 Research
 Open access
 Published:
Coupled fixed point theorems for nonlinear contractions without mixed monotone property
Fixed Point Theory and Applications volume 2012, Article number: 170 (2012)
Abstract
In this paper, we show the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented.
MSC:47H10, 54H25.
1 Introduction
Let X be an arbitrary nonempty set. A fixed point for a self mapping f:X\to X is a point x\in X such that fx=x. The applications of fixed point theorems are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering and economics in dealing with problems arising in approximation theory, potential theory, game theory, mathematical economics, theory of differential equations, theory of integral equations, theory of matrix equations etc. (see, e.g., [1–6]). For example, fixed point theorems are incredibly useful when it comes to prove the existence of various types of Nash equilibria (see, e.g., [1]) in economics. Fixed point theorems are also helpful for proving the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the JouleThomson effect (see, e.g., [7]).
One of the very popular tools of a fixed point theory is the Banach contraction principle which first appeared in 1922. It states that if (X,d) is a complete metric space and T:X\to X is a contraction mapping (i.e., d(Tx,Ty)\le kd(x,y) for all x,y\in X, where k is a nonnegative number such that k<1), then T has a unique fixed point. Several mathematicians have been dedicated to improvement and generalization of this principle (see [8–14]).
Especially, in 2004, Ran and Reurings [15] showed the existence of fixed points of nonlinear contraction mappings in metric spaces endowed with a partial ordering and presented applications of their results to matrix equations. Since 2004 some authors have studied fixed point theorems in partially ordered metric spaces (see [16–19] and references therein). Subsequently, Nieto and RodríguezLópez [18] extended the results in [15] for nondecreasing mappings and obtained a unique solution for a firstorder ordinary differential equation with periodic boundary conditions (see also [19]).
One of the interesting and crucial concepts, a coupled fixed point theorem, was introduced by Guo and Lakshmikantham [20]. In 2006 Bhaskar and Lakshmikantham [21] introduced the notion of the mixed monotone property of a given mapping. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. They also established the classical coupled fixed point theorems and gave some of their applications. The main results of Bhaskar and Lakshmikantham are as follows.
Theorem 1.1 (Bhaskar and Lakshmikantham [21])
Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists a k\in [0,1) with
for all x,y,u,v\in X for which x\u2ab0u and y\u2aafv. If there exists {x}_{0},{y}_{0}\in X such that
then there exists x,y\in X such that x=F(x,y) and y=F(y,x).
Theorem 1.2 (Bhaskar and Lakshmikantham [21])
Let (X,\u2aaf) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose that X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence with \{{x}_{n}\}\to x, then {x}_{n}\u2aafx for all n\ge 1,

(ii)
if \{{y}_{n}\} is a nonincreasing sequence with \{{y}_{n}\}\to y, then {y}_{n}\u2ab0y for all n\ge 1.
Let F:X\times X\to X be a mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) with
for all x,y,u,v\in X for which x\u2ab0u and y\u2aafv. If there exists {x}_{0},{y}_{0}\in X such that
then there exists x,y\in X such that x=F(x,y) and y=F(y,x).
Because of the important role of Theorems 1.1 and 1.2 in nonlinear differential equations, nonlinear integral equations and differential inclusions, many authors have studied the existence of coupled fixed points of the given mappings in several spaces and applications (see [22–31] and references therein).
In this paper, we establish the existence of a coupled fixed point of the given mapping in complete metric spaces without the mixed monotone property. We also give some illustrative examples to illustrate our main theorems. Furthermore, we find the necessary condition to guarantee the uniqueness of the coupled fixed point. Our results improve and extend some coupled fixed point theorems of Bhaskar and Lakshmikantham [21] and others. As an application, we apply the main results to the setting of partially ordered metric spaces and also present some applications to integral equations.
2 Preliminaries
In this section, we give some definitions, examples and remarks which are useful for main results in this paper.
Throughout this paper, P(X) denotes a collection of subsets of X, and (X,\u2aaf) denotes a partially ordered set with the partial order ⪯. By x\u2ab0y, we mean y\u2aafx. A mapping f:X\to X is said to be nondecreasing (resp., nonincreasing) if for all x,y\in X, x\u2aafy implies f(x)\u2aaff(y) (resp., f(y)\u2aaff(x)).
Definition 2.1 (Bhaskar and Lakshmikantham [21])
Let (X,\u2aaf) be a partially ordered set and F:X\times X\to X. The mapping F is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument, that is, for any x,y\in X,
and
Definition 2.2 (Bhaskar and Lakshmikantham [21])
Let X be a nonempty set. An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X\to X if x=F(x,y) and y=F(y,x).
Example 2.3 Let X=[0,\mathrm{\infty}) and F:X\times X\to X be defined by
for all x,y\in X. It is easy to see that F has a unique coupled fixed point (0,0).
Example 2.4 Let X=P([0,\mathrm{\infty})) and F:X\times X\to X be defined by
for all A,B\in X. We can see that a coupled fixed point of F is (\tilde{A},\tilde{B}), where \tilde{A} and \tilde{B} are disjoint sets.
Next, we give the notion of an Finvariant set which is due to Samet and Vetro [32].
Definition 2.5 (Samet and Vetro [32])
Let (X,d) be a metric space and F:X\times X\to X be a given mapping. Let M be a nonempty subset of {X}^{4}. We say that M is an Finvariant subset of {X}^{4} if and only if, for all x,y,z,w\in X,

