Fixed Point Theorems in Ordered Banach Spaces and Applications to Nonlinear Integral Equations

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 𝑋 be a real Banach space, ordered by a cone 𝐾. A cone 𝐾 is a closed convex subset of 𝑋 with 𝜆𝐾⊆𝐾(𝜆≥0), and 𝐾∩(−𝐾)={0}. As usual 𝑥≤𝑦⇔𝑦−𝑥∈𝐾.

Definition 1.1. Let 𝑋 be an ordered Banach space with order ≤. A mapping 𝑇∶𝐷(𝑇)⊆𝑋→𝑋 is said to be isotone increasing if for all 𝑥,𝑦∈𝐷(𝑇), we have that 𝑥≤𝑦 implies 𝑇𝑥≤𝑇𝑦.

Definition 1.2. Let 𝑀 be a nonempty subset of an ordered Banach space 𝑋 with order ≤. Two mappings 𝑆,𝑇∶𝑀→𝑀 are said to be weakly isotone increasing if 𝑆𝑥≤𝑇𝑆𝑥 and 𝑇𝑥≤𝑆𝑇𝑥 hold for all 𝑥∈𝑀. Similarly, we say that 𝑆 and 𝑇 are weakly isotone decreasing if 𝑇𝑥≥𝑆𝑇𝑥 and 𝑆𝑥≥𝑇𝑆𝑥 hold for all 𝑥∈𝑀. The mappings 𝑆 and 𝑇 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 ℬ(𝑋) denote the collection of all nonempty bounded subsets of 𝑋 and 𝒲(𝑋) the subset of ℬ(𝑋) consisting of all weakly compact subsets of 𝑋. Also, let 𝐵_𝑟 denote the closed ball centered at 0 with radius 𝑟.

Definition 1.3 (see  [5]). A function 𝜓∶ℬ(𝑋)→ℝ₊ is said to be a measure of weak noncompactness if it satisfies the following conditions.(1)The family ker(𝜓)={𝑀∈ℬ(𝑋)∶𝜓(𝑀)=0} is nonempty and ker(𝜓) is contained in the set of relatively weakly compact sets of 𝑋.(2)𝑀₁⊆𝑀₂⇒𝜓(𝑀₁)≤𝜓(𝑀₂). (3)𝜓(co(𝑀))=𝜓(𝑀), where co(𝑀) is the closed convex hull of 𝑀.(4)𝜓(𝜆𝑀₁+(1−𝜆)𝑀₂)≤𝜆𝜓(𝑀₁)+(1−𝜆)𝜓(𝑀₂) for 𝜆∈[0,1].(5)If (𝑀_𝑛)_𝑛≥1 is a sequence of nonempty weakly closed subsets of 𝑋 with 𝑀₁bounded and 𝑀₁⊇𝑀₂⊇⋯⊇𝑀_𝑛⊇⋯ such that lim_𝑛→∞𝜓(𝑀_𝑛)=0,then 𝑀_∞∶=⋂^∞_𝑛=1𝑀_𝑛 is nonempty.

The family ker𝜓 described in (1) is said to be the kernel of the measure of weak noncompactness 𝜓. Note that the intersection set 𝑀_∞ from (5) belongs to ker𝜓 since 𝜓(𝑀_∞)≤𝜓(𝑀_𝑛) for every 𝑛 and lim_𝑛→∞𝜓(𝑀_𝑛)=0. Also, it can be easily verified that the measure 𝜓 satisfies 𝜓𝑀^𝑤=𝜓(𝑀), (1.1) where 𝑀^𝑤 is the weak closure of 𝑀.

