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 Open Access
Coupled fixed point theorems for singlevalued operators in bmetric spaces
 MonicaFelicia Bota^{1},
 Adrian Petruşel^{1}Email author,
 Gabriela Petruşel^{2} and
 Bessem Samet^{3}
https://doi.org/10.1186/s1366301504823
© Bota et al. 2015
 Received: 29 August 2015
 Accepted: 4 December 2015
 Published: 22 December 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.
Keywords
 singlevalued operator
 vectorvalued metric
 fixed point
 ordered metric space
 coupled fixed point
 integral equation
 UlamHyers stability
MSC
 47H10
 54H25
1 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.
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.
 (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
 (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
 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.
2 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 \(y\in X\).
Our first main result is the following.
Theorem 2.2
 (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 \(y_{0}\geq T(y_{0},x_{0})\).
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 \((x_{0},y_{0})\leq_{P} (x_{1},y_{1})\).
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 \(T(y_{1},x_{1})\leq T(y_{0},x_{0})\).
This implies that \((T^{n}(x_{0},y_{0}))\) and \((T^{n}(y_{0},x_{0}))\) are Cauchy sequences in X.
Similarly we can verify that \((T^{n}(y_{0},x_{0}))\) is also a Cauchy sequence.
Finally, we claim that \((x^{*},y^{*})\) is a coupled fixed point for T.
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].
3 A vector approach in ordered bmetric spaces
Definition 3.1
 (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
 (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$$d\bigl(f(x),f(y)\bigr)\leq A d(x,y) \quad \textit{for each } (x,y)\in X_{\leq}. $$
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 \((f(x_{0}),f(x_{1}))=(x_{1},x_{2})\in X_{\leq}\).
Using the assumption (3) from the hypothesis we obtain \(\operatorname{Fix}(f)\neq\emptyset\).
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
 (2′):

\(f:(X,\le)\to(X,\le)\) is monotone increasing
 (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
 (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
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
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
 (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 thatfor all (\(x\geq u\) and \(y\leq v\)) or (\(u\geq x\) and \(v\leq y\)).$$ d\bigl(T(x,y),T(u,v)\bigr) \leq k_{1}d(x,u)+k_{2}d(y,v) $$
Proof
Concerning the UlamHyers stability problem for a system of operatorial equations we have the concept.
Definition 3.8
By Theorem 3.6,we get the following UlamHyers stability result for the coupled fixed point problem.
Theorem 3.9
Proof
The conclusion follows by Theorem 3.6, applied for the fixed point problem \((x,y)=F(x,y)\), where \(F ( x,y ):= ( T ( x,y ),T ( y,x ) )\). □
4 An application
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 \(t\in[0,T]\).
Then we have the following existence and uniqueness result.
Theorem 4.1
 (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 thator$$ \left \{ \textstyle\begin{array}{l} x_{0}(t)\le g(t)+\int_{0}^{T}G(s,t)f(s,x_{0}(s),y_{0}(s))\, ds, \\ y_{0}(t)\ge g(t)+\int_{0}^{T}G(s,t)f(s,y_{0}(s),x_{0}(s))\, ds \end{array}\displaystyle \right . $$(14)for all \(t\in[0,T]\).$$ \left \{ \textstyle\begin{array}{l} x_{0}(t)\ge g(t)+\int_{0}^{T}G(s,t)f(s,x_{0}(s),y_{0}(s))\, ds, \\ y_{0}(t)\le g(t)+\int_{0}^{T}G(s,t)f(s,y_{0}(s),x_{0}(s))\, ds \end{array}\displaystyle \right . $$(15)
Then there exists a unique solution \((x^{*},y^{*})\) of the system (13).
Proof
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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