#### Abstract

We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.

#### 1. Introduction

The problem of existence of common fixed points to a pair of nonlinear mappings is now a classical theme. The applications to differential and integral equations made it more interesting. A considerable importance has been attached to common fixed point theorems in ordered sets [1, 2]. In a recent paper, Dhage [3] proved some common fixed point theorems for pairs of condensing mappings in an ordered Banach space. More recently, Hussain et al. [4] extended the results of Dhage to 1-set contractive mappings. In the present paper we pursue the investigations started in the aforementioned papers and prove some new common fixed point theorems under weaker assumptions. Also we present some common fixed point results using the weak topology of a Banach space. The use of the concepts of ws-compactness and ww-compactness increases the usefulness of our results in many practical situations especially when we work in nonreflexive Banach spaces. We will illustrate this fact by proving the existence of nonnegative integrable solutions for an implicit integral equation. For the remainder of this section we gather some notations and preliminary facts. Let be a real Banach space, ordered by a cone . A cone is a closed convex subset of with , and . As usual .

*Definition 1.1. *Let be an ordered Banach space with order . A mapping is said to be isotone increasing if for all , we have that implies .

*Definition 1.2. *Let be a nonempty subset of an ordered Banach space with order . Two mappings are said to be weakly isotone increasing if and hold for all . Similarly, we say that and are weakly isotone decreasing if and hold for all . The mappings and are said to be weakly isotone if they are either weakly isotone increasing or weakly isotone decreasing.

In our considerations the following definition will play an important role. Let denote the collection of all nonempty bounded subsets of and the subset of consisting of all weakly compact subsets of . Also, let denote the closed ball centered at with radius .

*Definition 1.3 (see [5]). *A function is said to be a measure of weak noncompactness if it satisfies the following conditions.(1)The family is nonempty and is contained in the set of relatively weakly compact sets of .(2). (3), where is the closed convex hull of .(4) for .(5)If is a sequence of nonempty weakly closed subsets of with bounded and such that then is nonempty.

The family described in (1) is said to be the kernel of the measure of weak noncompactness . Note that the intersection set from (5) belongs to since for every and . Also, it can be easily verified that the measure satisfies where is the weak closure of .

A measure of weak noncompactness is said to be * regular* if
*subadditive* if
*homogeneous* if
and *set additive* (or *have the maximum property*) if

The first important example of a measure of weak noncompactness has been defined by De Blasi [6] as follows: for each .

Notice that is regular, homogeneous, subadditive, and set additive (see [6]).

By a measure of noncompactness on a Banach space we mean a map which satisfies conditions (1)–(5) in Definition 1.3 relative to the strong topology instead of the weak topology. The concept of a measure of noncompactness was initiated by the fundamental papers of Kuratowski [7] and Darbo [8]. Measures of noncompactness are very useful tools in nonlinear analysis [9] especially the so-called Kuratowski measure of noncompactness [7] and Hausdorff (or ball) measure of noncompactness [10].

*Definition 1.4. *Let be a Banach space and a measure of (weak) noncompactness on . Let be a mapping. If is bounded and for every nonempty bounded subset of with , we have, , then is called -condensing. If there exists , , such that is bounded and for each nonempty bounded subset of , we have , then is called *k*--contractive.

*Remark 1.5. *Clearly, every --contractive map with is -condensing and every -condensing map is 1--contractive.

*Definition 1.6 (see [11]). *A map is said to be ws-compact if it is continuous and for any weakly convergent sequence in the sequence has a strongly convergent subsequence in .

*Remark 1.7. *The concept of ws-compact mappings arises naturally in the study of both integral and partial differential equations (see [11–19]).

*Definition 1.8. *A map is said to be ww-compact if it is continuous and for any weakly convergent sequence in the sequence has a weakly convergent subsequence in .

*Definition 1.9. *Let be a Banach space. A mapping is called a nonlinear contraction if there exists a continuous and nondecreasing function such that
for all , where for .

*Remark 1.10. *It is easy to prove that a nonlinear contraction mapping is a nonlinear set-contraction with respect to the Kuratowskii measure of noncompactness. We will prove that the same property holds for the De Blasi measure of weak noncompactness provided that the nonlinear contraction mapping is ww-compact.

