Fixed Point Theory and Applications
VolumeΒ 2009Β (2009), Article IDΒ 791514, 7 pages
doi:10.1155/2009/791514
Research Article

Approximate Fixed Point Theorems for the Class of Almost 𝑆 - 𝐾 𝐾 𝑀 π’ž Mappings in Abstract Convex Uniform Spaces

Tong-Huei Chang,Β Chi-Ming Chen,Β and Yueh-Hung Huang

Department of Applied Mathematics, National Hsinchu University of Education, Hsin-Chu, Taiwan

Received 25 February 2009; Accepted 11 June 2009

Academic Editor: HichemΒ Ben-El-Mechaiekh

Copyright Β© 2009 Tong-Huei Chang et al. 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 use a concept of abstract convexity to define the almost 𝑆 - 𝐾 𝐾 𝑀 π’ž property, al- 𝑆 - 𝐾 𝐾 𝑀 π’ž ( 𝑋 , π‘Œ ) family, and almost Ξ¦ -spaces. We get some new approximate fixed point theorems and fixed point theorems in almost Ξ¦ -spaces. Our results extend some results of other authors.

1. Introduction and Preliminaries

In 1929, Knaster et al. [1] proved the well-known 𝐾 𝐾 𝑀 theorem for an 𝑛 -simplex. Ky Fan's generalization of the 𝐾 𝐾 𝑀 theorem to infinite dimensional topological vector spaces in 1961 [2] proved to be a very versatile tool in modern nonlinear analysis with many far-reaching applications.

Chang and Yen [3] undertook a systematic study of the 𝐾 𝐾 𝑀 property, and Chang et al. [4] generalized this property as well as the notion of a 𝐾 𝐾 𝑀 ( 𝑋 , π‘Œ ) family of [4] to the wider concepts of the 𝑆 - 𝐾 𝐾 𝑀 property and its related 𝑆 - 𝐾 𝐾 𝑀 ( 𝑋 , π‘Œ , 𝑍 ) family.

Among the many contributions in the study of the 𝐾 𝐾 𝑀 property and related topics, we mention the work by Amini et al. [5] where the classes of 𝐾 𝐾 𝑀 and 𝑆 - 𝐾 𝐾 𝑀 mappings have been introduced in the framework of abstract convex spaces. The authors of [5] also define a concept of convexity that contains a number of other concepts of abstract convexities and obtain fixed point theorems for multifunctions verifying the 𝑆 - 𝐾 𝐾 𝑀 property on Ξ¦ -spaces that extend results of Ben-El-Mechaiekh et al. [6] and Horvath [7], motivated by the works of Ky Fan [2] and Browder [8]. We refer for the study of these notions to Ben-El-Mechaiekh et al. [9], and more recently, to Park [10], and Kim and Park [11].

In this paper, we use a concept of abstract convexity to define the almost 𝑆 - 𝐾 𝐾 𝑀 π’ž property, the corresponding notion of almost 𝑆 - 𝐾 𝐾 𝑀 π’ž ( 𝑋 , π‘Œ ) family as well as the concept of almost Ξ¦ -spaces.

Let 𝑋 and π‘Œ be two sets, and let 𝑇 ∢ 𝑋 β†’ 2 π‘Œ be a set-valued mapping. We will use the following notations in the sequel;

(i) 𝑇 ( π‘₯ ) = { 𝑦 ∈ π‘Œ ∢ 𝑦 ∈ 𝑇 ( π‘₯ ) } , (ii) 𝑇 ( 𝐴 ) = βˆͺ π‘₯ ∈ 𝐴 𝑇 ( π‘₯ ) , (iii) 𝑇 βˆ’ 1 ( 𝑦 ) = { π‘₯ ∈ 𝑋 ∢ 𝑦 ∈ 𝑇 ( π‘₯ ) } , (iv) 𝑇 βˆ’ 1 ( 𝐡 ) = { π‘₯ ∈ 𝑋 ∢ 𝑇 ( π‘₯ ) ∩ 𝐡 β‰  πœ™ } , and(v)if 𝐷 is a nonempty subset of 𝑋 , then ⟨ 𝐷 ⟩ denotes the class of all nonempty finite subsets of 𝐷 .

For the case where 𝑋 and π‘Œ are two topological spaces, a set-valued map 𝑇 ∢ 𝑋 β†’ 2 π‘Œ is said to be closed if its graph 𝒒 𝑇 = { ( π‘₯ , 𝑦 ) ∈ 𝑋 Γ— π‘Œ ∢ 𝑦 ∈ 𝑇 ( 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 .