A measure of weak noncompactness 𝜓 is said to be regular if 𝜓(𝑀)=0iff𝑀isrelativelyweaklycompact, (1.2)subadditive if 𝜓𝑀₁+𝑀₂≤𝜓𝑀₁+𝜓𝑀₂, (1.3)homogeneous if 𝜓(𝜆𝑀)=|𝜆|𝜓(𝑀),𝜆∈ℝ, (1.4) and set additive (or have the maximum property) if 𝜓𝑀₁∪𝑀₂=max𝜓𝑀₁,𝜓𝑀₂. (1.5)

The first important example of a measure of weak noncompactness has been defined by De Blasi [6] as follows: 𝑤(𝑀)=inf𝑟>0∶thereexists𝑊∈𝒲(𝑋)with𝑀⊆𝑊+𝐵_𝑟, (1.6) for each 𝑀∈ℬ(𝑋).

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

By a measure of noncompactness on a Banach space 𝑋 we mean a map 𝜓∶ℬ(𝑋)→ℝ₊ 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 𝑋 be a Banach space and 𝜓 a measure of (weak) noncompactness on 𝑋. Let 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 be a mapping. If 𝐴(𝐷(𝐴)) is bounded and for every nonempty bounded subset 𝑀 of 𝐷(𝐴) with 𝜓(𝑀)>0, we have, 𝜓(𝐴(𝑀)))<𝜓(𝑀), then 𝐴 is called 𝜓-condensing. If there exists 𝑘, 0≤𝑘≤1, such that 𝐴(𝐷(𝐴)) is bounded and for each nonempty bounded subset 𝑀 of 𝐷(𝐴), we have 𝜓(𝐴(𝑀))≤𝑘𝜓(𝑀), then 𝐴 is called k-𝜓-contractive.

Remark 1.5. Clearly, every 𝑘-𝜓-contractive map with 𝑘<1 is 𝜓-condensing and every 𝜓-condensing map is 1-𝜓-contractive.

Definition 1.6 (see [11]). A map 𝐴∶𝐷(𝐴)→𝑋 is said to be ws-compact if it is continuous and for any weakly convergent sequence (𝑥_𝑛)_𝑛∈ℕ in 𝐷(𝐴) the sequence (𝐴𝑥_𝑛)_𝑛∈ℕ has a strongly convergent subsequence in 𝑋.

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 𝐴∶𝐷(𝐴)→𝑋 is said to be ww-compact if it is continuous and for any weakly convergent sequence (𝑥_𝑛)_𝑛∈ℕ in 𝐷(𝐴) the sequence (𝐴𝑥_𝑛)_𝑛∈ℕ has a weakly convergent subsequence in 𝑋.

Definition 1.9. Let 𝑋 be a Banach space. A mapping 𝑇∶𝐷(𝑇)⊆𝑋→𝑋 is called a nonlinear contraction if there exists a continuous and nondecreasing function 𝜑∶ℝ⁺→ℝ⁺ such that 𝑇𝑥−𝑇𝑦≤𝜑𝑥−𝑦 (1.7) for all 𝑥,𝑦∈𝐷(𝑇), where 𝜑(𝑟)<𝑟 for 𝑟>0.

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 𝑇 be a ww-compact nonlinear contraction mapping on a Banach X. Then for each bounded subset 𝑀 of 𝑋 one has 𝑤(𝑇𝑀)≤𝜙(𝑤(𝑀)). (1.8) Here, 𝑤 is the De Blasi measure of weak noncompactness.

Proof. Let 𝑀 be a bounded subset of 𝑋 and 𝑟>𝑤(𝑀). There exist 0≤𝑟₀<𝑟 and a weakly compact subset 𝑊 of 𝑋 such that 𝑀⊆𝑊+𝐵_𝑟0. Since 𝑇 is a 𝜙-nonlinear contraction, then 𝑇𝑀⊆𝑇𝑊+𝐵_𝜙(𝑟0)⊆𝑇𝑊^𝑤+𝐵_𝜙(𝑟0). (1.9) Moreover, since 𝑇 is ww-compact, then 𝑇𝑊^𝑤 is weakly compact. Accordingly, 𝑤(𝑇𝑀)≤𝜙𝑟₀≤𝜙(𝑟). (1.10) Letting 𝑟→𝑤(𝑀) and using the continuity of 𝜙 we deduce that 𝑤(𝑇𝑀)≤𝜙(𝑤(𝑀)). (1.11)

