similarly we can verify that \((T^{n}(y_{0},x_{0}))\) is also a Cauchy sequence.
Since X is a complete bmetric space and \((T^{n}(x_{0},y_{0}))\) and \((T^{n}(y_{0},x_{0}))\) are Cauchy sequences we see that there exist \(x^{*},y^{*}\in X\) such that
Finally, we claim that \((x^{*},y^{*})\) is a coupled fixed point for T.
Since T is continuous at any \((x,y)\in X\times X\), for any \(\varepsilon>0\) there exists \(\delta>0\) such that, for any \((u,v)\in X\times X\) with
Since \(T^{n}(x_{0},y_{0})\to x^{*}\) and \(T^{n}(y_{0},x_{0})\to y^{*}\), for \(\eta:=\min\{\frac{\varepsilon}{2s},\frac{\delta}{2s} \}>0\) there exist \(n_{0}, m_{0}\in\mathbb{N}\) such that, for every \(n\geq n_{0}\), \(m\geq m_{0}\), we have
Now, for \(n\in\mathbb{N}\) with \(n\geq\max\{n_{0},m_{0}\}\) we have
For the second part of the proof, we will take into account the following inequality:
Letting \(p\to\infty\) and taking into account that d is continuous, we obtain the conclusion:
Similarly, we obtain
and thus
This completes the proof. □
The above results extend some theorems given in [26] for the case of metric spaces. For another contraction type condition and a different approach see [27].
A vector approach in ordered bmetric spaces
Let X be a nonempty set endowed with a partial order relation denoted by ≤. We denote
If \(f:X\to X\) is an operator then we denote the Cartesian product of f with itself as follows:
Definition 3.1
Let X be a nonempty set. Then \((X,d,\le)\) is called an ordered generalized bmetric space if:

(i)
\((X,d)\) is a generalized bmetric space in the sense of Perov;

(ii)
\((X,\le)\) is a partially ordered set.
The following result will be an important tool in our approach.
Theorem 3.2
Let \((X,d,\leq)\) be an ordered generalized complete bmetric space with \(s\ge1\) and let \(f:X\rightarrow X\) be an operator. We suppose that:

(1)
for each \((x,y)\notin X_{\leq}\) there exists \(z(x,y):=z\in X\) such that \((x,z),(y,z)\in X_{\leq}\);

(2)
\(X_{\leq}\in I(f\times f)\);

(3)
\(f:(X,d)\rightarrow(X,d)\) has closed graph;

(4)
there exists \(x_{0}\in X\) such that \((x_{0},f(x_{0}))\in X_{\leq}\);

(5)
there exists a matrix \(A\in M_{mm}(\mathbb{R}_{+})\) for which sA converges to zero, such that
$$d\bigl(f(x),f(y)\bigr)\leq A d(x,y) \quad \textit{for each } (x,y)\in X_{\leq}. $$
Then f is a Picard operator, i.e., \(\operatorname{Fix}(f)=\{ x^{*} \}\) and \(f^{n}(x)\to x^{*}\), as \(n\to\infty\), for every \(x\in X\).
Proof
Let \(x_{0}\in X\) and define \(x_{1}:=f(x_{0})\). Using the condition (4) from the hypothesis we have \((x_{0},f(x_{0}))\in X_{\leq}\). Let \(x_{n+1}:=f(x_{n})\), for \(n\in\mathbb{N}^{*}\). We know that \((x_{0},x_{1})\in X_{\leq}\). By (2) we have \((f(x_{0}),f(x_{1}))=(x_{1},x_{2})\in X_{\leq}\).
We have
Let \(x_{3}:=f(x_{2})\). Since \((x_{1},x_{2})\in X_{\leq}\), we have \((f(x_{1}),f(x_{2}))\in X_{\leq}\). Thus \((x_{2},x_{3})\in X_{\leq}\) and
Thus, for the sequence \(x_{n}:=f^{n}(x_{0})\) of successive approximations of f starting from \(x_{0}\) we have
We prove next that the sequence \((x_{n})\) is Cauchy. We have
Since sA is convergent to zero, by Theorem 1.2, we see that \((IsA)\) is nonsingular and
Thus, by the above relation, we get
Notice now that, by Remark 1.3, the matrix A converges to zero too, which implies that \(d(x_{n},x_{n+p})\to0\) as \(n\to\infty\). Hence the sequence \((x_{n})\) is Cauchy. Since the bmetric space is complete, there exists \(x^{*}\in X\) such that \(x_{n}\to x^{*}\) as \(n\to\infty\).
Using the assumption (3) from the hypothesis we obtain \(\operatorname{Fix}(f)\neq\emptyset\).
For the uniqueness of the fixed point, we suppose that there exists \(y\in X\) such that \(y=f(y)\) and we estimate
From the fact that \((IA)\) is nonsingular we have \(d(x^{*},y)=0\). Hence \(x^{*}=y\).
If \((x,x_{0})\in X_{\leq}\) then, by (2), we have \((f^{n}(x),f^{n}(x_{0}))\in X_{\leq}\), \(\forall n\in\mathbb{N}\). Thus \(f^{n}(x)\to x^{*}\), \(n\to\infty\).
If \((x,x_{0})\notin X_{\leq}\), by (1), there exists \(z(x,x_{0}):=z\in X_{\leq}\) such that \((x,z), (x_{0},z)\in X_{\leq}\). By the fact that \((x_{0},z)\in X_{\leq}\) we have \((f^{n}(x_{0}),f^{n}(z))\in X_{\leq}\), which implies that \(f^{n}(z)\to x^{*}\), \(n\to \infty\). This together with \((x,z)\in X_{\leq}\) implies that \(f^{n}(x)\to x^{*}\), \(n\to\infty\). □
Remark 3.3
In particular, if one of the following classical assumptions holds:
 (2′):