Definition 1.1. An abstract convex space ( 𝐸 , π’ž ) consists of a nonempty topological space 𝐸 , and a family π’ž of subsets of 𝐸 such that 𝐸 and βˆ… belong to π’ž and π’ž is closed under arbitrary intersection. This kind of abstract convexity was widely studied; see [5, 9, 12, 13].
Suppose that 𝐴 is a nonempty subset of an abstract convex space ( 𝐸 , π’ž ) . Then
(i)a natural definition of π’ž -convex hull of 𝐴 is 𝑐 π‘œ π’ž ( 𝐴 ) = ∩ { 𝐡 ∈ π’ž ∢ 𝐴 βŠ‚ 𝐡 } , a n d ( 1 . 1 ) (ii)we say that 𝐴 is π’ž -convex if for each 𝐡 ∈ ⟨ 𝐴 ⟩ , 𝑐 π‘œ π’ž ( 𝐡 ) βŠ‚ 𝐴 .

Remark 1.2. It is clear that if 𝐴 ∈ π’ž , then 𝐴 is π’ž -convex. That is, each member of π’ž is π’ž -convex.

Definition 1.3. We list some properties of a uniform space. A uniformity [14] for a set 𝐸 is a nonempty family 𝒰 of subsets of 𝐸 Γ— 𝐸 such that(i)each member of 𝒰 contains the diagonal Ξ” where the diagonal Ξ” denotes the set of all pairs ( π‘₯ , π‘₯ ) for π‘₯ in 𝐸 ;(ii)if π‘ˆ ∈ 𝒰 , then π‘ˆ βˆ’ 1 ∈ 𝒰 ;(iii)if π‘ˆ ∈ 𝒰 , then 𝑉 ∘ 𝑉 βŠ‚ π‘ˆ for some 𝑉 ∈ 𝒰 ;(iv)if π‘ˆ , 𝑉 ∈ 𝒰 , then π‘ˆ ∩ 𝑉 ∈ 𝒰 ;(v)if π‘ˆ ∈ 𝒰 and π‘ˆ βŠ‚ 𝑉 βŠ‚ 𝐸 Γ— 𝐸 , then 𝑉 ∈ 𝒰 . The pair ( 𝐸 , π’ž ) is called a uniform space. Every member 𝑉 in 𝒰 is called an entourage. An entourage is said to be symmetric if ( π‘₯ , 𝑦 ) ∈ 𝑉 whenever ( 𝑦 , π‘₯ ) ∈ 𝑉 .

Definition 1.4. If ( 𝐸 , π’ž ) is an abstract convex space with a uniformity 𝒰 , then we say that ( 𝐸 , π’ž ) is an abstract convex uniform space.

Definition 1.5. Let 𝐴 be a nonempty subset of an abstract convex uniform space ( 𝐸 , π’ž ) which has a uniformity 𝒰 , and 𝒰 has a symmetric basis 𝒩 . Then 𝐴 is called almost π’ž -convex if, for any 𝐾 ∈ ⟨ 𝐴 ⟩ and for any 𝑉 ∈ 𝒩 , there exists a mapping β„Ž 𝐾 , 𝑉 ∢ 𝐾 β†’ 𝐴 such that β„Ž 𝐾 , 𝑉 ( π‘₯ ) ∈ 𝑉 [ π‘₯ ] for all π‘₯ ∈ 𝐾 and 𝑐 π‘œ π’ž ( β„Ž 𝐾 , 𝑉 ( 𝐾 ) ) βŠ‚ 𝐴 . Moreover, we call the mapping β„Ž 𝐾 , 𝑉 ∢ 𝐾 β†’ 𝐴 a π’ž -convex-inducing mapping.

Remark 1.6. It is clear that every π’ž -convex set must be almost π’ž -convex, but the converse is not true. And in general, the π’ž -convex-inducing mapping is not unique. If π‘ˆ , 𝑉 ∈ 𝒩 and π‘ˆ βŠ‚ 𝑉 , then β„Ž 𝐴 , π‘ˆ ∢ 𝐴 β†’ 𝑋 can be regarded as β„Ž 𝐴 , 𝑉 ∢ 𝐴 β†’ 𝑋 . If 𝐴 βŠ‚ 𝐡 , then β„Ž 𝐴 , π‘ˆ ∢ 𝐴 β†’ 𝑋 can be regarded as β„Ž 𝐡 , π‘ˆ ∢ 𝐡 β†’ 𝑋 .