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

Theorem 1.12. Let 𝑋 be an ordered Banach space and 𝜓 a set additive measure of noncompactness on 𝑋. Let 𝑀 be a nonempty closed convex subset of 𝑋 and 𝑆,𝑇∶𝑀→𝑀 be two continuous mappings satisfying the following: (i)𝑇 is 1-𝜓-contractive, (ii)𝑆 is 𝜓-condensing, (iii)𝑆 and 𝑇 are weakly isotone. Then 𝑇 and 𝑆 have a common fixed point.

Proof. Let 𝑥∈𝑀 be fixed. Consider the sequence {𝑥_𝑛} defined by 𝑥₀=𝑥,𝑥_2𝑛+1=𝑆𝑥_2𝑛,𝑥_2𝑛+2=𝑇𝑥_2𝑛,𝑛=0,1,2,…. (1.12) Suppose first that 𝑆 and 𝑇 are weakly isotone increasing on 𝑀. Then from (1.12) it follows that 𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤⋯. (1.13) Set 𝐴={𝑥₁,𝑥₂,…,𝑥_𝑛,…},𝐴₁={𝑥₂,𝑥₄,…,𝑥_2𝑛,…}, and 𝐴₂={𝑥₁,𝑥₃,…,𝑥_2𝑛+1,…}. Clearly 𝐴=𝐴₁∪𝐴₂,𝐴₂=𝑆𝐴₁∪𝑥₁,𝐴₁=𝑇𝐴₂. (1.14) We show that 𝐴₁ is relatively compact. Suppose the contrary, then 𝜓(𝐴₁)>0. Since 𝑇 is 1-𝜓-contractive and 𝑆 is 𝜓-condensing, it follows from (1.14) that 𝜓𝐴₂=𝜓𝑆𝐴₁∪𝑥₁=𝜓𝑆𝐴₁<𝜓𝐴₁,𝜓𝐴₁=𝜓𝑇𝐴₂≤𝜓𝐴₂. (1.15) This is impossible. Thus 𝐴₁ is relatively compact. The continuity of 𝑆 implies that 𝐴₂=𝑆(𝐴₁)∪{𝑥₁} is relatively compact. Consequently 𝐴 is relatively compact. Since {𝑥_𝑛} is monotone increasing in 𝐴, then it is convergent. Let 𝑥^∗ be its limit. Using the continuity of 𝑆 and 𝑇 we get 𝑥^∗=lim_𝑛→∞𝑥_2𝑛+1=lim_𝑛→∞𝑆𝑥_2𝑛=𝑆lim_𝑛→∞𝑥_2𝑛=𝑆𝑥^∗,𝑥^∗=lim_𝑛→∞𝑥_2𝑛+2=lim_𝑛→∞𝑇𝑥_2𝑛+1=𝑇lim_𝑛→∞𝑥_2𝑛+1=𝑇𝑥^∗. (1.16) To complete the proof we consider the case where 𝑆 and 𝑇 are weakly isotone decreasing on 𝑀. In this case, the sequence {𝑥_𝑛} is monotone decreasing and then converges to a common fixed point of 𝑆 and 𝑇.

Remark 1.13. In [3, Theorem 2.1] 𝑇 is assumed to be set condensing, while 𝑆 is assumed to be affine, 𝑋 is assumed to be reflexive and 𝐼−𝑇 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 𝑀 be a nonempty closed convex subset of a Banach space 𝑋 and let 𝑆,𝑇∶𝑀→𝑀 be two completely continuous mappings. Also assume 𝑇 and 𝑆 are weakly isotone mappings. Then 𝑇 and 𝑆 have a common fixed point.

