Set Contractions and KKM Mappings in Banach Spaces

Abstract

Some fixed point theorems for generalized set contraction maps and KKM type ones in Banach spaces are presented. Moreover, a new generalized set contraction is introduced. As an application, some coincidence theorems for KKM type set contractions are obtained.

1. Introduction

Let 𝐸 be a Banach space and 𝒫_𝑝(𝐸)=𝐴⊂𝐸∶𝐴isanonemptyandhasaproperty𝑝. (1) Thus 𝒫_bd(𝐸), 𝒫_cl(𝐸), 𝒫_cv(𝐸), 𝒫_cp(𝐸), 𝒫_co(𝐸), 𝒫_cl,bd(𝐸), 𝒫_cp,cv(𝐸), and 𝒫_rcp(𝐸) denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of 𝐸, respectively [1]. Let 𝐸₁ and 𝐸₂ be two Banach spaces. A multivalued mapping 𝑇∶𝐸₁→𝒫_𝑝(𝐸₂) is said to be (i)upper semicontinuous if and only if for every closed subset 𝐵 of 𝐸₂, the set 𝑇⁻(𝐵)={𝑥∈𝐸₁∶𝑇(𝑥)∩𝐵≠∅} is a closed subset of 𝐸₁,(ii)closed if its graph Gr(𝑇)={(𝑥,𝑦)∈𝐸₁×𝐸₂∶𝑦∈𝑇(𝑥)} is a closed subset of 𝐸₁×𝐸₂, (iii)compact if 𝑇(𝐸₁) is a compact subset of 𝐸₂. The first fixed point theorem for multivalued mappings is due to Kakutani in Banach spaces, in 1941 [2]. He proved a generalization of Brouwer’s fixed point theorem to the multivalued mappings.

Theorem 1 (see [2]). Let 𝐾 be a compact subset of a Banach space 𝐸 and let 𝐹∶𝐾→𝒫_{𝑐𝑣,𝑐𝑝}(𝐾) be an upper semicontinuous multivalued operator. Then 𝐹 has a fixed point.

The above theorem has been extended in the literature by generalizing or modifying the domain space 𝐸, domain set 𝐾, and the nature of the multivalued operator 𝐹. Here, compactness plays an essential role. The following definition of measure of noncompactness on a bounded subset of the Banach space 𝐸 is given by Dhage in 2010.

Definition 2 (see [1]). A function 𝜇∶𝒫_bd(𝐸)→ℝ⁺ is called a measure of noncompactness if it satisfies (i)∅≠𝜇⁻¹(0)⊂𝒫_rcp(𝐸), (ii)if 𝐴⊆𝐵 then 𝜇(𝐴)≤𝜇(𝐵), (iii)𝜇(𝐴)=𝜇(𝐴), where 𝐴 denotes the closure of 𝐴, (iv)𝜇(Conv(𝐴))=𝜇(𝐴), where Conv(𝐴) denotes the convex hull of 𝐴, (v)if {𝐴_𝑛} is a decreasing sequence of sets in 𝒫_cl,bd(𝐸) satisfying lim_𝑛→∞𝜇(𝐴_𝑛)=0, then the limiting set 𝐴_∞=∩^∞_𝑛=1𝐴_𝑛 is nonempty.

Definition 3 (see [1]). A multivalued mapping 𝑇∶𝐸→𝒫_cl,bd(𝐸) is called 𝒟-set Lipschitz if there exists a continuous nondecreasing function 𝜓∶ℝ⁺→ℝ⁺ such that 𝜇(𝑇(𝐴))≤𝜓(𝜇(𝐴)) for all 𝐴∈𝒫_cl,bd(𝐸) with 𝑇(𝐴)∈𝒫_cl,bd(𝐸), where 𝜓(0)=0. Sometimes we call the function 𝜓 to be a 𝒟-function of 𝑇 on 𝐸. In the spatial case, when 𝜓(𝑟)=𝑘𝑟, 𝑘>0, 𝑇 is called a 𝑘-set Lipschitz mapping and if 𝑘<1, then 𝑇 is called a 𝑘-set contraction on 𝐸. Further, if 𝜓(𝑟)<𝑟 for 𝑟>0, then 𝑇 is called a nonlinear 𝒟-set contraction on 𝐸.

Dhage proved a generalization of Theorem 1 under weaker upper semicontinuity conditions in 2010 [1].