(i)
(x,y,z,w)\in M\u27fa(w,z,y,x)\in M;

(ii)
(x,y,z,w)\in M\u27f9(F(x,y),F(y,x),F(z,w),F(w,z))\in M.
Here, we introduce the new property which is useful for our main results.
Definition 2.6 Let (X,d) be a metric space and M be a subset of {X}^{4}. We say that M satisfies the transitive property if and only if, for all x,y,z,w,a,b\in X,
Remark 2.7 We can easily check that the set M={X}^{4} is trivially Finvariant, which satisfies the transitive property.
Example 2.8 Let X=\{0,1,2,3\} endowed with the usual metric and F:X\times X\to X be defined by
It easy to see that M={\{1,2\}}^{4}\subseteq {X}^{4} is Finvariant, which satisfies the transitive property.
Example 2.9 Let X=\mathbb{R} endowed with the usual metric and F:X\times X\to X be defined by
It easy to see that M={[(\mathrm{\infty},1)\cup (1,\mathrm{\infty})]}^{4}\subseteq {X}^{4} is Finvariant, which satisfies the transitive property.
The following example plays a key role in the proof of our main results in a partially ordered set.
Example 2.10 Let (X,d) be a metric space endowed with a partial order ⪯. Let F:X\times X\to X be a mapping satisfying the mixed monotone property, that is, for all x,y\in X, we have
and
Define a subset M\subseteq {X}^{4} by
Then M is an Finvariant subset of {X}^{4}, which satisfies the transitive property.
3 Coupled fixed point theorems without the mixed monotone property
Theorem 3.1 Let (X,d) be a complete metric space and M be a nonempty subset of {X}^{4}. Assume that there is a function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with 0=\phi (0)<\phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t for each t>0, and also suppose that F:X\times X\to X is a mapping such that
for all (x,y,u,v)\in M. Suppose that either

