Table of Contents
ISRN Applied Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 696917, 13 pages
http://dx.doi.org/10.5402/2011/696917
Research Article

Some Results for the Family KKM(X,Y) and the Ξ¦-Mapping in FC-Spaces

Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610103, China

Received 15 July 2011; Accepted 24 August 2011

Academic Editor: A.Β Bellouquid

Copyright Β© 2011 Rong-Hua He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first establish a fixed point theorem for a π‘˜-set contraction map on the family 𝐾𝐾𝑀(𝑋,𝑋), which does not need to be a compact map. Next, we present the 𝐾𝐾𝑀 type theorems, matching theorems, coincidence theorems, and minimax theorems on the family 𝐾𝐾𝑀(𝑋,π‘Œ) and the Ξ¦-mapping in FC-spaces. Our results improve and generalize some recent results.

1. Introduction

In 1929, Knaster et al. [1] first established the well-known 𝐾𝐾𝑀 theorem in finite-dimensional spaces. In 1961, Fan [2] generalized the 𝐾𝐾𝑀 theorem to infinite dimensional topological vector spaces and gave some applications in several directions. Later, Chang and Yen [3] introduced the family 𝐾𝐾𝑀(𝑋,π‘Œ) and got some results about fixed point theorems, coincidence theorems, and its applications on this family. Recently, Lin and Chen [4] studied the coincidence theorems for two families of multivalued functions, Chen and Chang [5] obtained some results for the family 𝐾𝐾𝑀(𝑋,π‘Œ) and the Ξ¦-mapping in Hausdorff topological vector spaces. For the related results, the reader may consult [4–10]. In this paper, we first establish a fixed point theorem for a π‘˜-set contraction map on the family 𝐾𝐾𝑀(𝑋,𝑋), which does not need to be a compact map. Next, we present the 𝐾𝐾𝑀 type theorems, matching theorems, coincidence theorems, and minimax theorems on the family 𝐾𝐾𝑀(𝑋,π‘Œ) and the Ξ¦-mapping in 𝐹𝐢-spaces. Our results improve and generalize the corresponding results in [5, 8, 9].

2. Preliminaries

Let π‘Œ be a nonempty set. We denote by 2π‘Œ and βŸ¨π‘ŒβŸ© the family of all subsets of π‘Œ and the family of all nonempty finite subsets of π‘Œ, respectively. For each π΄βˆˆβŸ¨π‘‹βŸ©, we denote by |𝐴| the cardinality of 𝐴. Let Δ𝑛 denote the standard n-dimensional simplex with the vertices {𝑒0,…,𝑒𝑛}. If 𝐽 is a nonempty subset of {0,1,…,𝑛}, we will denote by Δ𝐽 the convex hull of the vertices {π‘’π‘—βˆΆπ‘—βˆˆπ½}.

Let 𝑋 and π‘Œ be two sets, and π‘‡βˆΆπ‘‹β†’2π‘Œ be a set-valued mapping. We will use the following notations in the following material: (i)𝑇(π‘₯)={π‘¦βˆˆπ‘ŒβˆΆπ‘¦βˆˆπ‘‡(π‘₯)},(ii)⋃𝑇(𝐴)=π‘₯βˆˆπ΄π‘‡(π‘₯),(iii)π‘‡βˆ’1(𝑦)={π‘₯βˆˆπ‘‹βˆΆπ‘¦βˆˆπ‘‡(π‘₯)},(iv)π‘‡βˆ’1β‹‚(𝐡)={π‘₯βˆˆπ‘‹βˆΆπ‘‡(π‘₯)π΅β‰ βˆ…},(v)𝐢(𝑋,π‘Œ) denote the family of single-valued continuous mappings from 𝑋 to π‘Œ.

For topological spaces 𝑋 and π‘Œ, π‘‡βˆΆπ‘‹β†’2π‘Œ is said to be closed if its graph 𝒒𝑇={(π‘₯,𝑦)βˆˆπ‘‹Γ—π‘ŒβˆΆπ‘¦βˆˆπ‘‡(π‘₯)} is closed. 𝑇 is said to be compact if the image 𝑇(𝑋) of 𝑋 under 𝑇 is contained in a compact subset of π‘Œ. A subset 𝐷 of 𝑋 is said to be compactly open (resp., compactly closed) if for each nonempty compact subset 𝐾 of 𝑋, 𝐷⋂𝐾 is open (resp., closed) in 𝐾. The compact closure of 𝐴 and the compact interior of 𝐴 (see [5]) are defined, respectively, by ccl𝐴={π΅βŠ‚π‘‹βˆΆπ΄βŠ‚BandBiscompactlyclosedinξšπ‘‹},cint𝐴={π΅βŠ‚π‘‹βˆΆπ΅βŠ‚π΄and𝐡iscompactlyopenin𝑋}.(2.1)

It is easy to see that ccl(𝑋⧡𝐴)=𝑋⧡cint𝐴,intπ΄βŠ‚cintπ΄βŠ‚π΄,π΄βŠ‚cclπ΄βŠ‚cl𝐴, 𝐴 is compactly open (resp., compactly closed) in 𝑋 if and only if cint𝐴=𝐴 (resp., ccl𝐴=𝐴). For each nonempty compact subset 𝐾 of 𝑋, β‹‚ccl𝐴𝐾=cl𝐾⋂(𝐴𝐾) and β‹‚cint𝐴𝐾=int𝐾⋂(𝐴𝐾), where cl𝐾⋂(𝐴𝐾) (resp., int𝐾⋂𝐾(𝐴)) denotes the closure (resp., interior) of 𝐴⋂𝐾 in 𝐾.

A set-valued mapping π‘‡βˆΆπ‘‹β†’2π‘Œ is said to be transfer compactly closed valued on 𝑋 (see [5]) if for each π‘₯βˆˆπ‘‹ and π‘¦βˆ‰π‘‡(π‘₯), there exists π‘₯ξ…žβˆˆπ‘‹ such that π‘¦βˆ‰ccl𝑇(π‘₯ξ…ž). 𝑇 is said to be transfer compactly open valued on 𝑋 if for each π‘₯βˆˆπ‘‹ and π‘¦βˆˆπ‘‡(π‘₯), there exists π‘₯ξ…žβˆˆπ‘‹ such that π‘¦βˆˆcint𝑇(π‘₯ξ…ž). 𝑇 is said to have the compactly local intersection property on 𝑋 if for each nonempty compact subset 𝐾 of 𝑋 and for each π‘₯βˆˆπ‘‹ with 𝑇(π‘₯)β‰ βˆ…, there exists an open neighborhood 𝑁(π‘₯) of π‘₯ in 𝑋 such β‹‚β‹‚πΎπ‘§βˆˆπ‘(π‘₯)𝑇(𝑧)β‰ βˆ….

Let 𝐴𝑖(𝑖=1,…,π‘š) is transfer compactly open valued, then β‹‚π‘šπ‘–=1cint𝐴𝑖⋂=cintπ‘šπ‘–=1𝐴𝑖. It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.

Throughout this paper, all topological spaces are assumed to be Hausdorff. The following notion of a finitely continuous topological space (in short, 𝐹𝐢-space) was introduced by Ding in [11].