Theorem 4. Let 𝑋 be a nonempty, closed, convex, and bounded subset of a Banach space 𝐸 and let 𝑇∶𝑋→𝒫_{𝑐𝑙,𝑐𝑣}(𝑋) be a closed and nonlinear 𝒟-set contraction. Then 𝑇 has a fixed point.

Lemma 5 (see [3]). If 𝜓 is a 𝒟-function with 𝜓(𝑟)<𝑟 for 𝑟>0, then lim_𝑛→∞𝜓^𝑛(𝑡)=0 for all 𝑡∈[0,∞).

Recall that a function 𝜑∶ℝ⁺→ℝ⁺ is called a comparison function if 𝜑 is increasing and lim_𝑛→∞𝜑^𝑛(𝑡)=0 for all 𝑡∈ℝ⁺ [4]. As a consequence, 𝜑(𝑡)<𝑡 for any 𝑡>0, 𝜑 is continuous at 0, and 𝜑(0)=0.

A function 𝜑∶ℝ⁺→ℝ⁺ is called a (c)-comparison function if 𝜑 is increasing and there exist 𝑘₀∈ℕ, 𝑎∈(0,1) and a convergent series of non-negative terms ∑^∞_𝑘=1𝜈_𝑘 such that 𝜑^𝑘+1(𝑡)≤𝑎𝜑^𝑘(𝑡)+𝜈_𝑘, (2) for 𝑘≥𝑘₀ and any 𝑡∈[0,∞) [5]. If 𝜑∶ℝ⁺→ℝ⁺ is a (c)-comparison function, then 𝜑 is a comparison function [5]. So, we can define 𝒞-set contraction as follows.

Definition 6. A multivalued mapping 𝑇∶𝐸→𝒫_cl,bd(𝐸) is called 𝒞-set contraction if there exists a continuous (c)-comparison function 𝜑 such that 𝜇(𝑇(𝐴))≤𝜑(𝜇(𝐴)) for all 𝐴∈𝒫_cl,bd(𝐸) with 𝑇(𝐴)∈𝒫_cl,bd(𝐸).

Let (𝑋,𝑑) be a metric space. The Hausdorff metric 𝐻_𝑑 induced by the metric 𝑑 and defined as follows 𝐻_𝑑(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴). (3)

Definition 7 (see [6]). Let 𝜀>0. A multivalued contractive (𝜀-contractive) map is a map 𝐹∶𝑋⊸𝒫_cl,bd(𝑋) such that for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦 (and for some 𝜀>0, 𝑑(𝑥,𝑦)<𝜀, resp.), 𝐻_𝑑(𝐹(𝑥),𝐹(𝑦))<𝑑(𝑥,𝑦).

Theorem 8 (see [7]). Let 𝑋 be a nonempty compact and connected metric space and let 𝐹∶𝑋→𝒫_𝑐𝑝(𝑋) be a multivalued 𝜀-contractive map, then 𝐹 has a fixed point.

We need the following definitions of KKM theory in the sequel [8].

Assume that 𝑋 is a convex subset of a topological vector space and 𝑌 is a topological space. If 𝐺∶𝑋→𝒫_𝑝(𝑌) and 𝐹∶𝑋→𝒫_𝑝(𝑌) are two multivalued maps and for each 𝐴∈⟨𝑋⟩, 𝐹(Conv(𝐴))⊂𝐺(𝐴), then 𝐺 is called generalized KKM mapping with respect to 𝐹, where ⟨𝑋⟩ denote the family of all nonempty finite subsets of 𝑋. More generally, if 𝐹∶𝑋→𝒫_𝑝(𝑌) satisfies the requirement that for any generalized KKM mapping 𝐺∶𝑋→𝒫_𝑝(𝑌) with respect to 𝐹 and the family {𝐺(𝑥)∶𝑥∈𝑋} has the finite intersection property, then 𝐹 is said to have the KKM property. Let ()=KKM𝑋,𝑌𝐹∶𝑋→𝒫_𝑝(𝑌)∶𝐹hastheKKMproperty. (4)

Lemma 9 (see [8]). Let 𝑋 be a nonempty convex subset of Hausdorff topological vector space 𝐸. Then 𝑇|_𝐷∈𝐾𝐾𝑀(𝐷,𝑌) whenever 𝑇∈𝐾𝐾𝑀(𝑋,𝑌) and 𝐷 is a nonempty convex subset of 𝑋.