(a)
F is continuous or

(b)
if for any two sequences \{{x}_{n}\}, \{{y}_{n}\} with ({x}_{n+1},{y}_{n+1},{x}_{n},{y}_{n})\in M,
\{{x}_{n}\}\to x,\phantom{\rule{2em}{0ex}}\{{y}_{n}\}\to y
for all n\ge 1, then (x,y,{x}_{n},{y}_{n})\in M for all n\ge 1.
If there exists ({x}_{0},{y}_{0})\in X\times X such that (F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M and M is an Finvariant set which satisfies the transitive property, then there exists x,y\in X such that x=F(x,y) and y=F(y,x), that is, F has a coupled fixed point.
Proof From F(X\times X)\subseteq X, we can construct two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that
for all n\in \mathbb{N}. If there exists {n}^{\star}\in \mathbb{N} such that {x}_{{n}^{\star}1}={x}_{{n}^{\star}} and {y}_{{n}^{\star}1}={y}_{{n}^{\star}}, then
Thus, ({x}_{{n}^{\star}1},{y}_{{n}^{\star}1}) is a coupled fixed point of F. This finishes the proof. Therefore, we may assume that {x}_{n1}\ne {x}_{n} or {y}_{n1}\ne {y}_{n} for all n\in \mathbb{N}.
Since (F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})=({x}_{1},{y}_{1},{x}_{0},{y}_{0})\in M and M is an Finvariant set, we get
Again, using the fact that M is an Finvariant set, we have
By repeating this argument, we get
for all n\in \mathbb{N}. Denote {\delta}_{n1}:=d({x}_{n},{x}_{n1})+d({y}_{n},{y}_{n1})>0 for all n\in \mathbb{N}.
Now, we show that
for all n\in \mathbb{N}. Since ({x}_{n},{y}_{n},{x}_{n1},{y}_{n1})\in M for all n\in \mathbb{N}, from (3.1), it follows that
Since M is an Finvariant set and ({x}_{n},{y}_{n},{x}_{n1},{y}_{n1})\in M for all n\in \mathbb{N}, we get ({y}_{n1},{x}_{n1},{y}_{n},{x}_{n})\in M for all n\in \mathbb{N}. From (3.1) and ({y}_{n1},{x}_{n1},{y}_{n},{x}_{n})\in M for all n\in \mathbb{N}, we get
Adding (3.3) and (3.4), we get
for all n\in \mathbb{N}. From (3.5) and \phi (t)<t for all t>0, we have
for all n\in \mathbb{N}, that is, \{{\delta}_{n}\} is a monotone decreasing sequence. Therefore, {lim}_{n\to \mathrm{\infty}}{\delta}_{n}=\delta for some \delta \ge 0.
Now, we show that \delta =0. Suppose that \delta >0. Taking n\to \mathrm{\infty} of both sides of (3.5), from {lim}_{r\to {t}^{+}}\phi (r)<t for all r>0, it follows that
which is a contradiction. Thus, \delta =0 and
Next, we prove that \{{x}_{n}\} and \{{y}_{n}\} are Cauchy sequences. Suppose that at least one, \{{x}_{n}\} or \{{y}_{n}\}, is not a Cauchy sequence. Then there exists \u03f5>0 and two subsequences of integers {n}_{k} and {m}_{k} with {n}_{k}>{m}_{k}\ge k such that
for all k\in \{1,2,\dots \}. Further, corresponding to {m}_{k}, we can choose {n}_{k} in such a way that it is the smallest integer with {n}_{k}>{m}_{k}\ge k satisfying (3.7). Then we have
Using (3.7), (3.8) and the triangle inequality, we have
Letting k\to \mathrm{\infty} and using (3.6), we have {lim}_{k\to \mathrm{\infty}}{r}_{k}=\u03f5>0.
Since {n}_{k}>{m}_{k} and M satisfies the transitive property, we get
From (3.1) and (3.10), we get
and
Adding (3.11) and (3.12), we get
for all k\in \{1,2,\dots \}. Taking k\to \mathrm{\infty} of both sides of (3.13), from {lim}_{r\to {t}^{+}}\phi (r)<t for all r>0, it follows that
which is a contradiction. Therefore, \{{x}_{n}\} and \{{y}_{n}\} are Cauchy sequences. Since X is complete, there exists x,y\in X such that
Finally, we show that x=F(x,y) and y=F(y,x). If the assumption (a) holds, then we have
and
Therefore, x=F(x,y) and y=F(y,x), that is, F has a coupled fixed point.
Suppose that (b) holds. We obtain that a sequence \{{x}_{n}\} converges to x and a sequence \{{y}_{n}\} converges to y for some x,y\in X. By the assumption, we have (x,y,{x}_{n},{y}_{n})\in M for all n\in \mathbb{N}. Since (x,y,{x}_{n},{y}_{n})\in M for all n\in \mathbb{N}, by the triangle inequality and (3.1), we get
Taking n\to \mathrm{\infty}, we have d(F(x,y),x)=0, and so x=F(x,y). Similarly, we can conclude that y=F(x,y). Therefore, F has a coupled fixed point. This completes the proof. □
Now, we give an example to validate Theorem 3.1.
Example 3.2 Let X=\mathbb{R} endowed with the usual metric d(x,y)=xy for all x,y\in X and endowed with the usual partial order defined by x\u2aafy\u27fayx\in [0,\mathrm{\infty}). Define a continuous mapping F:X\times X\to X by
for all (x,y)\in X\times X. Let {y}_{1}=2 and {y}_{2}=3. Then we have {y}_{1}\u2aaf{y}_{2}, but F(x,{y}_{1})\u2aafF(x,{y}_{2}), and so the mapping F does not satisfy the mixed monotone property.
Now, let \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) be a function defined by \phi (t)=\frac{2}{3}t for all t\in [0,\mathrm{\infty}). Then we obtain 0=\phi (0)<\phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t for any t>0. By simple calculation, we see that for all x,y,u,v\in X,
Therefore, if we apply Theorem 3.1 with M={X}^{4}, we know that F has a unique coupled fixed point, that is, a point (2,2) is a unique coupled fixed point.
Remark 3.3 Although the mixed monotone property is an essential tool in the partially ordered metric spaces to show the existence of coupled fixed points, the mappings do not have the mixed monotone property in a general case as in the above example. Therefore, Theorem 3.1 is interesting, as a new auxiliary tool, in showing the existence of a coupled fixed point.
If we take the mapping \phi (t)=kt for some k\in [0,1) in Theorem 3.1, then we get the following:
Corollary 3.4 Let (X,d) be a complete metric space and M be a nonempty subset of {X}^{4}. Suppose that F:X\times X\to X is a mapping such that there exists k\in [0,1) such that
for all (x,y,u,v)\in M. Suppose that either