Definition 2.1. (π‘Œ,πœ‘π‘) is said to be an 𝐹𝐢-space if π‘Œ is a topological space and for each 𝑁={𝑦0,…,𝑦𝑛}βˆˆβŸ¨π‘ŒβŸ© where some elements in 𝑁 may be same, there exists a continuous mapping πœ‘π‘βˆΆΞ”π‘›β†’π‘Œ. If 𝐴 and 𝐡 are two subsets of π‘Œ, 𝐡 is said to be an 𝐹𝐢-subspace of π‘Œ relative to 𝐴 if for each 𝑁={𝑦0,…,𝑦𝑛}βˆˆβŸ¨π‘ŒβŸ© and for any {𝑦𝑖0,…,π‘¦π‘–π‘˜}βŠ‚π‘βˆ©π΄, πœ‘π‘(Ξ”π‘˜)βŠ‚π΅, where Ξ”π‘˜=co({𝑒𝑖𝑗:𝑗=0,…,π‘˜}).
If 𝐴=𝐡, then 𝐡 is called an 𝐹𝐢-subspace of π‘Œ. Clearly, each 𝐹𝐢-subspace of π‘Œ is also an 𝐹𝐢-space.
For a subset 𝐴 of 𝑋, we can define the 𝐹𝐢-hull of 𝐴 (see [12]) as follows: ⋂𝐹𝐢(𝐴)={π΅βŠ‚π‘‹βˆΆπ΄βŠ‚π΅and𝐡is𝐹𝐢-subspaceof𝑋}.

Definition 2.2 (see [8]). An 𝐹𝐢-space (𝑋,πœ‘π‘) is said to be a locally 𝐹𝐢-space, denoted by (𝑋,𝒰,πœ‘π‘), if 𝑋 is a uniform topological space with a uniform structure 𝒰 having an open base 𝛽 of symmetric entourages such that for each π‘₯βˆˆπ‘‹ and for each π‘ˆβˆˆπ’°, π‘ˆ[π‘₯]={π‘¦βˆˆπ‘‹βˆΆ(𝑦,π‘₯)βˆˆπ‘ˆ} is an 𝐹𝐢-subspace of 𝑋.

Lemma 2.3 (see [5]). Let 𝑋 and π‘Œ be two topological spaces, and let πΉβˆΆπ‘‹β†’2π‘Œ be a set-valued mapping. Then the following conditions are equivalent:(1)𝐹 has the compactly local intersection property,(2)for each compact subset 𝐾 of 𝑋 and for each π‘¦βˆˆπ‘Œ, there exists an open subset 𝑂𝑦 of 𝑋 such that π‘‚π‘¦β‹‚πΎβŠ‚πΉβˆ’1(𝑦) and ⋃𝐾=π‘¦βˆˆπ‘Œ(𝑂𝑦⋂𝐾),(3)for any compact subset 𝐾 of 𝑋, there exists a set-valued mapping π‘ƒβˆΆπ‘‹β†’2π‘Œ such that 𝑃(π‘₯)βŠ‚πΉ(π‘₯) for each π‘₯βˆˆπ‘‹, π‘ƒβˆ’1(𝑦) is open in 𝑋 and π‘ƒβˆ’1β‹‚(𝑦)πΎβŠ‚πΉβˆ’1(𝑦) for each π‘¦βˆˆπ‘Œ, and ⋃𝐾=π‘¦βˆˆπ‘Œ(π‘ƒβˆ’1β‹‚(𝑦)𝐾),(4)for each compact subset 𝐾 of 𝑋 and for each π‘₯∈𝐾, there exists π‘¦βˆˆπ‘Œ such that π‘₯∈cintπΉβˆ’1⋂𝐾(𝑦) and ⋃𝐾=π‘¦βˆˆπ‘Œ(cintπΉβˆ’1β‹‚(𝑦)𝐾),(5)πΉβˆ’1 is transfer compactly open valued on π‘Œ,(6)⋃𝑋=π‘¦βˆˆπ‘ŒcintπΉβˆ’1(𝑦).

Now, we introduce the following Definitions 2.4 and 2.6.

Definition 2.4. Let π‘Œ be a topological space and (𝑋,πœ‘π‘) be an 𝐹𝐢-space. A set-valued mapping π‘‡βˆΆπ‘Œβ†’2𝑋 is called a Ξ¦-mapping if there exists a set-valued mapping π‘‡βˆΆπ‘Œβ†’2𝑋 such that(i)for each π‘¦βˆˆπ‘Œ, 𝑇(𝑦) is a nonempty 𝐹𝐢-subspace of 𝑋 relative to 𝐹(𝑦),(ii)𝐹 satisfies one of the conditions (1)–(6) in Lemma 2.3.
The mapping 𝐹 is said to be a companion mapping of 𝑇.

Remark 2.5. If π‘‡βˆΆπ‘Œβ†’2𝑋 be a Ξ¦-mapping, then for each nonempty subset π‘Œ1 of π‘Œ, 𝑇|π‘Œ1βˆΆπ‘Œ1β†’2𝑋 is also a Ξ¦-mapping.

The class 𝐾𝐾𝑀 was introduced by Ding [8]. Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space and let π‘Œ be a topological space. If 𝐹,π‘‡βˆΆπ‘‹β†’2π‘Œ are two set-valued mappings such that for each π‘βˆˆβŸ¨π‘‹βŸ© and for each {𝑒𝑖0,…,π‘’π‘–π‘˜}βŠ‚{𝑒0,…,𝑒𝑛},𝑇(πœ‘π‘(Ξ”π‘˜β‹ƒ))βŠ‚π‘˜π‘—=0𝐹(π‘₯𝑖𝑗), then 𝐹 is said to be a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇. Let π‘‡βˆΆπ‘‹β†’2π‘Œ be a set-valued mapping such that if πΉβˆΆπ‘‹β†’2π‘Œ is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇, then the family {𝐹π‘₯∢π‘₯βˆˆπ‘‹} has the finite intersection property, where 𝐹π‘₯ denotes the closure of 𝐹π‘₯, then 𝑇 is said to have the 𝐾𝐾𝑀 property. Write 𝐾𝐾𝑀(𝑋,π‘Œ)=π‘‡βˆΆπ‘‹βŸΆ2π‘ŒβˆΆπ‘‡hasthe𝐾𝐾𝑀propertyξ€Ύ.(2.2)

Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space and 𝐡(𝑋) be the family of nonempty bouned subsets. Let 𝒫 = {π‘ƒβˆΆπ‘ƒ is a family of seminorms which determines the topology on 𝑋. Let β„›+ be the set of all nonnegative real numbers. If {π‘βˆˆπ‘ƒβˆΆπ‘ƒβˆˆπ’«} is a family of seminorms which determines the topology on 𝑋}, then for each π‘ƒβˆˆπ’« and Ξ©βˆˆπ‘‹, we define the set-measure of noncompactness π›Όπ‘βˆΆ2𝑋→ℛ+ by 𝛼𝑝(Ξ©)=inf{πœ€>0∢Ωcanbecoveredbyafinitenumberofsetsandeach𝑝-diameterofthesetsislessthanπœ€},(2.3) where 𝑝-diameter of a set 𝐷=sup{𝑝(π‘₯βˆ’π‘¦)∢π‘₯,π‘¦βˆˆπ·}.

Definition 2.6. Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space, and a mapping π‘‡βˆΆπ‘‹β†’2𝑋 is said to be a π‘˜-set contraction map, if there exists π‘ƒβˆˆπ’« such that for each π‘βˆˆπ‘ƒ, 𝛼𝑝(𝑇(Ξ©))β‰€π‘˜π›Όπ‘(Ξ©) with 0<π‘˜<1 for each bounded subset Ξ© of 𝑋 and 𝑇(𝑋) is bounded.