Corollary 1.16. Let 𝑀 be a nonempty closed convex subset of a Banach space 𝑋 and 𝑆,𝑇∶𝑋→𝑋 be two continuous mappings. Assume 𝑇 is nonexpansive, that is, ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝑋, and 𝑆 is a nonlinear contraction. Also assume that 𝑆 and 𝑇 are weakly isotone mappings, 𝑇(𝑀)⊆𝑀 and 𝑆(𝑀)⊆𝑀. Then 𝑇 and 𝑆 have a common fixed point in 𝑀.

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

Note that in Corollary 1.17 we do not have uniqueness. However, if we assume 𝑇 is shrinking, we obtain uniqueness. Recall that 𝑇∶𝐷(𝑇)⊆𝑋→𝑋 is shrinking if 𝑇𝑥−𝑇𝑦<𝑥−𝑦whenever𝑥,𝑦∈𝐷(𝑇)with𝑥≠𝑦. (1.17)

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

Proof. From Corollary 1.17 it follows that there exists 𝑥∈𝑀 such that 𝑆𝑥=𝑇𝑥=𝑥. Now from (1.17) we infer that 𝑇 cannot have two different fixed points. This implies that 𝑆 and 𝑇 have a unique common fixed point.

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

Theorem 1.20. Let 𝑋 be an ordered Banach space and 𝜓 a set additive measure of weak noncompactness on 𝑋. Let 𝑀 be a nonempty closed convex subset of 𝑋 and 𝑆,𝑇∶𝑀→𝑀 be two sequentially weakly continuous mappings satisfying the following: (i)𝑇 is 1-𝜓-contractive, (ii)𝑆 is 𝜓-condensing,(iii)𝑆 and 𝑇 are weakly isotone. Then 𝑇 and 𝑆 have a common fixed point in 𝑀.

Proof. Let 𝑥∈𝑀 be fixed. Consider the sequence {𝑥_𝑛} defined by 𝑥₀=𝑥,𝑥_2𝑛+1=𝑆𝑥_2𝑛,𝑥_2𝑛+2=𝑇𝑥_2𝑛,𝑛=0,1,2,…. (1.18) Suppose first that 𝑆 and 𝑇 are weakly isotone increasing on 𝑀. Then from (1.18) it follows that 𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤⋯. (1.19) As in the proof of Theorem 1.12 we set 𝐴={𝑥₁,𝑥₂,…,𝑥_𝑛,…}, 𝐴₁={𝑥₂,𝑥₄,…,𝑥_2𝑛,…}, and 𝐴₂={𝑥₁,𝑥₃,…,𝑥_2𝑛+1,…}. Clearly 𝐴=𝐴₁∪𝐴₂,𝐴₂=𝑆𝐴₁∪𝑥₁,𝐴₁=𝑇𝐴₂. (1.20) We show that 𝐴₁ is relatively weakly compact. Suppose the contrary, then 𝜓(𝐴₁)>0. Since 𝑇 is 1-𝜓-contractive and 𝑆 is 𝜓-condensing, it follows from (1.20) that 𝜓𝐴₂=𝜓𝑆𝐴₁∪𝑥₁=𝜓𝑆𝐴₁<𝜓𝐴₁,𝜓𝐴₁=𝜓𝑇𝐴₂≤𝜓𝐴₂. (1.21) This is impossible. Thus 𝐴₁ is relatively weakly compact. The weak sequential continuity of 𝑆 implies that 𝐴₂=𝑆(𝐴₁)∪{𝑥₁} 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 𝑀 be a nonempty closed convex subset of a Banach space 𝑋 and let 𝑆,𝑇∶𝑀→𝑀 be two weakly completely continuous mappings. Also assume 𝑇 and 𝑆 are weakly isotone mappings. Then 𝑇 and 𝑆 have a common fixed point in 𝑀.