Lemma 1.11. *Let be a ww-compact nonlinear contraction mapping on a Banach X. Then for each bounded subset of one has
**
Here, is the De Blasi measure of weak noncompactness.*

*Proof. * Let be a bounded subset of and . There exist and a weakly compact subset of such that . Since is a -nonlinear contraction, then
Moreover, since is ww-compact, then is weakly compact. Accordingly,
Letting and using the continuity of we deduce that

The following theorem is a sharpening of [3, Theorem 2.1] and [4, Theorem 3.1].

Theorem 1.12. *Let be an ordered Banach space and a set additive measure of noncompactness on . Let be a nonempty closed convex subset of and be two continuous mappings satisfying the following: *(i)* is --contractive, *(ii)* is -condensing, *(iii)* and are weakly isotone. **Then and have a common fixed point.*

*Proof. *Let be fixed. Consider the sequence defined by
Suppose first that and are weakly isotone increasing on . Then from (1.12) it follows that
Set , and . Clearly
We show that is relatively compact. Suppose the contrary, then . Since is --contractive and is -condensing, it follows from (1.14) that
This is impossible. Thus is relatively compact. The continuity of implies that is relatively compact. Consequently is relatively compact. Since is monotone increasing in , then it is convergent. Let be its limit. Using the continuity of and we get
To complete the proof we consider the case where and are weakly isotone decreasing on . In this case, the sequence is monotone decreasing and then converges to a common fixed point of and .

*Remark 1.13. *In [3, Theorem 2.1] is assumed to be set condensing, while is assumed to be affine, is assumed to be reflexive and is demiclosed in [4, Theorem 3.1].

*Remark 1.14. *As an application of Theorem 1.12, the modified versions of Theorems 3.6–3.12 in [4] can be proved similarly.

As easy consequences of Theorem 1.12 we obtain the following results.

Corollary 1.15. *Let be a nonempty closed convex subset of a Banach space and let be two completely continuous mappings. Also assume and are weakly isotone mappings. Then and have a common fixed point.*

Corollary 1.16. *Let be a nonempty closed convex subset of a Banach space and be two continuous mappings. Assume is nonexpansive, that is, for all , and is a nonlinear contraction. Also assume that and are weakly isotone mappings, and . Then and have a common fixed point in .*

Corollary 1.17. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of noncompactness on . Let be two continuous mappings satisfying the following: *(i)* and ,*(ii)* is a nonexpansive mapping, *(iii)* is -condensing, *(iv)* and are weakly isotone.**Then and have a common fixed point in .*

Note that in Corollary 1.17 we do not have uniqueness. However, if we assume is shrinking, we obtain uniqueness. Recall that is shrinking if

Corollary 1.18. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of noncompactness on . Let be two continuous mappings satisfying the following: *(i)* and ,*(ii)* is a shrinking mapping, *(iii)* is -condensing, *(iv)* and are weakly isotone.**Then and have a unique common fixed point in .*

*Proof. *From Corollary 1.17 it follows that there exists such that . Now from (1.17) we infer that cannot have two different fixed points. This implies that and have a unique common fixed point.

Corollary 1.19. *Let be a nonempty closed convex subset of a Banach space and be two mappings satisfying the following: *(i)* and ,*(ii)* is a shrinking mapping, *(iii)* is a nonlinear contraction,*(iv)* and are weakly isotone.**Then and have a unique common fixed point in which is the unique fixed point of .*

Theorem 1.20. *Let be an ordered Banach space and a set additive measure of weak noncompactness on . Let be a nonempty closed convex subset of and be two sequentially weakly continuous mappings satisfying the following: *(i)* is --contractive, *(ii)* is -condensing,*(iii)* and are weakly isotone. **Then and have a common fixed point in .*

*Proof. * Let be fixed. Consider the sequence defined by
Suppose first that and are weakly isotone increasing on . Then from (1.18) it follows that
As in the proof of Theorem 1.12 we set , , and . Clearly
We show that is relatively weakly compact. Suppose the contrary, then . Since is --contractive and is -condensing, it follows from (1.20) that
This is impossible. Thus is relatively weakly compact. The weak sequential continuity of implies that is relatively weakly compact. The same reasoning as in the proof of Theorem 1.12 gives the desired result.