3. Main Results

In order to prove our main results, we need the following Lemmas. The following results are [8, Lemma 3.1(i) and Theorem 3.1].

Lemma 3.1. Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space and let π‘Œ be a topological space. Then we have π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ) if and only if π‘‡βˆ£π·βˆˆπΎπΎπ‘€(𝐷,π‘Œ) for each nonempty subset 𝐷 of 𝑋.

Lemma 3.2. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space. If π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ) is a compact mapping, then for each open entourage π‘ˆπ‘–βˆˆπ’° there exists π‘₯π‘–βˆˆπ‘‹ such that 𝑇(π‘₯𝑖)β‹‚π‘ˆπ‘–[π‘₯𝑖]β‰ βˆ….

Lemma 3.3. Let π‘Œ be a compact topological space and (𝑋,πœ‘π‘) be an 𝐹𝐢-space. Let π‘‡βˆΆπ‘Œβ†’2𝑋 be a Ξ¦-mapping. Then there exists a continuous function π‘“βˆΆπ‘Œβ†’π‘‹ such that for each π‘¦βˆˆπ‘Œ, 𝑓(𝑦)βˆˆπ‘‡(𝑦); that is, 𝑇 has a continuous selection.

Proof. Since 𝑇 is a Ξ¦-mapping, there exists a companion mapping πΉβˆΆπ‘Œβ†’2𝑋 such that(i)for each π‘¦βˆˆπ‘Œ, for each 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βŸ©, and for any {π‘₯𝑖0,…,π‘₯π‘–π‘˜β‹‚}βˆˆπ‘πΉ(𝑦),πœ‘π‘(Ξ”π‘˜)βŠ‚π‘‡(𝑦);(ii)β‹ƒπ‘Œ=π‘₯βˆˆπ‘‹cintπΉβˆ’1(π‘₯).
Since π‘Œ be a compact, there exists 𝑀={π‘₯0,…,π‘₯π‘š}βˆˆβŸ¨π‘‹βŸ© such that β‹ƒπ‘Œ=π‘šπ‘–=0cintπΉβˆ’1(π‘₯𝑖). Let {πœ“π‘–}π‘šπ‘–=0 be the continuous partition of unity subordinated to the open covering {cintπΉβˆ’1(π‘₯𝑖)}π‘šπ‘–=0 of π‘Œ, then for each π‘–βˆˆ{0,1,…,π‘š} and π‘¦βˆˆπ‘Œ, we have πœ“π‘–(𝑦)β‰ 0βŸΊπ‘¦βˆˆcintπΉβˆ’1ξ€·π‘₯π‘–ξ€ΈβŠ‚πΉβˆ’1ξ€·π‘₯π‘–ξ€ΈβŸΉπ‘₯π‘–βˆˆπΉ(𝑦).(3.1) Define a mapping πœ“βˆΆπ‘Œβ†’Ξ”π‘› by πœ“(𝑦)=Ξ£π‘šπ‘–=0πœ“π‘–(𝑦)𝑒𝑖, then πœ“ is continuous and for each π‘¦βˆˆπ‘Œ, πœ“(𝑦)=Ξ£π‘—βˆˆπ½(𝑦)πœ“π‘—(𝑦)π‘’π‘—βˆˆΞ”π½(𝑦), where 𝐽(𝑦)={π‘—βˆˆ{0,1,…,π‘š}βˆΆπœ“π‘—(𝑦)β‰ 0}. By (3.1), we have {π‘₯π‘—β‹‚βˆΆπ‘—βˆˆπ½(𝑦)}∈⟨𝐹(𝑦)⟩{π‘₯0,…,π‘₯π‘š}. From the definition of Ξ¦-mapping, we obtain for each π‘¦βˆˆπ‘Œ, πœ‘π‘(Δ𝐽(𝑦))βŠ‚π‘‡(𝑦). It follows that 𝑓(𝑦)=πœ‘π‘βˆ˜πœ“(𝑦)βˆˆπœ‘π‘ξ€·Ξ”π½(𝑦)ξ€ΈβŠ‚π‘‡(𝑦).(3.2) This shows that for each π‘¦βˆˆπ‘Œ, 𝑓(𝑦)βˆˆπ‘‡(𝑦), that is, 𝑇 has a continuous selection. This completes the proof.

Lemma 3.4. Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space and let π‘Œ be a topological space. If π‘‡βˆΆπ‘‹β†’2π‘Œ is a Ξ¦-mapping, then π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ).

Proof. Since 𝑇 is a Ξ¦-mapping, we have that for each 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βŸ© and πœ‘π‘(Δ𝑛) in 𝑋, π‘‡βˆ£πœ‘π‘(Δ𝑛)βˆΆπœ‘π‘(Δ𝑛)β†’π‘Œ is also a Ξ¦-mapping. By Lemma 3.3, π‘‡βˆ£πœ‘π‘(Δ𝑛) has a continuous selection function. Then π‘‡βˆ£πœ‘π‘(Δ𝑛)βˆˆπΎπΎπ‘€(πœ‘π‘(Δ𝑛),π‘Œ). It follows from Lemma 3.1 that π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ). This completes the proof.

Lemma 3.5. Let π‘Œ and 𝑍 be two topological spaces and (𝑋,πœ‘π‘) be an 𝐹𝐢-space. If π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ) and π‘“βˆˆπΆ(π‘Œ,𝑍), then π‘“π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑍).

Proof. Let πΉβˆΆπ‘‹β†’2𝑍 be a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑓𝑇 such that 𝐹π‘₯ is closed in 𝑍 for each π‘₯βˆˆπ‘‹. Take 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βŸ© and for any {𝑒𝑖0,…,π‘’π‘–π‘˜}βŠ‚{𝑒0,…,𝑒𝑛}, we have 𝑓𝑇(πœ‘π‘(Ξ”π‘˜β‹ƒ))βŠ‚π‘˜π‘—=0𝐹(π‘₯𝑖𝑗), and hence 𝑇(πœ‘π‘(Ξ”π‘˜β‹ƒ))βŠ‚π‘˜π‘—=0π‘“βˆ’1𝐹(π‘₯𝑖𝑗). So π‘“βˆ’1𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇. Since π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ), then {π‘“βˆ’1𝐹π‘₯∢π‘₯βˆˆπ‘‹} has the finite intersection property, and hence {𝐹π‘₯∢π‘₯βˆˆπ‘‹} has also the finite intersection property and π‘“π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑍). This completes the proof.

Remark 3.6. Lemmas 3.3, 3.4, and 3.5 generalize Lemmas 2, 3, and 5 in [5], respectively.

Corollary 3.7. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space and π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ) is compact and closed. Then 𝑇 has a fixed point in 𝑋.

Proof. Since 𝑇 is compact, we have 𝐾=𝑇(𝑋) is compact in 𝑋. Without loss of generality, assume that {𝑉𝑖}π‘–βˆˆπΌ be a basis of symmetric open entourages for the uniformity 𝒰. It follows from the Lemma 3.2 that for each π‘‰βˆˆ{𝑉𝑖}π‘–βˆˆπΌ there exists π‘₯π‘–βˆˆπ‘‹ such that 𝑇(π‘₯𝑖)⋂𝑉[π‘₯𝑖]β‰ βˆ…. Hence, for each π‘‰βˆˆ{𝑉𝑖}π‘–βˆˆπΌ there exists π‘¦π‘–βˆˆπ‘‡(π‘₯𝑖) and π‘¦π‘–βˆˆπ‘‰[π‘₯𝑖]. Since {𝑦𝑖}∈𝐾 and 𝐾 is compact, we may assume that {𝑦𝑖} converges to some π‘¦βˆˆπΎ, and then π‘₯𝑖 also converges to 𝑦. Since 𝑇 is closed, we have π‘¦βˆˆπ‘‡(𝑦). This completes the proof.