Corollary 1.22. Let 𝑀 be a nonempty closed convex subset of a Banach space 𝑋 and let 𝑆,𝑇∶𝑋→𝑋 be two sequentially weakly continuous mappings. Assume 𝑇 is nonexpansive and 𝑆 is a nonlinear contraction. Also assume that 𝑆 and 𝑇 are weakly isotone mappings, 𝑇(𝑀)⊆𝑀 and 𝑆(𝑀)⊆𝑀. Then 𝑇 and 𝑆 have a common fixed point in 𝑀.

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

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

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

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

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 𝑋 be an ordered Banach space and 𝜓 a set additive measure of weak noncompactness on 𝑋. Let 𝑀 be a nonempty closed convex subset of 𝑋 and 𝑆,𝑇∶𝑀→𝑀 two continuous mappings satisfying the following: (i)𝑇 is 1-𝜓-contractive,(ii)𝑆 is ws-compact and 𝜓-condensing,(iii)𝑆 and 𝑇 are weakly isotone. Then 𝑇 and 𝑆 have a common fixed point.

Proof. Let 𝑥∈𝑀 be fixed. Consider the sequence {𝑥_𝑛} defined by 𝑥₀=𝑥,𝑥_2𝑛+1=𝑆𝑥_2𝑛,𝑥_2𝑛+2=𝑇𝑥_2𝑛,𝑛=0,1,2,…. (1.22) Suppose first that 𝑆 and 𝑇 are weakly isotone increasing on 𝑀. Then from (1.22) it follows that 𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤⋯. (1.23) Set 𝐴={𝑥₁,𝑥₂,…,𝑥_𝑛,…},𝐴₁={𝑥₂,𝑥₄,…,𝑥_2𝑛,…}, and 𝐴₂={𝑥₁,𝑥₃,…,𝑥_2𝑛+1,…}. Clearly 𝐴=𝐴₁∪𝐴₂,𝐴₂=𝑆𝐴₁∪𝑥₁,𝐴₁=𝑇𝐴₂. (1.24) Similar reasoning as in Theorem 1.20 gives that 𝐴₁ is relatively weakly compact. From the ws-compactness of 𝑆 it follows that 𝐴₂=𝑆(𝐴₁)∪{𝑥₁} is relatively compact. Now the continuity of 𝑇 yields that 𝐴₁=𝑇(𝐴₂) is relatively compact. Consequently, 𝐴=𝐴₁∪𝐴₂ 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 𝑆,𝑇∶𝑋→𝑋 be two continuous mappings. Assume 𝑇 is nonexpansive and ww-compact and 𝑆 is a ws-compact nonlinear contraction. Also assume that 𝑆 and 𝑇 are weakly isotone mappings. Then 𝑇 and 𝑆 have a common fixed point.

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

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

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

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

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

Note that if 𝑋 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 𝑋 be a Banach lattice and 𝜓 a set additive measure of weak noncompactness on 𝑋. Let 𝑀 be a nonempty closed convex subset of 𝑋 and 𝑆,𝑇∶𝑀→𝑀 two continuous mappings satisfying the following: (i)𝑇 is 1-𝜓-contractive,(ii)𝑆 is 𝜓-condensing, (iii)𝑆 and 𝑇 are weakly isotone. Then 𝑇 and 𝑆 have a common fixed point in 𝑀.