As easy consequences of Theorem 1.20 we obtain the following results.

Corollary 1.21. *Let be a nonempty closed convex subset of a Banach space and let be two weakly completely continuous mappings. Also assume and are weakly isotone mappings. Then and have a common fixed point in .*

Corollary 1.22. *Let be a nonempty closed convex subset of a Banach space and let be two sequentially weakly continuous mappings. Assume is nonexpansive and is a nonlinear contraction. Also assume that and are weakly isotone mappings, and . Then and have a common fixed point in .*

Corollary 1.23. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of weak noncompactness on . Let be two sequentially weakly continuous mappings satisfying the following: *(i)* is a nonexpansive mapping, *(ii)* is -condensing,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a common fixed point.*

*Proof. *This follows from Theorem 1.20 on the basis of Lemma 1.11.

Corollary 1.24. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of weak noncompactness on . Let be two sequentially weakly continuous mappings satisfying the following: *(i)* is a shrinking mapping,*(ii)* is -condensing,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a unique common fixed point in .*

Corollary 1.25. *Let be a nonempty closed convex subset of a Banach space and let be two sequentially weakly continuous mappings satisfying the following: *(i)* is a shrinking mapping,*(ii)* is a nonlinear contraction,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a unique common fixed point in which is the unique fixed point of .*

*Remark 1.26. *It is worth noting that, in some applications, the weak sequential continuity is not easy to be verified. The ws-compactness seems to be a good alternative (see [16] and the references therein). In the following we provide a version of Theorem 1.20 where the weak sequential continuity is replaced with ws-compactness.

Theorem 1.27. *Let be an ordered Banach space and a set additive measure of weak noncompactness on . Let be a nonempty closed convex subset of and two continuous mappings satisfying the following: *(i)* is --contractive,*(ii)* is ws-compact and -condensing,*(iii)* and are weakly isotone. **Then and have a common fixed point.*

*Proof. * Let be fixed. Consider the sequence defined by
Suppose first that and are weakly isotone increasing on . Then from (1.22) it follows that
Set ,, and . Clearly
Similar reasoning as in Theorem 1.20 gives that is relatively weakly compact. From the ws-compactness of it follows that is relatively compact. Now the continuity of yields that is relatively compact. Consequently, is relatively compact. The rest of the proof runs as in the proof of Theorem 1.12.

As easy consequences of Theorem 1.27 we obtain the following results.

Corollary 1.28. *Let be two continuous mappings. Assume is nonexpansive and ww-compact and is a ws-compact nonlinear contraction. Also assume that and are weakly isotone mappings. Then and have a common fixed point.*

*Proof. *This follows from Theorem 1.27 on the basis of Lemma 1.11.

Corollary 1.29. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of weak noncompactness on . Let be two continuous mappings satisfying the following the following: *(i)* is a ww-compact nonexpansive mapping,*(ii)* is ws-compact and -condensing,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a common fixed point in .*

*Proof. * This follows From Theorem 1.27 on the basis of Lemma 1.11.

Corollary 1.30. *Let be a nonempty closed convex subset of a Banach space and a set additive measure of weak noncompactness on . Let be two continuous mappings satisfying the following: *(i)* is a ww-compact shrinking mapping,*(ii)* is ws-compact and -condensing,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a unique common fixed point in .*

Corollary 1.31. *Let be a nonempty closed convex subset of a Banach space and let two continuous mappings satisfying the following: *(i)* is a ww-compact and shrinking mapping,*(ii)* is a ws-compact nonlinear contraction,*(iii)* and are weakly isotone,*(iv)* and .**Then and have a unique common fixed point in which is the unique fixed point of .*

Note that if is a Banach Lattice, then the ws-compactness in Theorem 1.27 can be removed, as it is shown in the following result.

Theorem 1.32. *Let be a Banach lattice and a set additive measure of weak noncompactness on . Let be a nonempty closed convex subset of and two continuous mappings satisfying the following: *(i)* is --contractive,*(ii)* is -condensing, *(iii)* and are weakly isotone. **Then and have a common fixed point in .*

*Proof. *Let be fixed. Consider the sequence defined by
Suppose first that and are weakly isotone increasing on . Then from (1.25) it follows that
Set ,, and . Clearly
Similar reasoning as in Theorem 1.20 gives that is relatively weakly compact. Thus has a weakly convergent subsequence, say . Since in a Banach lattice every weakly convergent increasing sequence is norm-convergent we infer that is norm-convergent. Consequently is relatively compact. Similar reasoning as in the proof of Theorem 1.12 gives the desired result.