Chen and Chang obtained some fixed point theorems for KKM type set contraction mappings in various spaces [9–12]. In 2010, Amini-Harandi et al. introduced generalized set contraction on topological spaces [13].

In Section 2, we present some fixed point theorems for generalized set contractions which are 𝜀-contractive (KKM 𝜀-contractive) multivalued maps. In the first step of Section 3, we introduce a new type of generalized set contraction and then prove that the results of Section 2 hold for them. Section 4 is devoted to some KKM coincidence theorems as applications of these results.

2. Generalized Set Contractions

In this section by applying Theorem 8, we obtain some fixed point theorems for 𝜀-contractive multivalued maps which are either generalized set contraction or KKM type ones. In all cases, the multivalued maps are not necessarily compact values. We consider measurement of noncompactness in Definition 2.

Definition 10 (see [13]). A multivalued mapping 𝐹∶𝐸→𝒫_𝑝(𝐸) is said to be a generalized set contraction, if for each 𝜖>0 there exists 𝛿>0 such that for 𝐴∈𝒫_cl,bd(𝐸) with 𝜖≤𝜇(𝐴)<𝜖+𝛿, there exists 𝑛∈𝑁 such that 𝜇(𝐹^𝑛(𝐴))<𝜖.

Lemma 11 (see [13]). Let 𝑋 be a topological space and let 𝜇 be a measure of noncompactness on 𝑋. Suppose that 𝐹 is a generalized set contraction on 𝑋. Then for every subset 𝐴 of 𝑋 for which 𝐹(𝐴)⊂𝐴 and 𝜇(𝐴)<∞, one has lim_𝑛→∞𝜇(𝐹^𝑛(𝐴))=0.

Proposition 12. Let 𝐸 be a Hausdorff topological space. If {𝑋_𝑛} is a decreasing sequence of closed and connected sets in 𝒫_{𝑐𝑙,𝑏𝑑}(𝐸) such that lim_𝑛→∞𝜇(𝑋_𝑛)=0, then ∩^∞_𝑛=1𝑋_𝑛 is nonempty, compact, and connected.

Proof. Clearly, 𝐾=∩^∞_𝑛=1𝑋_𝑛 is a nonempty, closed, and compact subset of 𝐸. Let 𝐴 and 𝐵 be two nonempty, disjoint, and closed sets so that 𝐾=𝐴∪𝐵. We can find disjoint open sets 𝑈 and 𝑉 around 𝐴 and 𝐵, respectively. For every 𝑛∈ℕ, 𝑋_𝑛⧵(𝑈∪𝑉) is nonempty. If not, then (𝑈∩𝑋_𝑛) and (𝑉∩𝑋_𝑛) are nonempty and 𝑋_𝑛=(𝑈∩𝑋_𝑛)∪(𝑉∩𝑋_𝑛), which cannot happen. The collection of {𝑋_𝑛⧵(𝑈∪𝑉)} is also a decreasing sequence of nonempty closed sets. Since 𝜇(𝑋_𝑛⧵(𝑈∪𝑉))≤𝜇(𝑋_𝑛) then 𝜇(𝑋_𝑛⧵(𝑈∪𝑉))→0 as 𝑛→∞. Hence, ∩^∞_𝑛=0(𝑋_𝑛⧵(𝑈∪𝑉)) is nonempty, that is, 𝐾∩(𝑈∪𝑉)^𝑐≠∅, which is a contradiction.

Theorem 13. Let 𝑋 be a nonempty, bounded, closed, and connected subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑐𝑜}(𝑋) is an 𝜀-contractive and generalized set contraction, then 𝐹 has a fixed point and the set of fixed points of 𝐹 is compact.