Proof. Let 𝑥∈𝑀 be fixed. Consider the sequence {𝑥_𝑛} defined by 𝑥₀=𝑥,𝑥_2𝑛+1=𝑆𝑥_2𝑛,𝑥_2𝑛+2=𝑇𝑥_2𝑛,𝑛=0,1,2,…. (1.25) Suppose first that 𝑆 and 𝑇 are weakly isotone increasing on 𝑀. Then from (1.25) it follows that 𝑥₁≤𝑥₂≤⋯≤𝑥_𝑛≤⋯. (1.26) Set 𝐴={𝑥₁,𝑥₂,…,𝑥_𝑛,…},𝐴₁={𝑥₂,𝑥₄,…,𝑥_2𝑛,…}, and 𝐴₂={𝑥₁,𝑥₃,…,𝑥_2𝑛+1,…}. Clearly 𝐴=𝐴₁∪𝐴₂,𝐴₂=𝑆𝐴₁∪𝑥₁,𝐴₁=𝑇𝐴₂. (1.27) Similar reasoning as in Theorem 1.20 gives that 𝐴₁ is relatively weakly compact. Thus {𝑥_𝑛} has a weakly convergent subsequence, say {𝑥_𝑛𝑘}. Since in a Banach lattice every weakly convergent increasing sequence is norm-convergent we infer that {𝑥_𝑛𝑘} is norm-convergent. Consequently 𝐴₁ 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 𝑝(𝑡,𝑥(𝑡))=¹₀𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈[0,1]. (2.1)

Integral equations like (2.1) were studied in [20] in 𝐿²[0,1] and in [4] in 𝐿^𝑝[0,1] with 1<𝑝<+∞. In this section, we look for a nonnegative solution to (2.1) in 𝐿¹[0,1]. For the remainder we gather some definitions and results from the literature which will be used in the sequel. Recall that a function 𝑓∶[0,1]×ℝ→ℝ is said to be a Carathéodory function if (i)for any fixed 𝑢∈ℝ, the function 𝑡→𝑓(𝑡,𝑢) is measurable from [0,1] to ℝ,(ii)for almost any 𝑥∈[0,1], the function 𝑓(𝑡,⋅)∶ℝ→ℝ is continuous.

Let 𝐦[0,1] be the set of all measurable functions 𝑥∶[0,1]→ℝ. If 𝑓 is a Carathéodory function, then 𝑓 defines a mapping 𝐍_𝑓∶𝐦[0,1]→𝐦[0,1] by 𝐍_𝑓(𝑥)(𝑡)∶=𝑓(𝑡,𝑥(𝑡)). This mapping is called the superposition (or Nemytskii) operator generated by 𝑓. The next two lemmas are of foremost importance for our subsequent analysis.

Lemma 2.1 (see [21, 22]). Let 𝑓∶[0,1]×ℝ→ℝ be a Carathéodory function. Then the superposition operator 𝐍_𝑓 maps continuously 𝐿¹[0,1] into itself if and only if there exist a constant 𝑏≥0 and a function 𝑎(⋅)∈𝐿¹₊[0,1] such that 𝑓(𝑡,𝑢)≤𝑎(𝑡)+𝑏|𝑢|, (2.2) where 𝐿¹₊[0,1] denotes the positive cone of the space 𝐿¹[0,1].

Lemma 2.2 (see [14]). If 𝑓∶[0,1]×ℝ→ℝ is a Carathéodory function and 𝐍_𝑓 maps continuously 𝐿¹[0,1] into itself, then 𝐍_𝑓 is ww-compact.

Remark 2.3. Although the Nemytskii operator 𝐍_𝑓 is ww-compact, generally it is not weakly continuous. In fact, only linear functions generate weakly continuous Nemytskii operators in 𝐿¹ 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)𝜁∶[0,1]×[0,1]→ℝ is strongly measurable and ∫¹₀𝜁(⋅,𝑠)𝑦(𝑠)𝑑𝑠∈𝐿¹[0,1] whenever 𝑦∈𝐿¹[0,1] and there exists a function 𝜃∶[0,1]→ℝ belonging to 𝐿^∞[0,1] such that 0≤𝜁(𝑡,𝑠)≤𝜃(𝑡) for all (𝑡,𝑠)∈[0,1]×[0,1]. The function 𝑓∶[0,1]×ℝ→ℝ is a Carathéodory function and there exist a constant 𝑏>0 and a function 𝑎(⋅)∈𝐿¹₊[0,1] such that |𝑓(𝑡,𝑢)|≤𝑎(𝑡)+𝑏|𝑢| for all 𝑡∈[0,1] and 𝑢∈ℝ. Moreover, 𝑓(𝑡,𝑥(𝑡))≥0 whenever 𝑥∈𝐿¹₊[0,1];(b)the function 𝑝∶[0,1]×ℝ→ℝ is nonexpansive with respect to the second variable, that is, |𝑝(𝑡,𝑢)−𝑝(𝑡,𝑣)|≤|𝑢−𝑣| for all 𝑡∈[0,1] and 𝑢,𝑣∈ℝ;(c)for all 𝑟>0 and for all 𝑥∈𝐵⁺_𝑟 we have 0≤𝜃(𝑡)‖𝑎‖+𝑏𝜃(𝑡)¹₀𝑝(𝑠,𝑥(𝑠))𝑑𝑠≤𝑝(𝑡,𝑥(𝑡))≤𝑥(𝑡), (2.3) where 𝐵⁺_𝑟={𝑥∈𝐿¹[0,1]∶0≤𝑥(𝑡)≤𝑟forall𝑡∈[0,1]},(d)𝑏‖𝜃‖_∞<1.

