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Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations
Fixed Point Theory and Applications volume 2013, Article number: 261 (2013)
Abstract
The aim of this paper is to obtain coupled fixed point theorems for self-mappings defined on an ordered closed and convex subset of a quasi-Banach space. Our method of proof is different and constructive in nature. As an application of our coupled fixed point results, we establish corresponding coupled coincidence point results without any type of commutativity of underlying maps. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.
MSC:47H10, 54H25, 55M20.
Dedication
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
1 Introduction
It is well known that the Banach contraction principle is one of the most important results in classical functional analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics (see [1–15]). The study of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [16], and then it attracted many researchers; see, for example, [3, 5] and references therein. Recently, Bhaskar and Lakshmikantham [12] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [17] presented nice generalizations of these results. Alsulami et al. [3] further extended the work of Luong and Thuan to coupled coincidences in partial metric spaces. For more related work on coupled fixed points and coincidences, we refer the readers to recent results in [13–30].
In recent years, several authors have obtained coupled fixed point results for various classes of mappings on the setting of many generalized metric spaces. The concept of metric-type space appeared in some works, such as Czerwik [8], Khamsi [9] and Khamsi and Hussain [10]. Metric-type space is a symmetric space with some special properties. A metric-type space can also be regarded as a triplet , where is a symmetric space and is a real number such that
for any . In this paper, we adopt a different and constructive method to prove some coupled fixed and coincidence point results for contractive mappings defined on an ordered closed convex subset of a quasi-Banach space. Moreover, an application to integral equations is given to illustrate the usability of the obtained results.
2 Preliminaries
The aim of this section is to present some notions and results used in the paper.
Definition 2.1 Let X be a non-empty set and . is a symmetric space (also called E-space) if and only if it satisfies the following conditions:
-
(W1)
if and only if ;
-
(W2)
for any .
Symmetric spaces differ from more convenient metric spaces in the absence of triangle inequality. Nevertheless, many notions can be defined similar to those in metric spaces. For instance, in a symmetric space , the limit point of a sequence is defined by
Also, a sequence is said to be a Cauchy sequence if, for every given , there exists a positive integer such that for all . A symmetric space is said to be complete if and only if each of its Cauchy sequences converges to some .
Definition 2.2 Let X be a nonempty set and let be a given real number. A function is said to be of metric type if and only if for all the following conditions are satisfied:
-
(1)
if and only if ;
-
(2)
;
-
(3)
.
A triplet is called a metric-type space.
We observe that a metric-type space is included in the class of symmetric spaces. So the notions of convergent sequence, Cauchy sequence and complete space are defined as in symmetric spaces.
Next, we give some examples of metric-type spaces.
Example 2.3 [4]
The space with ()
together with the function , defined by
where , is a metric-type space. By an elementary calculation, we obtain .
Example 2.4 [4]
The space () of all real functions such that
is a metric-type space if we take
for each . The constant K is again equal to .
For more examples of metric-type (or b-metric) spaces, we refer to [8, 10, 11]. We recall that a quasi-norm defined on a real vector space X is a mapping such that:
-
(1)
if and only if ;
-
(2)
for and ;
-
(3)
for all , where is a constant independent of x, y.
A triplet is called a quasi-Banach space.
What makes quasi-Banach spaces different from the more classical Banach spaces is the triangle inequality. In quasi-Banach spaces, the triangle inequality is allowed to hold approximately: for some constant . This relaxation leads to a broad class of spaces. Lebesgue spaces are Banach spaces for and only quasi-Banach spaces for .
Remark 2.5 Let be a quasi-Banach space, then the mapping defined by for all is a metric-type (b-metric). In general, a metric-type (b-metric) function d is not continuous (see [23, 26]).
The following result is useful for some of the proofs in the paper.
Lemma 2.6 Let be a metric-type space and let . Then
From Lemma 2.6, we deduce the following lemma.
Lemma 2.7 Let be a sequence in a metric-type space such that
for some λ, , and each . Then and are two Cauchy sequences in X.