Theorem 3.8. Let (𝑋,πœ‘π‘) be a bounded 𝐹𝐢-space. Assume that π‘‡βˆΆπ‘‹β†’2𝑋 be a π‘˜-set contraction map, 0<π‘˜<1. Then 𝑋 contains a precompact 𝐹𝐢-subspace.

Proof. Since 𝑇 is a π‘˜-set contraction map, 0<π‘˜<1, there exists π‘ƒβˆˆπ’« such that for each π‘βˆˆπ‘ƒ, we have 𝛼𝑝(𝑇(𝐴))β‰€π‘˜π›Όπ‘(𝐴) for each subset 𝐴 of 𝑋. Take π‘¦βˆˆπ‘‹. Let 𝑋0=𝑋,𝑋1𝑇𝑋=𝐹𝐢0ξ€Έξšξƒͺ,𝑋{𝑦}𝑛+1𝑇𝑋=πΉπΆπ‘›ξ€Έξšξƒͺ{𝑦}foreachπ‘›βˆˆπ‘.(3.3) Then(1)𝑋𝑛 is 𝐹𝐢-subspace of 𝑋 for each π‘›βˆˆπ‘,(2)𝑋𝑛+1βŠ‚π‘‹π‘› for each π‘›βˆˆπ‘,(3)𝑇(𝑋𝑛)βŠ‚π‘‹π‘›+1 for each π‘›βˆˆπ‘,(4)𝛼𝑝(𝑋𝑛+1)≀𝛼𝑝(𝑇(𝑋𝑛))β‰€π‘˜π›Όπ‘(𝑋𝑛)β‰€β‹―β‰€π‘˜π‘›+1𝛼𝑝(𝑋0) for each π‘›βˆˆπ‘.
Thus 𝛼𝑝(𝑋𝑛)β†’0, as π‘›β†’βˆž, and hence π‘‹βˆž=⋂𝑛β‰₯1𝑋𝑛 is a nonempty precompact 𝐹𝐢-subspace of 𝑋. This completes the proof.

Remark 3.9. Theorem 3.8 generalized [5, Theorem 1] from a nonempty bounded convex subset of a Hausdorff topological vector space to a bounded 𝐹𝐢-space. In the process of the proof Theorem 3.8, we call the set π‘‹βˆž a precompact-inducing 𝐹𝐢-subspace of 𝑋.

The following result is a fixed point theorem for a π‘˜-set contraction map on the family 𝐾𝐾𝑀(𝑋,π‘Œ), which does not need to be a compact map.

Theorem 3.10. Let 𝑋 be a bounded 𝐹𝐢-subspace of a locally 𝐹𝐢-space (𝐸,𝒰,πœ‘π‘), and let π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋) be a π‘˜-set contraction map, 0<π‘˜<1 and closed with 𝑇(𝑋)βŠ‚π‘‹. Then 𝑇 has a fixed point in 𝑋.

Proof. By the same process of the proof Theorem 3.8, we get a precompact-inducing 𝐹𝐢-subspace π‘‹βˆž of 𝑋. Since 𝑇(𝑋)βŠ‚π‘‹ and 𝑇(𝑋𝑛+1)βŠ‚π‘‡(𝑋𝑛)βŠ‚π‘‡(𝑋) for each π‘›βˆˆπ‘, we have 𝑇(𝑋𝑛+1)βŠ‚π‘‡(𝑋𝑛)βŠ‚π‘‹ for each π‘›βˆˆπ‘. Since 𝛼𝑝(𝑇(𝑋𝑛))β†’0 as π‘›β†’βˆž, we have 𝑇(π‘‹βˆžβ‹‚)=𝑛β‰₯1𝑇(𝑋𝑛) is a nonempty compact subset of 𝑋.
Since π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋) and π‘‹βˆž is a nonempty 𝐹𝐢-subspace of 𝑋, by Lemma 3.1, we have π‘‡βˆ£π‘‹βˆžβˆˆπΎπΎπ‘€(π‘‹βˆž,𝑋).
Let {𝑉𝑖}π‘–βˆˆπΌ be a basis of symmetric open entourages for the uniformity 𝒰, then there exists π‘‰βˆˆ{𝑉𝑖}π‘–βˆˆπΌ such that π‘‰βˆˆπ’°. We now claim that for π‘ˆπ‘–βˆˆπ’° there exists π‘₯π‘–βˆˆπ‘‹βˆž such that 𝑇(π‘₯𝑖)β‹‚(π‘₯𝑖+π‘ˆπ‘–)β‰ βˆ…. If it is false, then there exists π‘ˆβˆˆπ’° such that ⋂𝑇(π‘₯)(π‘₯+π‘ˆ)=βˆ… for all π‘₯βˆˆπ‘‹βˆž. Let 𝐾=𝑇(π‘‹βˆž). Define πΉβˆΆπ‘‹βˆžβ†’2𝐾 by ξ‚€1𝐹(π‘₯)=𝐾⧡π‘₯+2π‘ˆξ‚foreachπ‘₯βˆˆπ‘‹βˆž,(3.4) then 𝐹(π‘₯) is a compact for each π‘₯βˆˆπ‘‹βˆž. Next, let 𝑉[π‘₯]=π‘₯+(1/2)π‘ˆ, then 𝐹(π‘₯)=𝐾⧡𝑉[π‘₯], we prove that 𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘‹βˆž.
Suppose, on contrary, 𝐹 is not a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘‹βˆž. Then there exists 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βˆžβŸ© and {𝑒𝑖0,…,π‘’π‘–π‘˜}βŠ‚{𝑒0,…,𝑒𝑛} such that π‘‡βˆ£π‘‹βˆžξ€·πœ‘π‘ξ€·Ξ”π‘˜βŠ„ξ€Έξ€Έπ‘˜ξšπ‘—=0𝐹π‘₯𝑖𝑗,(3.5) where Ξ”π‘˜=co({π‘’π‘–π‘—βˆΆπ‘—=0,…,π‘˜}). Hence there exist π‘’βˆˆπœ‘π‘(Ξ”π‘˜) and π‘£βˆˆπ‘‡βˆ£π‘‹βˆž(𝑒)βŠ‚π‘‡(π‘‹βˆž)=𝐾 such that β‹ƒπ‘£βˆ‰π‘˜π‘—=0𝐹(π‘₯𝑖𝑗). From the definition of 𝐹 it follows that π‘£βˆˆπ‘‰[π‘₯𝑖𝑗] for all π‘—βˆˆ{0,…,π‘˜}. Noting that π‘‰βˆˆπ‘ˆ is symmetric, we have {π‘₯𝑖𝑗:𝑗=0,…,π‘˜}βŠ‚π‘‰[𝑣]. Since 𝑉[𝑣] is 𝐹𝐢-subspace of 𝑋, we have π‘’βˆˆπœ‘π‘(Ξ”π‘˜)βŠ‚π‘‰[𝑣]. By the symmetry of 𝑉, we obtain π‘£βˆˆπ‘‰[𝑒] and π‘£βˆˆπ‘‡|π‘‹βˆžβ‹‚(𝑒)𝑉[𝑒]βŠ‚π‘‡βˆ£π‘‹βˆžβ‹‚(𝑒)π‘ˆ[𝑒]βŠ‚π‘‡βˆ£π‘‹βˆžβ‹‚(𝑒)(𝑒+π‘ˆ) which contradicts the fact ⋂𝑇(π‘₯)(π‘₯+π‘ˆ)=βˆ… for all π‘₯π‘–βˆˆπ‘‹βˆž. Therefore 𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘‹βˆž.
Since π‘‡βˆ£π‘‹βˆžβˆˆπΎπΎπ‘€(π‘‹βˆž,𝑋) and 𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘‹βˆž, the family {𝐹(π‘₯)∢π‘₯βˆˆπ‘‹βˆž} has the finite intersection property, and so we conclude that β‹‚π‘₯βˆˆπ‘‹βˆžπΉ(π‘₯)β‰ βˆ…. Choose β‹‚πœ‚βˆˆπ‘₯βˆˆπ‘‹βˆžπΉ(π‘₯), then πœ‚βˆˆπΎβ§΅π‘‰[π‘₯] for all π‘₯βˆˆπ‘‹βˆž. Since β‹‚πœ‚βˆˆπ‘₯βˆˆπ‘‹βˆžπΉ(π‘₯)βŠ‚πΎ=𝑇(π‘‹βˆž)βŠ‚π‘‹βˆž+(1/4)π‘ˆ, hence there exists π‘₯0βˆˆπ‘‹βˆž such that πœ‚βˆˆπ‘₯0+(1/2)π‘ˆ=𝑉[π‘₯0]. But πœ‚βˆˆπΎβ§΅π‘‰[π‘₯0], a contradiction. Therefore, we have prove that for each π‘ˆπ‘–βˆˆ{𝑉𝑖}π‘–βˆˆπΌ, there exists π‘₯π‘–βˆˆπ‘‹βˆž such that 𝑇(π‘₯𝑖)β‹‚(π‘₯𝑖+π‘ˆπ‘–)β‰ βˆ…. Let π‘¦π‘–βˆˆπ‘‡(π‘₯𝑖)β‹‚(π‘₯𝑖+π‘ˆπ‘–). Since {𝑦𝑖}βŠ‚πΎ and 𝐾 is compact, we may assume that {𝑦𝑖} converges to some π‘¦βˆˆπΎ, and then π‘₯𝑖 also converges to 𝑦. Since 𝑇 is closed, we have π‘¦βˆˆπ‘‡(𝑦). This completes the proof.