#### 2. Application to Implicit Integral Equations

The purpose of this section is to study the existence of integrable nonnegative solutions of the integral equation given by

Integral equations like (2.1) were studied in [20] in and in [4] in with . In this section, we look for a nonnegative solution to (2.1) in . For the remainder we gather some definitions and results from the literature which will be used in the sequel. Recall that a function is said to be a Carathéodory function if (i)for any fixed , the function is measurable from to ,(ii)for almost any , the function is continuous.

Let be the set of all measurable functions . If is a Carathéodory function, then defines a mapping by . This mapping is called the superposition (or Nemytskii) operator generated by . The next two lemmas are of foremost importance for our subsequent analysis.

Lemma 2.1 (see [21, 22]). *Let be a Carathéodory function. Then the superposition operator maps continuously into itself if and only if there exist a constant and a function such that
**
where denotes the positive cone of the space .*

Lemma 2.2 (see [14]). *If is a Carathéodory function and maps continuously into itself, then is ww-compact.*

*Remark 2.3. * Although the Nemytskii operator is ww-compact, generally it is not weakly continuous. In fact, only linear functions generate weakly continuous Nemytskii operators in spaces (see, for instance, [23, Theorem 2.6]).

The problem of existence of nonnegative integrable solutions to (2.1) will be discussed under the following assumptions: (a) is strongly measurable and whenever and there exists a function belonging to such that for all . The function is a Carathéodory function and there exist a constant and a function such that for all and . Moreover, whenever ;(b)the function is nonexpansive with respect to the second variable, that is, for all and ;(c)for all and for all we have where ,(d).

*Remark 2.4. * If , then from assumption (b) it follows that whenever .

*Remark 2.5. * From assumption (b) it follows that
Using Lemma 2.2 together with Lemma 2.1 we infer that is ww-compact.

Theorem 2.6. *Assume that the conditions (a–d) are satisfied. Then the implicit integral equation (2.1) has at least one nonnegative solution in .*

* Proof. *The problem (2.1) may be written in the shape:
where and are the nonlinear operators given by

Note that and may be written in the shape:

Here and are the superposition operators associated to the functions and , respectively, and denotes the linear integral defined by

Let be the real defined by
and be the closed convex subset of defined by

Note that, for any , the functions and belong to which is a consequence of the assumptions (a, b). Moreover, from assumption (c) it follows that for each we have
This implies that . Also, in view of assumption (b) we have for all and for all
Hence
Thus
Therefore, .

Our strategy consists in applying Corollary 1.29 to find a nonnegative common fixed point for and in . For the sake of simplicity, the proof will be displayed into four steps.

*Step 1. * and are continuous. Indeed, the assumption and Lemma 2.1 guarantee that and map continuously into itself. To complete the proof it remains only to show that is continuous. This follows immediately from the hypothesis .

*Step 2. * is ws-compact. Indeed, let be a weakly convergent sequence of . Using Lemma 2.2, the sequence has a weakly convergent subsequence, say . Let be the weak limit of . Accordingly, keeping in mind the boundedness of the mapping we get

The use of the dominated convergence theorem allows us to conclude that the sequence converges in .

*Step 3. * maps bounded sets of into weakly compact sets. To see this, let be a bounded subset of and let such that for all . For we have

Consequently,
for all measurable subsets of . Taking into account the fact that any set consisting of one element is weakly compact and using Corollary 11 in [24, page 294] we get , where is the Lebesgue measure of . Applying Corollary 11 in [24, page 294] once again we infer that the set is weakly compact.

*Step 4. * and are weakly isotone decreasing. Indeed, using assumption we get
and from our assumptions we have
Hence and and therefore and are weakly isotone decreasing.

Note that the * Steps *1–4 show that the hypotheses of Corollary 1.29 are satisfied. Thus and have a common fixed point and therefore the problem (2.1) has at least one nonnegative solution in .

#### Acknowledgment

The first and second authors gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.