Proof. Let 𝑋₀=𝑋 and 𝑋_𝑛=𝐹^𝑛(𝑋) for all 𝑛∈ℕ. Since 𝐹 is an 𝜀-contractive map, then 𝐹 is continuous and 𝐹(𝐴)⊆𝐹(𝐴). On the other hand, 𝐹^𝑛+1(𝑋)⊆𝐹^𝑛(𝑋), so we have 𝐹(𝑋_𝑛)⊆𝑋_𝑛 and 𝑋_𝑛+1⊆𝑋_𝑛. Since 𝐹 is continuous with closed and connected values, then by [14, Lemma 1.6], 𝐹(𝑋) is connected. Hence, 𝑋_𝑛 which is closure of connected set 𝐹^𝑛(𝑋) for all 𝑛∈ℕ, is connected. But 𝜇(𝑋_𝑛)=𝜇(𝐹^𝑛(𝑋)), and by Lemma 11 we have 𝜇(𝑋_𝑛)→0. Therefore, by Proposition 12, 𝐾=∩^∞_𝑛=1𝑋_𝑛 is nonempty compact and connected. Since 𝐹(𝐾)⊆𝐾, then the desired conclusion followed by an application of Theorem 8 to the multivalued map 𝐹∶𝐾→𝒫_cp(𝐾).
Let 𝐵={𝑥∈𝑋∶𝑥∈𝐹(𝑥)}. We claim that 𝜖₀=𝜇(𝐵)=0. If 𝜇(𝐵)≠0 then 𝜇(𝐵)=𝜇(𝐹^𝑛(𝐵)). Since 𝐹 is a generalized set contraction, then for 𝜖₀>0, there exists 𝛿>0 such that for 𝐵⊆𝑋 with 𝜖₀≤𝜇(𝐵)<𝜖₀+𝛿, there exists 𝑛∈ℕ such that 𝜖₀=𝜇(𝐵)<𝜖₀, which is a contradiction. So 𝐵 is relatively compact, and since 𝐹 is continuous, then 𝐵 is a compact subset of 𝑋.

If 𝐹∶𝐸→𝒫_𝑝(𝐸) is a 𝑘-set contraction, then 𝐹 is a generalized set contraction, but the converse is not true [13]. Therefore, we have the following result.

Corollary 14. Let 𝑋 be a nonempty, bounded, closed, and connected subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑐𝑜}(𝑋) is an 𝜀-contractive and 𝑘-set contraction, then 𝐹 has a fixed point.

Corollary 15. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑏𝑑}(𝑋) is an 𝜀-contractive and generalized set contraction, then there exists a compact subset 𝐾 of 𝑋 such that 𝐹(𝐾)⊆𝐾.

Theorem 16. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∈𝐾𝐾𝑀(𝑋,𝑋) is an 𝜀-contractive and generalized set contraction with nonempty closed and bounded values, then 𝐹 has a fixed point.

Proof. By Corollary 15, there exists a compact subset 𝐾 of 𝑋 such that 𝐹(𝐾)⊆𝐾. Let 𝑌=𝐹(𝐾). Hence there exists a finite subset 𝐴 of 𝐾 such that 𝑌⊆∪_𝑥∈𝐴𝑁(𝑥,𝜀), where 𝑁(𝑥,𝜀)={𝑦∈𝐸∶‖𝑥−𝑦‖<𝜀}. Define a map 𝑇∶𝐾→𝒫_𝑝(𝑌) by 𝑇(𝑥)=𝑌⧵𝑁(𝑥,𝜖) for all 𝑥∈𝐾; then 𝑇(𝑥) is closed for each 𝑥∈𝐾 and ∩_𝑥∈𝐴𝑇(𝑥)=∅. Since 𝑌⊆𝐾⊆𝑋 and 𝐹∈KKM(𝑋,𝑋), then by Lemma 9, 𝐹|_𝐾∈KKM(𝐾,𝑌). Thus, 𝑇 is not a generalized KKM map with respect to 𝐹|_𝐾. Hence, there exists {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝐾 such that 𝐹(Conv({𝑥₀,𝑥₁,…,𝑥_𝑛}))⊈∪^𝑛_𝑖=0𝑇(𝑥_𝑖). Thus there exists 𝑦∈𝐹(Conv({𝑥₀,𝑥₁,…,𝑥_𝑛})) such that 𝑦∉∪^𝑛_𝑖=0𝑇(𝑥_𝑖), that is, 𝑦∈𝐹(𝑧) for some 𝑧∈Conv({𝑥₀,𝑥₁,…,𝑥_𝑛}) and 𝑦∈𝑁(𝑥_𝑖,𝜖), so 𝑥_𝑖∈𝑁(𝑦,𝜖) for 𝑖∈{0,1,…,𝑛}. Since 𝑧∈Conv({𝑥₀,𝑥₁,…,𝑥_𝑛})⊂𝐵(𝑦,𝜖), then 𝑦∈𝐹(𝑧)∩𝐵(𝑧,𝜖). Since 𝑌 is a compact subset of 𝑋, then 𝑦 converges to some 𝑥∈𝑌 as 𝜖→0. Consequently, 𝑧 converges to 𝑥 as 𝜖→0. Since 𝐹 is continuous, then by [14, Lemma 1.6] we have 𝑥∈𝐹(𝑥).