By Lemma 3.4 and Theorem 3.10, we can get the following result immediately.

Corollary 3.11. Let 𝑋 be a bounded 𝐹𝐢-subspace of a locally 𝐹𝐢-subspace (𝐸,𝒰,πœ‘π‘), and let π‘‡βˆΆπ‘‹β†’2𝑋 be a Ξ¦-mapping, π‘˜-set contraction, 0<π‘˜<1 and closed with 𝑇(𝑋)βŠ‚π‘‹. Then 𝑇 has a fixed point in 𝑋.

Remark 3.12. Theorem 3.10 generalizes [5, Theorem 2] from a nonempty bounded convex subset of a locally convex space to a bounded 𝐹𝐢-subspace of a locally 𝐹𝐢-space and [8, Theorem 3.10]. Corollary 3.11 generalizes [5, Corollary 1] in several aspects.

4. Applications

By Definition 4.3 of Ding [13], we have the following definition.

Definition 4.1. Let (𝑋,πœ‘π‘) be an 𝐹𝐢-space. π‘Œ be topological space. π‘“βˆΆπ‘‹Γ—π‘Œβ†’π‘… is said to be 𝐹𝐢-quasiconvex (resp., 𝐹𝐢-quasiconcave) if for each π‘¦βˆˆπ‘Œ and for each πœ†βˆˆπ‘…, the set {π‘₯βˆˆπ‘‹βˆΆπ‘“(π‘₯,𝑦)<πœ†} (resp., {π‘₯βˆˆπ‘‹βˆΆπ‘“(π‘₯,𝑦)>πœ†}) is an 𝐹𝐢-subspace of 𝑋.

Definition 4.2 (see [5]). Let 𝑋 and π‘Œ be two topological spaces, and let π‘“βˆΆπ‘‹Γ—π‘Œβ†’π‘…βˆͺ{βˆ’βˆž,+∞} be a function. Then 𝑓 is said to be transfer compactly lower semicontinuous (in short, transfer compactly l.s.c) in 𝑦 if for each π‘¦βˆˆπ‘Œ and π›Ύβˆˆπ‘… with π‘¦βˆˆ{π‘’βˆˆπ‘ŒβˆΆπ‘“(π‘₯,𝑒)>𝛾}, there exists π‘₯βˆˆπ‘‹ such that π‘¦βˆˆcint{π‘’βˆˆπ‘ŒβˆΆπ‘“(π‘₯,𝑒)>𝛾}. 𝑓 is said to be transfer compactly u.s.c in 𝑦 if βˆ’π‘“ is transfer compactly l.s.c in 𝑦.

Now, we establish the following 𝐾𝐾𝑀-type theorem for a π‘˜-set contraction map.

Theorem 4.3. Let (π‘Œ,πœ‘π‘) be a bounded 𝐹𝐢-space and let 𝑋 be a topological space. If 𝑇,πΉβˆΆπ‘Œβ†’2𝑋 are two set-valued mappings satisfying the following:(i)π‘‡βˆˆπΎπΎπ‘€(π‘Œ,𝑋) is a π‘˜-set contraction map, 0<π‘˜<1, with 𝑇(π‘Œ)βŠ‚π‘‹,(ii)for any π‘¦βˆˆπ‘Œ, 𝐹(𝑦) is compactly closed in 𝑋,(iii)𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇.
Then π‘‡ξ€·π‘Œβˆžξ€Έξ™ξ‚€ξ™ξ€½πΉ(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆžξ€Ύξ‚β‰ βˆ…,(4.1) where π‘Œβˆž is the precompact-inducing 𝐹𝐢-subspace of π‘Œ.

Proof. Let {𝑉𝑖}π‘–βˆˆπΌ be a basis of symmetric open entourages for the uniformity 𝒰. By the same process of the proof Theorem 3.10, we get a compact subset 𝑇(π‘Œβˆž) of π‘Œ, and 𝑇|π‘ŒβˆžβˆˆπΎπΎπ‘€(π‘Œβˆž,𝑋), since π‘‡βˆˆπΎπΎπ‘€(π‘Œ,𝑋).
Define π»βˆΆπ‘Œβˆžβ†’2𝑋 by 𝐻(𝑦)=π‘‡ξ€·π‘Œβˆžξ€Έξ™πΉ(𝑦)foreachπ‘¦βˆˆπ‘Œβˆž.(4.2) By condition (ii), 𝐻(𝑦) is compact in 𝑋, for each π‘¦βˆˆπ‘Œβˆž. We now claim that 𝐻 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘Œβˆž. Let π‘βˆˆβŸ¨π‘ŒβˆžβŸ© and for each {𝑒𝑖0,…,π‘’π‘–π‘˜}βŠ‚{𝑒0,…,𝑒𝑛}. By condition (iii), π‘‡βˆ£π‘Œβˆž(πœ‘π‘(Ξ”π‘˜))βŠ‚π‘‡(π‘Œβˆž)β‹‚(β‹ƒπ‘˜π‘—=0𝐹(𝑦𝑖𝑗⋃))=π‘˜π‘—=0𝐻(𝑦𝑖𝑗). Thus, we have shown that 𝐻 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘Œβˆž. Since π‘‡βˆ£π‘ŒβˆžβˆˆπΎπΎπ‘€(π‘Œβˆž,𝑋), the family {𝐻(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆž} has the finite intersection property. And, since 𝐻(𝑦) is compact, β‹‚π‘¦βˆˆπ‘Œβˆžπ»(𝑦)β‰ βˆ…, that is, 𝑇(π‘Œβˆž)β‹‚(β‹‚{𝐹(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆž})β‰ βˆ…. This completes the proof.