Remark 2.4. If 𝑝(𝑡,0)=0, then from assumption (b) it follows that 𝑝(𝑡,𝑥(𝑡))≤𝑥(𝑡) whenever 𝑥∈𝐿¹₊[0,1].

Remark 2.5. From assumption (b) it follows that 𝑝(𝑡,𝑥(𝑡))≤𝑝(𝑡,0)+|𝑥(𝑡)|. (2.4) Using Lemma 2.2 together with Lemma 2.1 we infer that 𝒩_𝑝 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 𝐿¹[0,1].

Proof. The problem (2.1) may be written in the shape: 𝑇𝑥=𝑆𝑥, (2.5) where 𝑇 and 𝑆 are the nonlinear operators given by (𝑇𝑥)(𝑡)=𝑝(𝑡,𝑥(𝑡)),(𝑆𝑥)(𝑡)=¹₀𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠)). (2.6)
Note that 𝑇 and 𝑆 may be written in the shape: 𝑇=𝒩_𝑝,𝑆=𝐾𝒩_𝑓. (2.7)
Here 𝒩_𝑝 and 𝒩_𝑓 are the superposition operators associated to the functions 𝑝 and 𝑓, respectively, and 𝐾 denotes the linear integral defined by 𝐾∶𝐿¹[0,1]⟶𝐿¹[0,1]∶𝑢(𝑡)⟼𝐾𝑢(𝑡)∶=𝜇¹₀𝜁(𝑡,𝑠)𝑢(𝑠)𝑑𝑠. (2.8)
Let 𝑟₀ be the real defined by 𝑟₀=‖𝑎‖‖𝜃‖1−𝑏‖𝜃‖ (2.9) and 𝑀 be the closed convex subset of 𝐿¹[0,1] defined by 𝑀=𝐵⁺_𝑟0=𝑥∈𝐿¹[0,1]∶0≤𝑥(𝑡)≤𝑟₀∀𝑡∈[0,1]. (2.10)
Note that, for any 𝑥∈𝐿¹[0,1], the functions 𝑇𝑥 and 𝑆𝑥 belong to 𝐿¹[0,1] which is a consequence of the assumptions (a, b). Moreover, from assumption (c) it follows that for each 𝑥∈𝑀 we have 0≤𝑝(𝑡,𝑥(𝑡))≤𝑥(𝑡)≤𝑟₀. (2.11) This implies that 𝑇(𝑀)⊆𝑀. Also, in view of assumption (b) we have for all 𝑥∈𝑀 and for all 𝑠∈[0,1]0≤𝑓(𝑠,𝑥(𝑠))≤𝑎(𝑠)+𝑏𝑥(𝑠)≤𝑎(𝑠)+𝑏𝑟₀. (2.12) Hence 0≤𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))≤𝜃(𝑡)𝑎(𝑠)+𝑏𝑟₀. (2.13) Thus 0≤¹₀𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠≤𝜃(𝑡)‖𝑎‖+𝑏𝑟₀≤‖𝜃‖_∞‖𝑎‖+𝑏𝑟₀=𝑟₀. (2.14) Therefore, 𝑆(𝑀)⊆𝑀.