3. Asymptotic Generalized Set Contractions

In this section, we define a new type of set contraction in Banach spaces. Then we prove that the results of Section 2 hold for them. Also, we conclude some fixed point theorems for nonlinear 𝒟-set contractions.

Definition 17. Let 𝑋 be a nonempty, closed, and bounded subset of a Banach space 𝐸. A multivalued mapping 𝐹∶𝑋→𝒫_𝑝(𝑋) is said to be an asymptotic generalized set contraction, if there exists a sequence {𝜑_𝑛} of functions from ℝ⁺ in to itself satisfies (i)for each 𝜀>0, there exists 𝛿>0 and 𝑚∈ℕ such that 𝜑_𝑚(𝑡)≤𝜀 for all 𝜀≤𝑡<𝜀+𝛿, (ii)𝜇(𝐹^𝑛(𝑋))<𝜑_𝑛(𝜇(𝑋)).

Theorem 18. Let 𝑋 be a nonempty, bounded, closed, and connected subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑐𝑜}(𝑋) is an 𝜀-contractive and asymptotic generalized set contraction, then 𝐹 has a fixed point.

Proof. Define a sequence {𝑋_𝑛} of sets in 𝒫_cl,bd(𝑋) such that 𝑋₀=𝑋 and 𝑋_𝑛=𝐹^𝑛(𝑋) for all 𝑛∈ℕ. As the proof of Theorem 13, 𝐹(𝑋_𝑛)⊆𝑋_𝑛, 𝑋_𝑛+1⊂𝑋_𝑛, and 𝑋_𝑛 is connected for all 𝑛∈ℕ. If there exists an integer 𝑁>0 such that 𝜇(𝑋_𝑁)=0, then 𝑋_𝑁 is a compact and connected set and invariant under 𝐹. Thus Theorem 8 implies that 𝐹∶𝑋_𝑁→𝒫_cp(𝑋_𝑁) has a fixed point. So we assume that 𝜇(𝑋_𝑛)≠0 for all 𝑛∈ℕ. Define 𝜀_𝑛=𝜇(𝑋_𝑛) and 𝑟=inf𝜀_𝑛. If 𝑟≠0, by Definition 17, there exists 𝑛₀∈ℕ, 𝛿_𝑟>0 and 𝑚∈ℕ such that 𝜑_𝑚(𝑡)≤𝑟 for all 𝑟≤𝑡<𝑟+𝛿_𝑟 and 𝑟≤𝜀_𝑛0<𝑟+𝛿_𝑟, so 𝜀_𝑛0+𝑚=𝜇𝑋_𝑛0+𝑚=𝜇𝐹^𝑛0+𝑚(𝑋)<𝜑_𝑚𝜇𝐹^𝑛0(𝑋)=𝜑_𝑚𝜇𝑋_𝑛0≤𝑟, (5) which is a contradiction. Hence 𝜇(𝑋_𝑛)⇀0 as 𝑛→∞. Now by Proposition 12, 𝐾=∩^∞_𝑛=1𝑋_𝑛 is nonempty, compact, and connected. Moreover 𝐹(𝐾)⊆𝐾. So by Theorem 8, the multivalued map 𝐹∶𝐾→𝒫_cp(𝐾) has a fixed point.

Corollary 19. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑏𝑑}(𝑋) is an 𝜀-contractive and asymptotic generalized set contraction, then there exists a compact subset 𝐾 of 𝑋 such that 𝐹(𝐾)⊆𝐾.

The proof of following theorem is similar to that of Theorem 16; hence it is omitted.

Theorem 20. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∈𝐾𝐾𝑀(𝑋,𝑋) is an 𝜀-contractive and asymptotic generalized set contraction with nonempty closed and bounded values, then 𝐹 has a fixed point.

Proposition 21. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑏𝑑}(𝑋) is a nonlinear 𝒟-set contraction, then 𝐹 is an asymptotic generalized set contraction.

Proof. Let 𝐹 be a nonlinear 𝒟-set contraction with 𝒟-function 𝜓. Define 𝜑_𝑛=𝜓^𝑛 for all 𝑛∈ℕ. Clearly, 𝜇(𝐹^𝑛(𝑋))≤𝜓^𝑛(𝜇(𝑋))=𝜑_𝑛(𝜇(𝑋)), on the other hand, by Lemma 5 we have 𝜑_𝑛(𝑡)→0 as 𝑛→∞. Thus 𝐹 is an asymptotic generalized set contraction.