\(f:(X,\le)\to(X,\le)\) is monotone increasing
or
 (2″):

\(f:(X,\le)\to(X,\le)\) is monotone decreasing.
Notice that the assertion (2) in Theorem 3.2 is more general.
Remark 3.4
Condition (4) from the above theorem is equivalent with
 (4′):

f has a lower or an upper fixed point in X.
Notice also that the above theorem extends to the case of bmetric spaces; a result of this type given in [28].
Definition 3.5
Let \((X,d)\) be a generalized bmetric space with constant \(s\ge1\) and \(f:X\to X\) be an operator. The fixed point equation
is called generalized UlamHyers stable if and only if there exists \(\psi:\mathbb{R}^{m}_{+}\rightarrow\mathbb{R}^{m}_{+}\) increasing, continuous in 0 and \(\psi(0)=0\), such that for each \(\varepsilon:=(\varepsilon_{1},\ldots,\varepsilon_{m})\) with \(\varepsilon_{i}>0\) for \(i\in\{1,\ldots,m \}\) and for each solution \(y^{\ast }\in X\) of the inequality
there exists a solution \(x^{\ast}\) of the fixed point equation (5) such that
In particular, if there exists \(C\in M_{m,m}(\mathbb{R_{+}})\) such that \(\psi(t):=C\cdot t\), for each \(t\in\mathbb{R}^{m}_{+}\), then the fixed point equation (5) is said to be UlamHyers stable.
Theorem 3.6
Let \((X,d)\) be an ordered generalized bmetric space with constant \(s\ge1\) and \(f:X\to X\) be an operator. Suppose that all the hypotheses of Theorem 3.2 hold. Then the fixed point equation (5) is UlamHyers stable.
Proof
By Theorem 3.2 we see that f is a Picard operator and hence \(\operatorname{Fix}(f)=\{x^{*} \}\). Thus, for any \(y^{*}\in X\) with \(d(y^{*},f(y^{*}))\le\varepsilon\) we have
Thus,
□
Let us consider now the case of coupled fixed point problems. Let \((X,d)\) be a bmetric space and \(T:X\times X\to X\) be an operator. Then, by definition, a coupled fixed point for T is a pair \((x^{*},y^{*})\in X\times X\) satisfying
We will apply the above results to the above coupled fixed point problem. Our main result concerning the coupled fixed point problem (7) is the following theorem.
Theorem 3.7
Let \((X,d,\le)\) be an ordered and complete bmetric space with constant \(s\ge1\) and let \(T:X\times X\rightarrow X\) be an operator. We suppose:

(i)
for each \(z,w\in X\times X\) which are not comparable with respect to the partial ordering ⪯ on \(X\times X\), there exists \(t\in X\times X\) (which may depend on z and w) such that t is comparable (with respect to the partial ordering ⪯) with both z and w;