(a)
F is continuous or

(b)
for any two sequences \{{x}_{n}\}, \{{y}_{n}\} with ({x}_{n+1},{y}_{n+1},{x}_{n},{y}_{n})\in M, if
\{{x}_{n}\}\to x,\phantom{\rule{2em}{0ex}}\{{y}_{n}\}\to y
for all n\in \mathbb{N}, then (x,y,{x}_{n},{y}_{n})\in M for all n\in \mathbb{N}.
If there exists ({x}_{0},{y}_{0})\in X\times X such that (F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M and M is an Finvariant set which satisfies the transitive property, then there exists x,y\in X such that x=F(x,y) and y=F(y,x), that is, F has a coupled fixed point.
Now, from Theorem 3.1, we have the following question:
(Q1) Is it possible to guarantee the uniqueness of the coupled fixed point of F?
Now, we give positive answers to this question.
Theorem 3.5 In addition to the hypotheses of Theorem 3.1, suppose that for all (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X such that (x,y,u,v)\in M and (z,t,u,v)\in M. Then F has a unique coupled fixed point.
Proof From Theorem 3.1, we know that F has a coupled fixed point. Suppose that (x,y) and (z,t) are coupled fixed points of F, that is, x=F(x,y), y=F(y,x), z=F(z,t) and t=F(t,z).
Now, we show that x=z and y=t. By the hypothesis, there exists (u,v)\in X\times X such that (x,y,u,v)\in M and (z,t,u,v)\in M. We put {u}_{0}=u and {v}_{0}=v and construct two sequences \{{u}_{n}\} and \{{v}_{n}\} by
for all n\in \mathbb{N}.
Since M is Finvariant and (x,y,{u}_{0},{v}_{0})=(x,y,u,v)\in M, we have
that is,
From (x,y,{u}_{1},{v}_{1})\in M, if we use again the property of Finvariant, then it follows that
and so
By repeating this process, we get
for all n\in \mathbb{N}. From (3.1) and (3.19), we have
Since M is Finvariant and (x,y,{u}_{n},{v}_{n})\in M for all n\in \mathbb{N}, we have
for all n\in \mathbb{N}. From (3.1) and (3.21), we get
Thus, from (3.20) and (3.22), we have
for all n\in \mathbb{N}. By repeating this process, we get
for all n\in \mathbb{N}. From \phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t, it follows that {lim}_{n\to \mathrm{\infty}}{\phi}^{n}(t)=0 for each t>0. Therefore, from (3.24), we have
Similarly, we can prove that
By the triangle inequality, for all n\in \mathbb{N}, we have
Taking n\to \mathrm{\infty} in (3.27) and using (3.25) and (3.26), we have d(x,z)+d(y,t)=0, and so x=z and y=t. Therefore, F has a unique coupled fixed point. This completes the proof. □
Next, we give a simple application of our results to coupled fixed point theorems in partially ordered metric spaces.
Corollary 3.6 Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Assume that there is a function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with 0=\phi (0)<\phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t for each t>0 and also suppose that F:X\times X\to X is a mapping such that F has the mixed monotone property and
for all x,y,u,v\in X for which x\u2ab0u and y\u2aafv. Suppose that either

(a)
F is continuous or

(b)
X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence with \{{x}_{n}\}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N},