Theorem 4.4. Let (π‘Œ,πœ‘π‘) be a bounded 𝐹𝐢-space and let 𝑋 be a topological space. If 𝑇,πΉβˆΆπ‘Œβ†’2𝑋 are two set-valued mappings satisfying the following:(i)π‘‡βˆˆπΎπΎπ‘€(π‘Œ,𝑋) is a π‘˜-set contraction map, 0<π‘˜<1, with 𝑇(π‘Œ)βŠ‚π‘‹,(ii)for any π‘¦βˆˆπ‘Œ, 𝐹(𝑦) is transfer compactly closed in 𝑋,(iii)𝐹 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇.
Then π‘‡ξ€·π‘Œβˆžξ€Έξ™ξ‚€ξ™ξ€½πΉ(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆžξ€Ύξ‚β‰ βˆ…,(4.3) where π‘Œβˆž is the precompact-inducing 𝐹𝐢-subspace of π‘Œ.

Proof. Define a mapping cclπΉβˆΆπ‘Œβˆžβ†’2𝑋 by (ccl𝐹)(𝑦)=ccl𝐹(𝑦) for each π‘¦βˆˆπ‘Œβˆž, it is easy to see that ccl𝐹 is also a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to π‘‡βˆ£π‘Œβˆž with compactly closed values. By the same process of the proof Theorem 3.10, we get a compact subset 𝑇(π‘Œβˆž) of π‘Œ. By Theorem 4.3, 𝑇(π‘Œβˆž)β‹‚(β‹‚{ccl𝐹(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆž})β‰ βˆ…. And since for any π‘¦βˆˆπ‘Œ, 𝐹(𝑦) is transfer compactly closed in 𝑋, by Lemma 2.2 [9], we have 𝑇(π‘Œβˆž)β‹‚(β‹‚{𝐹(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆž})=𝑇(π‘Œβˆž)β‹‚(β‹‚{ccl𝐹(𝑦)βˆΆπ‘¦βˆˆπ‘Œβˆž})β‰ βˆ…. This completes the proof.

Remark 4.5. Theorem 4.3 generalizes [5, Theorem 3] from a nonempty bounded convex subset of a Hausdorff topological vector space to a bounded 𝐹𝐢-space and [9, Theorem 3.1]. Theorem 4.4 generalizes [5, Theorem 3] in several aspects and [9, Theorem 3.2].

The following results are the generalization of the Ky Fan matching theorem and coincidence theorems.

Theorem 4.6. Let (𝑋,πœ‘π‘) be a bounded 𝐹𝐢-space. If 𝑇,π»βˆΆπ‘‹β†’2𝑋 are two set-valued mappings satisfying the following:(i)π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋) is a π‘˜-set contraction map, 0<π‘˜<1, with 𝑇(𝑋)βŠ‚π‘‹,(ii)for any π‘₯βˆˆπ‘‹, 𝐻(π‘₯) is compactly open in 𝑋,(iii)for the precompact-inducing 𝐹𝐢-subspace π‘‹βˆž of 𝑋, 𝑇(π‘‹βˆž)βŠ‚π»(π‘‹βˆž).
Then for the precompact-inducing 𝐹𝐢-subspace π‘‹βˆž of 𝑋 satisfying the following condition: π‘‡ξ€·π‘‹βˆžξ€Έξ™ξ‚€ξ™ξ‚{𝐻(π‘₯)∢π‘₯βˆˆπ‘€}β‰ βˆ…forsomeπ‘€βˆˆβŸ¨π‘‹βˆžβŸ©.(4.4)

Proof. Let {𝑉𝑖}π‘–βˆˆπΌ be a basis of symmetric open entourages for the uniformity 𝒰. By the same process of the proof Theorem 3.10, we get a compact subset 𝑇(π‘‹βˆž) of 𝑋, and 𝑇|π‘‹βˆžβˆˆπΎπΎπ‘€(π‘‹βˆž,𝑋), since π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋).
We claim that there exists π‘€βˆˆβŸ¨π‘‹βˆžβŸ© such that 𝑇(π‘‹βˆž)β‹‚(β‹‚{𝐻(π‘₯)∢π‘₯βˆˆπ‘€})β‰ βˆ…. On the contrary, assume that 𝑇(π‘‹βˆž)β‹‚(β‹‚{𝐻(π‘₯)∢π‘₯βˆˆπ‘€})=βˆ… for any π‘€βˆˆβŸ¨π‘‹βˆžβŸ©, then 𝑇(π‘‹βˆžβ‹‚)βŠ‚π‘₯βˆˆπ‘€π»π‘(π‘₯). Since π‘‹βˆž is 𝐹𝐢-subspace of 𝑋, π‘€βˆˆβŸ¨π‘‹βˆžβŸ© and for any {π‘₯𝑖0,…,π‘₯π‘–π‘˜}βŠ‚π‘€, we have 𝑇(πœ‘π‘(Ξ”π‘˜))βŠ‚π‘‡(π‘‹βˆžβ‹‚)βŠ‚π‘₯βˆˆπ‘€π»π‘(π‘₯), where Ξ”π‘˜=co({π‘’π‘–π‘—βˆΆπ‘—=0,…,π‘˜}). This implies that 𝐻𝑐 is a generalized 𝐹-𝐾𝐾𝑀 mapping with respect to 𝑇. By condition (ii), for any π‘₯βˆˆπ‘‹, 𝐻𝑐(π‘₯) is compactly closed in 𝑋. Follows Theorem 4.3, we have 𝑇(π‘‹βˆž)β‹‚(β‹‚{𝐻𝑐(π‘₯)∢π‘₯βˆˆπ‘‹βˆž})β‰ βˆ…, which implies 𝑇(π‘‹βˆž)ΜΈβŠ†β‹ƒπ‘₯βˆˆπ‘‹βˆžπ»(π‘₯), a contradiction to condition (iii). This completes the proof.

Theorem 4.7. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space. Assume that(i)π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋) is a π‘˜-set contraction map, 0<π‘˜<1, with 𝑇(𝑋)βŠ‚π‘‹,(ii)πΉβˆΆπ‘‹β†’2𝑋 is a Ξ¦-mapping.
Then there exists (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹ such that π‘¦βˆˆπ‘‡(π‘₯) and π‘₯∈𝐹(𝑦).