(ii)
T has the generalized mixed monotone property, i.e., for all (\(x\geq u\) and \(y\leq v\)) or (\(u\geq x\) and \(v\leq y\)) we have
$$\left \{ \textstyle\begin{array}{l} T(x,y)\geq T(u,v), \\ T(y,x)\leq T(v,u)\end{array}\displaystyle \right .\quad \textit{or} \quad \left \{ \textstyle\begin{array}{l} T(u,v)\geq T(x,y), \\ T(v,u)\leq T(y,x);\end{array}\displaystyle \right . $$ 
(iii)
\(T:X\times X\rightarrow X\) has closed graph;

(iv)
there exists \(z_{0}:=(z_{0}^{1},z_{0}^{2})\in X\times X\) such that
$$\left \{ \textstyle\begin{array}{l} z_{0}^{1}\geq T(z_{0}^{1},z_{0}^{2}), \\ z_{0}^{2}\leq T(z_{0}^{2},z_{0}^{1})\end{array}\displaystyle \right . \quad \textit{or} \quad \left \{ \textstyle\begin{array}{l} T(z_{0}^{1},z_{0}^{2})\geq z_{0}^{1} , \\ T(z_{0}^{2},z_{0}^{1})\leq z_{0}^{2};\end{array}\displaystyle \right . $$ 
(v)
there exist \(k_{1}, k_{2}\in\mathbb{R}_{+}\) with \(k_{1}+k_{2}<\frac{1}{s}\) such that
$$ d\bigl(T(x,y),T(u,v)\bigr) \leq k_{1}d(x,u)+k_{2}d(y,v) $$for all (\(x\geq u\) and \(y\leq v\)) or (\(u\geq x\) and \(v\leq y\)).
Then there exists a unique element \((x^{\ast},y^{\ast})\in X\times X\) such that
and the sequence of the successive approximations \((T^{n}(w_{0}^{1},w_{0}^{2}),T^{n}(w_{0}^{2},w_{0}^{1}))\) converges to \((x^{\ast},y^{\ast})\) as \(n\rightarrow\infty\), for all \((w_{0}^{1},w_{0}^{2})\in X\times X\).
Proof
Denote \(Z:= X\times X\) and define on Z, for \(z:=(x,y), w:=(u,v)\in Z\), the partial order relation
We denote
Let \(F:Z\to Z\) be an operator defined by
By our assumption (ii), we have \(Z_{\preceq}\in I(F\times F)\).
Indeed, by our hypotheses, it follows that Theorem 3.2 is applicable for the operator F. More precisely, F is a contraction with a matrix
i.e., we have
where the space Z is endowed with the vectorvalued bmetric \(\tilde{d}:Z\times Z\rightarrow \mathbb{R}_{+}^{2}\) by
□
Concerning the UlamHyers stability problem for a system of operatorial equations we have the concept.
Definition 3.8
Let \((X,d)\) be a bmetric space with constant \(s\ge1\) and let \(T_{1},T_{2}:X\times X\to X\) be two operators. The operatorial equation system
is called UlamHyers stable if and only if there exist \(c_{1},c_{2},c_{3},c_{4}>0\) such that, for each \(\varepsilon_{1},\varepsilon_{2}>0\) and for each solution pair \((u^{\ast},v^{\ast})\in X\times X\) of the inequations
there exists a solution \((x^{\ast},y^{\ast})\in X\times X\) of the fixed point system (9) such that
By Theorem 3.6,we get the following UlamHyers stability result for the coupled fixed point problem.
Theorem 3.9
Assume that all the assumptions of Theorem 3.7 are satisfied. Then the operatorial equations system
is UlamHyers stable.
Proof
The conclusion follows by Theorem 3.6, applied for the fixed point problem \((x,y)=F(x,y)\), where \(F ( x,y ):= ( T ( x,y ),T ( y,x ) )\). □
An application
We will discuss now an application of the previous result. Let us consider the following system of integral equations:
where \(t\in[0,T]\).
A solution of the above system is a pair \((x,y)\in C[0,T]\times C[0,T]\) satisfying the above relations for all \(t\in[0,T]\).
We consider \(X:=C[0,T]\) endowed with the partial order relation:
We will also consider the following bmetric on X:
Notice that d is a bmetric with constant \(s=2\) and d can be represented using the supremum (Cebîşev) type norm by \(d(x,y)=\(xy)^{2}\_{C}\).
Then we have the following existence and uniqueness result.
Theorem 4.1
Consider the integral system (13). We suppose:

(i)
\(g:[0,T]\to\mathbb{R}\) and \(f:[0,T]\times\mathbb{R}^{2}\to \mathbb{R}\) are continuous and \(G:[0,T]\times[0,T]\to\mathbb{R}_{+}\) is integrable with respect to the first variable.