Definition 2.8 (Mixed monotone property)
Let be a partially ordered set and . We say that the mapping F has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument. That is, for any ,
and
Definition 2.9 [13]
Let . We say that is a coupled fixed point of F if and .
Lakshmikantham and Ćirić [13] introduced the following concept of a mixed g-monotone mapping.
Definition 2.10 [13]
Let be a partially ordered set, and . We say that F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any ,
and
Note that if g is the identity mapping, then this definition reduces to Definition 2.8.
Definition 2.11 [13]
An element is called a coupled coincidence point of the mappings and if
3 Main results
Let be an ordered subset of a quasi-Banach space . Throughout this paper, we assume that the partial order ⪯ have the following properties:
-
(A)
If and , then ;
-
(B)
If and , then .
The following theorem is our first main result.
Theorem 3.1 Let be an ordered closed and convex subset of a quasi-Banach space where and is such that . Assume that is a mapping with the mixed monotone property on C and suppose that there exist non-negative real numbers α, β and γ with such that
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then there exist such that and , that is, F has a coupled fixed point in C.
Proof Let be such that and . Then
Define such that and . Similarly, and . We construct two sequences and such that
and
Let us prove that
Since
then (3.4) hold for . Suppose that (3.4) hold for . Since F has the mixed monotone property, so we have
and
Then, by mathematical induction, it follows that (3.4) hold for all .
By (3.2) and (3.3) we have
and
Thus
and
Therefore
and
Also,
Then
Similarly,
On the other hand, by (3.1) and (3.4), we have
and
Hence, by (3.5), (3.6), (3.7) and (3.8), we have
and
Thus
By Lemma 2.7 we conclude that and are two Cauchy sequences. Then there exist such that and .
At first, we assume that F is continuous. Hence
Similarly,
That is, F has a coupled fixed point.
Now we assume that (b) holds. Since and as , then (b) implies that and for all . Now, by (3.1) with , , , , we have
which implies
Taking the limit as in the above inequality, we have
Also, by (3.3) we get . Now we can write
and hence . That is, . Similarly, as required. □
If in Theorem 3.1 we take , we obtain following result.
Corollary 3.2 Let be an ordered closed and convex subset of a quasi-Banach space , where , and let be such that . Assume that is a mapping such that F has the mixed monotone property on X and there exists a non-negative real number γ with such that
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then there exist such that and , that is, F has a coupled fixed point.
The following theorem is our second main result.
Theorem 3.3 Let be an ordered closed and convex subset of a quasi-Banach space , where , and let be such that . Assume that is a mapping such that F has the mixed monotone property on X and there exists a non-negative real number α with such that
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F has a coupled fixed point.
Proof Let be such that and . Then
and
Define such that and . Similarly, and . We construct two sequences and such that
and
Let us prove that
As
so (3.12) hold for . Suppose that (3.12) hold for . As F has the mixed monotone property, so
and
Then, by mathematical induction, it follows that (3.12) holds for all .
By (3.10) and (3.11), we have
and
Thus
and
Therefore
and
Also, we have
which implies
Similarly, we have
Now, by (3.9), (3.13) and (3.15), (3.16), we have
Similarly,
Thus
By Lemma 2.7, we conclude that and are Cauchy sequences. Thus, there exist such that and .
Now, proceeding as in the proof of Theorem 3.1, we can prove that is a coupled fixed point of F. □
Since
so by Theorem 3.3 we obtain the following result.
Corollary 3.4 Let be an ordered closed and convex subset of a quasi-Banach space , where , and let be such that . Assume that , F has the mixed monotone property on C and for a non-negative real number α with , F satisfies following inequality:
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F has a coupled fixed point.
By taking in the above proved results, we can obtain the following couple fixed results in Banach spaces.
Corollary 3.5 Let be an ordered closed and convex subset of a Banach space , and let be a mapping such that F has the mixed monotone property on C. Suppose that there exist non-negative real numbers α, β and γ with such that
for all with and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F has a coupled fixed point.