(ii)
if \{{y}_{n}\} is a nonincreasing sequence with \{{y}_{n}\}\to y, then y\u2ab0{y}_{n} for all n\in \mathbb{N}.
If there exists {x}_{0},{y}_{0}\in X such that
then there exists x,y\in X such that x=F(x,y) and y=F(y,x), that is, F has a coupled fixed point.
Proof First, we define a subset M\subseteq {X}^{4} by
From Example 2.10, we can conclude that M is an Finvariant set which satisfies the transitive property. By (3.28), we have
for all x,y,u,v\in X with (x,y,u,v)\in M. Since {x}_{0},{y}_{0}\in X such that
we get
For the assumption (b), for any two sequences \{{x}_{n}\}, \{{y}_{n}\} such that \{{x}_{n}\} is a nondecreasing sequence in X with {x}_{n}\to x and \{{y}_{n}\} is a nonincreasing sequence in X with {y}_{n}\to y, we have
and
for all n\in \mathbb{N}. Therefore, we have (x,y,{x}_{n},{y}_{n})\in M for all n\in \mathbb{N}, and so the assumption (b) of Theorem 3.1 holds.
Now, since all the hypotheses of Theorem 3.1 hold, F has a coupled fixed point. This completes the proof. □
Corollary 3.7 In addition to the hypotheses of Corollary 3.6, suppose that for all (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X such that x\u2ab0u, y\u2aafv and z\u2ab0u, t\u2aafv. Then F has a unique coupled fixed point.
Proof First, we define a subset M\subseteq {X}^{4} by
From Example 2.10, we can conclude that M is an Finvariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.6.
Next, we show the uniqueness of a coupled fixed point of F. Since for all (x,y),(z,t)\in X\times X, there exists (u,v)\in X\times X such that x\u2ab0u, y\u2aafv and z\u2ab0u, t\u2aafv, we can conclude that (x,y,u,v)\in M and (z,t,u,v)\in M. Therefore, since all the hypotheses of Theorem 3.5 hold, F has a unique coupled fixed point. This completes the proof. □
Corollary 3.8 (Bhaskar and Lakshmikantham [21])
Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) with
for all x,y,u,v\in X for which x\u2ab0u and y\u2aafv. If there exists {x}_{0},{y}_{0}\in X such that
then there exists x,y\in X such that x=F(x,y) and y=F(y,x).
Proof Taking \phi (t)=kt for some k\in [0,1) in Corollary 3.6(a), we can get the conclusion. □
Corollary 3.9 (Bhaskar and Lakshmikantham [21])
Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Suppose that X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence with \{{x}_{n}\}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N},