(ii)
\(f(s,\cdot,\cdot)\) has the generalized mixed monotone property with respect to the last two variables for all \(s\in[0,T]\).

(iii)
There exists \(\alpha,\beta:[0,T]\to\mathbb{R}_{+}\) in \(L^{1}[0,T]\) such that, for each \(u_{1},u_{2},v_{1},v_{2}\in\mathbb{R}\) with \(u_{1}\le v_{1}\) and \(u_{2}\ge v_{2}\) (or reversely), we have
$$\bigl\vert f(s,u_{1},u_{2})f(s,v_{1},v_{2}) \bigr\vert \le\alpha(s)u_{1}v_{1}+\beta(s) u_{2}v_{2} \quad \textit{for each } s\in[0,T]. $$ 
(iv)
\(\max_{t\in[0,T]}(\int_{0}^{T} G(s,t)\alpha (s)\, ds)^{2}+ \max_{t\in[0,T]}(\int_{0}^{T} G(s,t) \beta(s)\, ds)^{2}<\frac{1}{4}\).

(v)
There exists \(x_{0},y_{0}\in C[0,T]\) such that
$$ \left \{ \textstyle\begin{array}{l} x_{0}(t)\le g(t)+\int_{0}^{T}G(s,t)f(s,x_{0}(s),y_{0}(s))\, ds, \\ y_{0}(t)\ge g(t)+\int_{0}^{T}G(s,t)f(s,y_{0}(s),x_{0}(s))\, ds \end{array}\displaystyle \right . $$(14)or
$$ \left \{ \textstyle\begin{array}{l} x_{0}(t)\ge g(t)+\int_{0}^{T}G(s,t)f(s,x_{0}(s),y_{0}(s))\, ds, \\ y_{0}(t)\le g(t)+\int_{0}^{T}G(s,t)f(s,y_{0}(s),x_{0}(s))\, ds \end{array}\displaystyle \right . $$(15)for all \(t\in[0,T]\).
Then there exists a unique solution \((x^{*},y^{*})\) of the system (13).
Proof
We will work in the space \((X,d)\) with
which is a bmetric space with \(s=2\).
We can prove that all the assumptions of Theorem 3.7 are satisfied. We define \(S:X\times X\to X\) by
Then system (13) can be written as a coupled fixed point problem for S:
Then, for all (\(x\geq u\) and \(y\leq v\)) or (\(u\geq x\) and \(v\leq y\)), we have
Thus, taking the maximum over \(t\in[0,T]\), we get
where \(k_{1}:=2 \max_{t\in[0,T]}(\int_{0}^{T} G(s,t)\alpha (s)\, ds)^{2}\) and \(k_{2}:=2 \max_{t\in[0,T]}(\int_{0}^{T} G(s,t) \beta(s)\, ds)^{2}\).
Hence, for all (\(x\geq u\) and \(y\leq v\)) or (\(u\geq x\) and \(v\leq y\)), we get
Since \(k_{1}+k_{2}<\frac{1}{2}\) (by the assumption (iv)), we see that all the assumptions of Theorem 3.7 are satisfied and the conclusion follows. □
Remark 4.2
Using Theorem 3.9, the UlamHyers stability of the integral equations system can be established, under similar assumptions. We also mention that the method can be extended to a system of Volterra type integral equations; see for such a result [27].
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Acknowledgements
For the first author, this paper was supported by a grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, project number PNIIIDPCE201130094. The second author thanks the Visiting Professor Programming at King Saud University for funding this work. The fourth author extends his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Prolific Research Group (PRG143610).
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MFB and GP carried out the studies on coupled fixed points for mixed monotone operators and drafted the manuscript. AP and BS carried out the vectorial approach to coupled fixed point theory and the application section. All the authors have equal contributions to this work. All authors read and approved the final manuscript.
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Bota, MF., Petruşel, A., Petruşel, G. et al. Coupled fixed point theorems for singlevalued operators in bmetric spaces. Fixed Point Theory Appl 2015, 231 (2015). https://doi.org/10.1186/s1366301504823
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DOI: https://doi.org/10.1186/s1366301504823
MSC
 47H10
 54H25
Keywords
 singlevalued operator
 vectorvalued metric
 fixed point
 ordered metric space
 coupled fixed point
 integral equation
 UlamHyers stability