Applying Proposition 21, Theorems 18 and 20, it is easy to conclude the following results.

Corollary 22. Let 𝑋 be a nonempty, bounded, closed, and connected subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑐𝑜}(𝑋) is an 𝜀-contractive and nonlinear 𝒟-set contraction, then 𝐹 has a fixed point.

Corollary 23. Let 𝑋 be a nonempty, closed, and bounded subset of Banach space 𝐸. If 𝐹∈𝐾𝐾𝑀(𝑋,𝑋) is an 𝜀-contractive and nonlinear 𝒟-set contraction with nonempty closed and bounded values, then 𝐹 has a fixed point.

Remark 24. Since every 𝒞-set contraction is a nonlinear 𝒟-set contraction, then Theorem 4, Corollaries 22 and 23 hold for these mappings.

4. Some Applications in KKM Theory

In this section we obtain two coincidence theorems for KKM type set contractions.

Theorem 25. Let 𝑋 be a nonempty, closed, bounded, and convex subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑏𝑑}(𝑋) and 𝐺∶𝑋→𝒫_𝑐𝑣(𝑋) are two multivalued mappings satisfying (i)𝐹∈𝐾𝐾𝑀(𝑋,𝑋), (ii)𝐹 is a generalized (an asymptotic) set contraction and 𝜀-contractive map, (iii)for each compact subset 𝐶 of 𝑋 and any 𝑦∈𝑋, 𝐺⁻(𝑦)∩𝐶 is open in 𝐶, then, there exists 𝑥₀,𝑦₀∈𝑋 such that 𝑦₀∈𝐹(𝑥₀) and 𝑥₀∈𝐺(𝑦₀).

Proof. By Corollary 15 (Corollary 19), there exists a compact subset 𝐾 of 𝑋 such that 𝐹(𝐾)⊂𝐾. Since 𝐺(𝑥)≠∅ and 𝐾 is compact, then 𝑋=∪_𝑥∈𝑋𝐺⁻(𝑋) and 𝐾=∪^𝑛_𝑖=1(𝐺⁻(𝑥_𝑖)∩𝐾) for some 𝑥₁,…,𝑥_𝑛∈𝑋. Define a map 𝑆∶𝑋→𝒫_𝑝(𝐾) by 𝑆(𝑥)=𝐾⧵(𝐺⁻(𝑥)∩𝐾) for each 𝑥∈𝑋, then ∩^𝑛_𝑖=1𝑆(𝑥_𝑖)=∅. Therefore, 𝑆 is not a generalized KKM map with respect to 𝐹. So there exists a finite subset 𝐴={𝑎₁,…𝑎_𝑚} of 𝑋 such that 𝐹(Conv(𝐴))⊈𝑆(𝐴). Hence, there exist 𝑥₀∈Conv(𝐴) and 𝑦₀∈𝐹(𝑥₀) such that 𝑦₀∉𝑆(𝐴). Thus 𝑦₀∈𝐺⁻(𝑎_𝑖)∩𝐾 and so 𝑎_𝑖∈𝐺(𝑦₀) for 𝑖=1,…,𝑚. Since 𝐺(𝑦₀) is convex, then Conv(𝐴)⊆𝐺(𝑦₀) and so 𝑥₀∈𝐺(𝑦₀).

By Proposition 21, Corollary 19, and slight modification of the proof of Theorem 25, we have the following theorem.

Theorem 26. Let 𝑋 be a nonempty, closed, bounded, and convex subset of Banach space 𝐸. If 𝐹∶𝑋→𝒫_{𝑐𝑙,𝑐𝑣}(𝑋) and 𝐺∶𝑋→𝒫_𝑐𝑣(𝑋) are two multivalued mappings satisfying (i)𝐹∈𝐾𝐾𝑀(𝑋,𝑋) is a nonlinear 𝒟-set contraction. (ii)for each compact subset 𝐶 of 𝑋 and any 𝑦∈𝑋, 𝐺⁻(𝑦)∩𝐶 is open in 𝐶, then, there exists 𝑥₀,𝑦₀∈𝑋 such that 𝑦₀∈𝐹(𝑥₀) and 𝑥₀∈𝐺(𝑦₀).