(ii)
if \{{y}_{n}\} is a nonincreasing sequence with \{{y}_{n}\}\to y, then {y}_{n}\u2ab0y for all n\in \mathbb{N}.
Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) with
for all x,y,u,v\in X for which x\u2ab0u and y\u2aafv. If there exists {x}_{0},{y}_{0}\in X such that
then there exists x,y\in X such that x=F(x,y) and y=F(y,x).
Proof Taking \phi (t)=kt for some k\in [0,1) in Corollary 3.6(b), we can get the conclusion. □
4 Applications
In this section, we apply our theorem to the existence theorem for a solution of the following nonlinear integral equations:
where T is a real number such that T>0 and f:[0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}.
Let X=C([0,T],\mathbb{R}) denote the space of \mathbb{R}valued continuous functions on the interval [0,T]. We endowed X with the metric d:X\times X\to \mathbb{R} defined by
It is clear that (X,d) is a complete metric space.
Now, we consider the following assumptions:
Definition 4.1 An element \alpha ,\beta \in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R}) is called a coupled lower and upper solution of the integral equation (4.1) if \alpha (t)\le \beta (t) and
and
for all t\in [0,T].
(⋆_{1}) f:[0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R} is continuous;
(⋆_{2}) for all t\in [0,T] and for all x,y,u,v\in \mathbb{R} for which x\ge u and y\le v, we have
where \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is continuous, nondecreasing and satisfies 0=\phi (0)<\phi (t)<t and {lim}_{r\to {t}^{+}}\phi (r)<t for each t>0.
Next, we give the existence theorem for a unique solution of the integral equations (4.1).
Theorem 4.2 Suppose that ({\star}_{1}) and ({\star}_{2}) hold. Then the integral equations (4.1) have the unique solution (\tilde{x},\tilde{y})\in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R}) if there exists a coupled lower and upper solution for (4.1).
Proof Define the mapping F:C([0,T],\mathbb{R})\times C([0,T],\mathbb{R})\to C([0,T],\mathbb{R}) by
Let M=\{(x,y,u,v)\in {X}^{4}:x(t)\ge u(t)\text{and}y(t)\le v(t)\text{for all}t\in [0,T]\}. It is obvious that M is an Finvariant subset of {X}^{4} which satisfies the transitive property. It is easy to see that (b) given in Theorem 3.1 is satisfied.
Next, we prove that F has a coupled fixed point (\tilde{x},\tilde{y})\in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R}).
Now, let (x,y,u,v)\in M. Using ({\star}_{2}), for all t\in [0,T], we have
which implies that
Therefore, we get
for all (x,y,u,v)\in M. This implies that the condition (3.1) of Theorem 3.1 is satisfied. Moreover, it is easy to see that there exists ({x}_{0},{y}_{0})\in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R}) such that (F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}),{x}_{0},{y}_{0})\in M and all conditions in Theorem 3.1 are satisfied. Therefore, we apply Theorem 3.1 and then we get the solution (\tilde{x},\tilde{y})\in C([0,T],\mathbb{R})\times C([0,T],\mathbb{R}). □
References
Border KC: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, New York; 1985.
Cataldo, A, Lee, EA, Liu, X, Matsikoudis, ED, Zheng, H: A constructive Fixed point theorem and the feedback semantics of timed systems. Technical Report UCB/EECS2006–4, EECS Dept., University of California, Berkeley (2006)
Guo Y: A generalization of Banach’s contraction principle for some nonobviously contractive operators in a cone metric space. Turk. J. Math. 2012, 36: 297–304.
Hyvärinen A: Fast and robust fixedpoint algorithms for independent component analysis. IEEE Trans. Neural Netw. 1999, 10(3):626–634. 10.1109/72.761722
Noumsi A, Derrien S, Quinton P: Acceleration of a content based image retrieval application on the RDISK cluster. IEEE International Parallel and Distributed Processing Symposium 2006.
Yantir A, Gulsan Topal S: Positive solutions of nonlinear mpoint BVP on time scales. Int. J. Differ. Equ. 2008, 3(1):179–194. 0973–6069
Badii M: Existence of periodic solutions for the thermistor problem with the JouleThomson effect. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2008, 54: 1–10. 10.1007/s1156500800415
Arvanitakis AD: A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc. 2003, 131: 3647–3656. 10.1090/S0002993903069375
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S00029939196902395599
Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 93
Sintunavarat W, Kumam P:Weak condition for generalized multivalued (f,\alpha ,\beta )weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042
Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multivalued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3
Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multivalued contraction mappings. Appl. Math. Lett. 2012, 25: 52–57. 10.1016/j.aml.2011.05.047
Sintunavarat W, Kumam P:Common fixed point theorems for generalized \mathcal{JH}voperator classes and invariant approximations. J. Inequal. Appl. 2011., 2011: Article ID 67
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Cho YJ, Saadati R, Wang S: Common fixed point theorems on generalized distance in order cone metric spaces. Comput. Math. Appl. 2011, 61: 1254–1260. 10.1016/j.camwa.2011.01.004
Graily E, Vaezpour SM, Saadati R, Cho YJ: Generalization of fixed point theorems in ordered metric spaces concerning generalized distance. Fixed Point Theory Appl. 2011., 2011: Article ID 30
Nieto JJ, Lopez RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205–2212. 10.1007/s1011400507690
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 1987, 11: 623–632. 10.1016/0362546X(87)900770
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point in partially ordered G metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
Cho YJ, He G, Huang NJ: The existence results of coupled quasisolutions for a class of operator equations. Bull. Korean Math. Soc. 2010, 47: 455–465.
Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly F contractive mappings in topological spaces. Appl. Math. Lett. 2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004
Cho YJ, Rhoades BE, Saadati R, Samet B, Shantawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8
Gordji ME, Cho YJ, Baghani H: Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
Sintunavarat W, Cho YJ, Kumam P: Coupled fixedpoint theorems for contraction mapping induced by cone ballmetric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
Sintunavarat W, Petruşel A, Kumam P:Common coupled fixed point theorems for {w}^{\ast}compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012. doi:10.1007/s12215–012–0096–0
Samet B, Vetro C: Coupled fixed point F invariant set and fixed point of N order. Ann. Funct. Anal. 2010, 1: 46–56.
Acknowledgements
This project was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRUCSEC No.55000613). The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the third 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: 20110021821).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Sintunavarat, W., Kumam, P. & Cho, Y.J. Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl 2012, 170 (2012). https://doi.org/10.1186/168718122012170
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122012170