Corollary 3.6 Let be an ordered closed and convex subset of a Banach space , and let be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number γ with such that
for all with and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F has a coupled fixed point.
Corollary 3.7 Let be an ordered closed and convex subset of a Banach space , and let be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number α with such that
for all with and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
(3.17)(3.18)
If there exist such that and , then F has a coupled fixed point.
Corollary 3.8 Let be an ordered closed and convex subset of a Banach space , and let be a mapping such that F has the mixed monotone property on C. Suppose that there exists a non-negative real number α with such that
for all with and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F has a coupled fixed point.
The following lemma is an easy consequence of the axiom of choice (see p.5 [25], AC5: For every function , there is a function g such that and for every , ).
Lemma 3.9 Let X be a nonempty set and be a mapping. Then there exists a subset such that and is one-to-one.
As an application of Theorem 3.1, we now establish a coupled coincidence point result.
Theorem 3.10 Let be a nonempty ordered subset of a quasi-Banach space , where , and let be such that . Assume that and are mappings where F has the mixed g-monotone property on C, is closed and convex and . Suppose that there exist non-negative real numbers α, β and a real number γ with such that
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then there exist such that and , that is, F and g have a coupled coincidence point in C.
Proof Using Lemma 3.9, there exists such that and is one-to-one. We define a mapping by
for all . As g is one-to-one on and , so G is well defined. Thus, it follows from (3.19) and (3.20) that
for all , for which and . Since F has the mixed g-monotone property, for all ,
and
which imply that G has the mixed monotone property. Also, there exist such that
This implies that there exist such that
Suppose that assumption (a) holds. Since F is continuous, G is also continuous. Using Theorem 3.1 to the mapping G, it follows that G has a coupled fixed point .
Suppose that assumption (b) holds. We conclude similarly that the mapping G has a coupled fixed point . Finally, we prove that F and g have a coupled coincidence point. Since is a coupled fixed point of G, we get
Since , there exists a point such that
It follows from (3.24) and (3.25) that
Combining (3.20) and (3.26), we get
Thus, is a required coupled coincidence point of F and g. This completes the proof. □
Similarly, as an application of Theorem 3.3, we can prove the following coupled coincidence point result.
Theorem 3.11 Let be a nonempty ordered subset of a quasi-Banach space , where , and let be such that . Assume that and are mappings where F has the mixed g-monotone property on C, is closed and convex and . Suppose that there exists a real number α with such that
for all , for which and . Also suppose that either
-
(a)
F is continuous, or
-
(b)
C has the following property:
If there exist such that and , then F and g have a coupled coincidence point in C.
4 Existence of a solution for a system of integral equations
We consider the space of continuous functions defined on endowed with the structure given by
for all . We endow with the partial order ⪯ given by
Clearly, the partial order ⪯ satisfies conditions A and B. Further, it is known that is regular [24], that is,
Motivated by the work in [1, 17, 18, 26], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous section.
Consider the integral equations in the following system.
We will consider system (4.1) under the following assumptions:
-
(i)
are continuous;
-
(ii)
is continuous;
-
(iii)
is continuous;
-
(iv)
there exist with such that for all and all with , we have
where
-
(v)
there exist continuous functions such that
-
(vi)
assume that
Theorem 4.1 Under assumptions (i)-(vi), system (4.1) has a solution in , where is defined above.
Proof We consider the operator defined by
for all , .
Clearly, F has the mixed monotone property [26].
Let with . Since F has the mixed monotone property, we have
Notice that
Thus,
Further, by (v), we get
All of the conditions of Corollary 3.5 are satisfied, so we deduce the existence of such that and . □
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and third authors acknowledge with thanks DSR, KAU for financial support. The second author is thankful for the support of Rasht Branch, Islamic Azad University. The authors would like to express their thanks to the referees for their helpful comments and suggestions.
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Hussain, N., Salimi, P. & Al-Mezel, S. Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations. Fixed Point Theory Appl 2013, 261 (2013). https://doi.org/10.1186/1687-1812-2013-261
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DOI: https://doi.org/10.1186/1687-1812-2013-261