Proof. By the same process of the proof Theorem 3.10, we get a compact subset 𝑇(π‘‹βˆž) of 𝑋, and 𝑇|π‘‹βˆžβˆˆπΎπΎπ‘€(π‘‹βˆž,𝑋), since π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋).
Let 𝐾=𝑇(π‘‹βˆž) is compact. Then 𝐹∣𝐾 is a Ξ¦-mapping, and by Lemma 3.3, 𝐹∣𝐾 has a continuous selection π‘“βˆΆπΎβ†’π‘‹. So, by Lemma 3.5, we have π‘“π‘‡βˆ£π‘‹βˆžβˆˆπΎπΎπ‘€(π‘‹βˆž,𝑋) and so, it follows from Theorem 3.10 that there exists π‘₯βˆˆπ‘‡(π‘‹βˆž) such that π‘₯βˆˆπ‘“π‘‡(π‘₯)βŠ‚πΉπ‘‡(π‘₯); that is, there exists π‘¦βˆˆπ‘‡(π‘₯) such that π‘₯∈𝐹(𝑦). This completes the proof

Theorem 4.8. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space, and let π‘Œ be a topological space. Assume that(i)π‘‡βˆˆπΎπΎπ‘€(𝑋,π‘Œ) is compact and closed,(ii)πΉβˆΆπ‘Œβ†’2𝑋 is a Ξ¦-mapping.
Then there exists (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘Œ such that π‘¦βˆˆπ‘‡(π‘₯) and π‘₯∈𝐹(𝑦).

Proof. Since 𝑇 is compact, we have 𝐾=𝑇(𝑋) is compact in π‘Œ. By condition (ii), we have 𝐹∣𝐾 is also a Ξ¦-mapping. By Lemma 3.3, 𝐹∣𝐾 has a continuous selection π‘“βˆΆπΎβ†’π‘‹. So, by Lemma 3.5, we have π‘“π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋), and so, it follows from Corollary 3.7 that there exists π‘₯βˆˆπ‘‹ such that π‘₯βˆˆπ‘“π‘‡(π‘₯)βŠ‚πΉπ‘‡(π‘₯); that is, there exists π‘¦βˆˆπ‘‡(π‘₯) such that π‘₯∈𝐹(𝑦). This completes the proof.

As a consequence of the above Theorem 4.6, we have the following generalized variational inequality.

Corollary 4.9. Let (𝑋,πœ‘π‘) be a bounded 𝐹𝐢-space, and let π‘‡βˆˆπΎπΎπ‘€(𝑋,𝑋) be a π‘˜-set contraction map, 0<π‘˜<1, with 𝑇(𝑋)βŠ‚π‘‹. If πœ‘,πœ“βˆΆπ‘‹Γ—π‘‹β†’π‘…βˆͺ{βˆ’βˆž,+∞} are two real-valued mappings satisfying the following:(i)πœ“(π‘₯,𝑦)≀0 for each (π‘₯,𝑦)βˆˆπ’’π‘‡,(ii)for fixed π‘₯βˆˆπ‘‹, the mapping π‘¦β†¦πœ‘(π‘₯,𝑦) is lower semicontinuous on 𝐾 for each compact subset 𝐾 of 𝑋,(iii)for fixed π‘¦βˆˆπ‘‹, for each 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βŸ© and 𝑁1={π‘₯𝑖0,…,π‘₯π‘–π‘˜}βŠ‚π‘, π‘₯βˆˆπœ‘π‘(Ξ”π‘˜) such that πœ“(π‘₯,𝑦)≀0 implies that there exists π‘₯βˆˆπ‘1 such that πœ‘(π‘₯,𝑦)≀0.
Then for the precompact-inducing 𝐹𝐢-subspace π‘‹βˆž of 𝑋, there exists π‘¦βˆˆπ‘‹βˆž such that πœ‘(π‘₯,𝑦)≀0 for each π‘₯βˆˆπ‘‹βˆž.

Proof. Define 𝐹,π‘†βˆΆπ‘‹β†’2𝑋 by 𝑆(π‘₯)={π‘¦βˆˆπ‘‹βˆΆπœ“(π‘₯,𝑦)≀0}foreach𝐹π‘₯βˆˆπ‘‹,(π‘₯)={π‘¦βˆˆπ‘‹βˆΆπœ‘(π‘₯,𝑦)≀0}foreachπ‘₯βˆˆπ‘‹.(4.5) By condition (i), we have π’’π’―βŠ‚π’’π‘†, and by condition (ii), 𝐹(π‘₯) is compactly closed for each π‘₯βˆˆπ‘‹. The condition (iii) implies that for each 𝑁={π‘₯0,…,π‘₯𝑛}βˆˆβŸ¨π‘‹βŸ© and 𝑁1={π‘₯𝑖0,…,π‘₯π‘–π‘˜}βŠ‚π‘, 𝑆(πœ‘π‘(Ξ”π‘˜β‹ƒ))βŠ‚π‘˜π‘—=0𝐹(π‘₯𝑖𝑗), and then 𝑇(πœ‘π‘(Ξ”π‘˜β‹ƒ))βŠ‚π‘˜π‘—=0𝐹(π‘₯𝑖𝑗); that is, 𝐹 is a generalized 𝐹-KKM mapping with respect to 𝑇.
Thus, all conditions of Theorem 4.3 are satisfied. By Theorem 4.3, and for the precompact-inducing 𝐹𝐢-subspace π‘‹βˆž of 𝑋, we have that 𝑇(π‘‹βˆž)β‹‚(β‹‚{𝐹(π‘₯)∢π‘₯βˆˆπ‘‹βˆž})β‰ βˆ…. Let π‘¦βˆˆπ‘‡(π‘‹βˆž)β‹‚(β‹‚{𝐹(π‘₯)∢π‘₯βˆˆπ‘‹βˆž}), and hence we have πœ‘(π‘₯,𝑦)≀0 for each π‘₯βˆˆπ‘‹βˆž. This completes the proof.

Applying Theorem 4.8 and Lemma 3.4, we have the following result.

Corollary 4.10. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space, and let π‘Œ be a topological space. If π‘‡βˆΆπ‘‹β†’2π‘Œ, πΉβˆΆπ‘Œβ†’2𝑋 are two Ξ¦-mappings, and 𝑇 is compact and closed, then there exists (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘Œ such that π‘¦βˆˆπ‘‡(π‘₯) and π‘₯∈𝐹(𝑦).

From Corollary 4.10, we have the following result.

Corollary 4.11. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space, and let (π‘Œ,πœ‘π‘€) be a compact 𝐹𝐢-space. If 𝑓,𝑔,𝑝,π‘žβˆΆπ‘‹Γ—π‘Œβ†’π‘… are four real-valued functions, and π‘Ž,𝑏 be two real numbers. Suppose the following conditions hold:(i)𝑔(π‘₯,𝑦)≀𝑓(π‘₯,𝑦) and 𝑝(π‘₯,𝑦)β‰€π‘ž(π‘₯,𝑦) for all π‘₯βˆˆπ‘‹,π‘¦βˆˆπ‘Œ,(ii)for each π‘₯βˆˆπ‘‹, 𝑓 be 𝐹𝐢-quasiconcave on π‘Œ and for each π‘¦βˆˆπ‘Œ,𝑝 be 𝐹𝐢-quasiconvex on 𝑋,(iii)for each π‘₯βˆˆπ‘‹, 𝑔 be transfer compactly lower semicontinuous in π‘₯ and for each π‘¦βˆˆπ‘Œ, π‘ž be transfer compactly upper semicontinuous in 𝑦,(iv)𝑓 is upper semicontinuous on π‘‹Γ—π‘Œ.
Then one of the following statements holds:(1)there exists πœ‡βˆˆπ‘‹ such that 𝑔(πœ‡,𝑦)β‰€π‘Ž for each π‘¦βˆˆπ‘Œ,(2)there exists πœˆβˆˆπ‘Œ such that π‘ž(π‘₯,𝜈)β‰₯𝑏 for each π‘₯βˆˆπ‘‹,(3)there exists (πœ‡,𝜈)βˆˆπ‘‹Γ—π‘Œ such that 𝑓(πœ‡,𝜈)β‰₯π‘Ž and 𝑝(πœ‡,𝜈)≀𝑏.