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

Step 1. 𝑆 and 𝑇 are continuous. Indeed, the assumption (𝑎) and Lemma 2.1 guarantee that 𝑁_𝑝 and 𝒩_𝑓 map continuously 𝐿¹[0,1] into itself. To complete the proof it remains only to show that 𝐾 is continuous. This follows immediately from the hypothesis (𝑏).

Step 2. 𝑆 is ws-compact. Indeed, let (𝜌_𝑛) be a weakly convergent sequence of 𝐿¹[0,1]. Using Lemma 2.2, the sequence (𝒩_𝑓(𝜌_𝑛)) has a weakly convergent subsequence, say (𝒩_𝑓(𝜌_𝑛𝑘)). Let 𝜌 be the weak limit of (𝒩_𝑓(𝜌_𝑛𝑘)). Accordingly, keeping in mind the boundedness of the mapping 𝜁(𝑡,⋅) we get

¹₀𝜁(𝑡,𝑠)𝑓𝑠,𝜌_𝑛𝑘(𝑠)𝑑𝑠→¹₀𝜁(𝑡,𝑠)𝑓𝑠,𝜌(𝑠)𝑑𝑠. (2.15)

The use of the dominated convergence theorem allows us to conclude that the sequence (𝑆𝜌_𝑛𝑘) converges in 𝐿¹[0,1].

Step 3. 𝑆 maps bounded sets of 𝐿¹[0,1] into weakly compact sets. To see this, let 𝒪 be a bounded subset of 𝐿₁[0,1] and let 𝐶>0 such that ‖𝑥‖≤𝐶 for all 𝑥∈𝒪. For 𝑥∈𝒪 we have |(𝑆𝑥)(𝑡)|𝑑𝑡≤¹₀𝜁(𝑓((𝑠≤𝑡,𝑠)𝑠,𝑥))𝑑𝑠₁₀𝜃(𝑡𝑎(𝑠)|𝑥(𝑠(𝑡𝑎‖).)(+𝑏)|)𝑑𝑠≤𝜃)(‖+𝑏𝐶 (2.16)
Consequently, |(𝑆𝑥)(𝑡_𝐸)|𝑑𝑡≤(‖𝑎‖+𝑏𝐶)_𝐸𝜃(𝑡)𝑑𝑡 (2.17) for all measurable subsets 𝐸 of [0,1]. 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 lim_|𝐸|→0∫_𝐸|𝜃(𝑡)|𝑑𝑡=0, where |𝐸| is the Lebesgue measure of 𝐸. Applying Corollary 11 in [24, page 294] once again we infer that the set 𝑆(𝒪) is weakly compact.

Step 4. 𝑆 and 𝑇 are weakly isotone decreasing. Indeed, using assumption (c) we get ¹₀𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠≥𝑝𝑡,¹₀𝜁(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (2.18) and from our assumptions we have ¹₀𝜁(𝑡,𝑠)𝑓𝑠,𝑝(𝑠,𝑥(𝑠))𝑑𝑠≤¹₀𝜁(𝑡,𝑠)𝑎(𝑠)𝑑𝑠+𝑏¹₀𝜁()𝑝((𝑠(𝑡𝑎‖(𝑡)𝑡,𝑠𝑠,𝑥))𝑑𝑠≤𝜃)‖+𝑏𝜃¹₀𝑝((𝑠((𝑡.𝑠,𝑥))𝑑𝑠≤𝑝𝑡,𝑥)) (2.19) Hence 𝑆𝑥≥𝑇𝑆𝑥 and 𝑇𝑥≥𝑆𝑇𝑥 and therefore 𝑆 and 𝑇 are weakly isotone decreasing.

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

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.