Proof. Let 𝑆,π‘‡βˆΆπ‘‹β†’2π‘Œ and 𝐻,πΉβˆΆπ‘Œβ†’2𝑋 be defined by 𝑆(π‘₯)={π‘¦βˆˆπ‘ŒβˆΆπ‘”(π‘₯,𝑦)βˆ’π‘Ž>0}foreach𝑇π‘₯βˆˆπ‘‹,(π‘₯)={π‘¦βˆˆπ‘ŒβˆΆπ‘“(π‘₯,𝑦)βˆ’π‘Žβ‰₯0}foreachπ‘₯βˆˆπ‘‹,𝐻(𝑦)={π‘₯βˆˆπ‘‹βˆΆπ‘ž(π‘₯,𝑦)βˆ’π‘<0}foreachπ‘¦βˆˆπ‘Œ,𝐹(𝑦)={π‘₯βˆˆπ‘‹βˆΆπ‘(π‘₯,𝑦)βˆ’π‘β‰€0}foreachπ‘¦βˆˆπ‘Œ.(4.6) By the condition (iii), π‘†βˆ’1 is transfer compactly open valued on π‘Œ and π»βˆ’1 is transfer compactly open valued on 𝑋. By the condition (i), we have 𝑆(π‘₯)βŠ‚π‘‡(π‘₯) for each π‘₯βˆˆπ‘‹ and 𝐻(𝑦)βŠ‚πΉ(𝑦) for each π‘¦βˆˆπ‘Œ. By the condition (ii), 𝑇(π‘₯) is 𝐹𝐢-subspace of π‘Œ for each π‘₯βˆˆπ‘‹, and so 𝐹(π‘₯) is 𝐹𝐢-subspace of 𝑋 for each π‘¦βˆˆπ‘Œ.
Suppose that the Statements (1) and (2) are false. Then 𝑆(π‘₯)β‰ βˆ… for each π‘₯βˆˆπ‘‹ and 𝐻(𝑦)β‰ βˆ… for each π‘¦βˆˆπ‘Œ. So, we conclude that 𝑇 is a Ξ¦-mapping with a companion mapping 𝑆 and 𝐹 is a Ξ¦-mapping with a companion mapping 𝐻. By the Condition (iv), 𝑇 is closed. Thus, all conditions of Corollary 4.9 are satisfied. By Corollary 4.9, there exists (πœ‡,𝜈)βˆˆπ‘‹Γ—π‘Œ such that πœˆβˆˆπ‘‡(πœ‡) and πœ‡βˆˆπΉ(𝜈); that is, 𝑓(πœ‡,𝜈)β‰₯π‘Ž and 𝑝(πœ‡,𝜈)≀𝑏. This completes the proof.

From Corollary 4.11, we have the following minimax theorem.

Corollary 4.12. Let (𝑋,𝒰,πœ‘π‘) be a locally 𝐹𝐢-space, and let (π‘Œ,πœ‘π‘€) be a compact 𝐹𝐢-space. If 𝑓,𝑔,𝑝,π‘žβˆΆπ‘‹Γ—π‘Œβ†’π‘… are four real-valued functions, and let π‘Ž,𝑏 be two real numbers. Suppose the following conditions hold(i)𝑔(π‘₯,𝑦)≀𝑓(π‘₯,𝑦)≀𝑝(π‘₯,𝑦)β‰€π‘ž(π‘₯,𝑦) for all π‘₯βˆˆπ‘‹,π‘¦βˆˆπ‘Œ,(ii)for each π‘₯βˆˆπ‘‹, let 𝑓 be 𝐹𝐢-quasiconcave on π‘Œ and for each π‘¦βˆˆπ‘Œ, let 𝑝 be 𝐹𝐢-quasiconvex on 𝑋,(iii)for each π‘₯βˆˆπ‘‹, let 𝑔 be transfer compactly lower semicontinuous in π‘₯ and for each π‘¦βˆˆπ‘Œ, let π‘ž be transfer compactly upper semicontinuous in 𝑦, (iv)𝑓 is upper semicontinuous on π‘‹Γ—π‘Œ.
Then infπ‘₯βˆˆπ‘‹supπ‘¦βˆˆπ‘Œπ‘”(π‘₯,𝑦)≀supπ‘¦βˆˆπ‘Œinfπ‘₯βˆˆπ‘‹π‘ž(π‘₯,𝑦).(4.7)

Proof. Let πœ€>0 and let π‘Ž=infπ‘₯βˆˆπ‘‹supπ‘¦βˆˆπ‘Œπ‘”(π‘₯,𝑦)βˆ’πœ€,𝑏=supπ‘¦βˆˆπ‘Œinfπ‘₯βˆˆπ‘‹π‘ž(π‘₯,𝑦)+πœ€.(4.8) Then for each π‘₯βˆˆπ‘‹, there exists π‘¦βˆˆπ‘Œ such that 𝑔(π‘₯,𝑦)>π‘Ž, and for each π‘¦βˆˆπ‘Œ, there exists π‘₯βˆˆπ‘‹ such that π‘ž(π‘₯,𝑦)<𝑏. Therefore, Conclusions (1) and (2) of Corollary 4.11 are false. So there exist (πœ‡,𝜈)βˆˆπ‘‹Γ—π‘Œ such that 𝑓(πœ‡,𝜈)β‰₯π‘Ž and 𝑝(πœ‡,𝜈)≀𝑏; that is 𝑓(πœ‡,𝜈)β‰₯infπ‘₯βˆˆπ‘‹supπ‘¦βˆˆπ‘Œπ‘”(π‘₯,𝑦)βˆ’πœ€,𝑝(πœ‡,𝜈)≀supπ‘¦βˆˆπ‘Œinfπ‘₯βˆˆπ‘‹π‘ž(π‘₯,𝑦)+πœ€.(4.9) So, by Condition (i), we have infπ‘₯βˆˆπ‘‹supπ‘¦βˆˆπ‘Œπ‘”(π‘₯,𝑦)βˆ’πœ€<supπ‘¦βˆˆπ‘Œinfπ‘₯βˆˆπ‘‹π‘ž(π‘₯,𝑦)+πœ€.(4.10) Since πœ€ is arbitrary positive number, by letting πœ€β†“0, we get infπ‘₯βˆˆπ‘‹supπ‘¦βˆˆπ‘Œπ‘”(π‘₯,𝑦)≀supπ‘¦βˆˆπ‘Œinfπ‘₯βˆˆπ‘‹π‘ž(π‘₯,𝑦).(4.11) This completes the proof.

Remark 4.13. Theorem 4.6 (resp., Theorems 4.7 and 4.8, Corollaries 4.9, 4.10, 4.11, and 4.12) generalize Theorem 4 (resp., Theorems 7, 6, 5, 8, 9, and 10) in [5] in several aspects.

Acknowledgment

The author’s research was supported by the Scientific Research Foundation of CUIT under Grant KYTZ201114.

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