Recently, Amini et al. [5] introduced the class of multifunctions with the 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž property in abstract convex spaces.

Definition 1.7 (see [5]). Let 𝑍 be a nonempty set, ( 𝑋 , π’ž ) an abstract convex space, and π‘Œ a topological space. If 𝑆 ∢ 𝑍 β†’ 2 𝑋 , 𝑇 ∢ 𝑋 β†’ 2 π‘Œ and 𝐹 ∢ 𝑍 β†’ 2 π‘Œ are three multifunctions satisfying 𝑇 ξ€· 𝑐 π‘œ π’ž ξ€Έ ( 𝑆 ( 𝐴 ) ) βŠ‚ βˆͺ π‘₯ ∈ 𝐴 𝐹 ( π‘₯ ) , f o r e a c h 𝐴 ∈ ⟨ 𝑍 ⟩ , ( 1 . 2 ) then 𝐹 is called a 𝑆 - 𝐾 𝐾 𝑀 π’ž mapping with respect to 𝑇 . If the multifunction 𝑇 ∢ 𝑋 β†’ 2 π‘Œ satisfies the requirement that for any 𝑆 - 𝐾 𝐾 𝑀 π’ž mapping 𝐹 with respect to 𝑇 , the family { 𝐹 ( π‘₯ ) ∢ π‘₯ ∈ 𝑍 } has the finite intersection property where 𝐹 ( π‘₯ ) denotes the closure of 𝐹 ( π‘₯ ) , then 𝑇 is said to have the 𝑆 - 𝐾 𝐾 𝑀 π’ž property with respect to π’ž . We define 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž ξ€½ ( 𝑍 , 𝑋 , π‘Œ ) ∢ = 𝑇 ∢ π‘Š β†’ 2 π‘Œ ∣ 𝑇 h a s t h e 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž p r o p e r t y w i t h r e s p e c t t o π’ž ξ€Ύ . ( 1 . 3 )

We extended the 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž property to the almost 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž property, as follows.

Definition 1.8. Let 𝑍 be a nonempty set, let 𝑋 be an almost π’ž -convex subset of an abstract convex uniform space ( 𝐸 , π’ž ) which has a uniformity 𝒰 and 𝒰 has a symmetric basis 𝒩 , and let π‘Œ be a topological space. If 𝑆 ∢ 𝑍 β†’ 2 𝑋 , 𝑇 ∢ 𝑋 β†’ 2 π‘Œ and 𝐹 ∢ 𝑍 β†’ 2 π‘Œ are three multifunctions satisfying for each 𝐴 ∈ ⟨ 𝑍 ⟩ , each 𝐡 ∈ ⟨ 𝑆 ( 𝐴 ) ⟩ , and each π‘ˆ ∈ 𝒩 , there exists a π’ž -convex-inducing mapping β„Ž 𝐡 , π‘ˆ ∢ 𝐡 β†’ π‘Š such that 𝑇 ξ€· 𝑐 π‘œ π’ž ξ€· β„Ž 𝐡 , π‘ˆ ( 𝐡 ) ξ€Έ ξ€Έ βŠ‚ 𝐹 ( 𝐴 ) , ( 1 . 4 ) then 𝐹 is called an almost 𝑆 - 𝐾 𝐾 𝑀 π’ž mapping with respect to 𝑇 . If the multifunction 𝑇 ∢ 𝑋 β†’ 2 π‘Œ satisfies the requirement that for any almost 𝑆 - 𝐾 𝐾 𝑀 π’ž mapping 𝐹 with respect to 𝑇 , the family { 𝐹 ( π‘₯ ) ∢ π‘₯ ∈ 𝑍 } has the finite intersection property, then 𝑇 is said to have the almost 𝑆 - 𝐾 𝐾 𝑀 π’ž property with respect to π’ž . We define π‘Ž 𝑙 βˆ’ 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž ξ€½ ( 𝑍 , 𝑋 , π‘Œ ) ∢ = 𝑇 ∢ π‘Š β†’ 2 π‘Œ ∣ 𝑇 h a s t h e a l m o s t 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž p r o p e r t y w i t h r e s p e c t t o π’ž ξ€Ύ . ( 1 . 5 )

From the above definitions, we have the following proposition of the π‘Ž 𝑙 βˆ’ 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑍 , 𝑋 , π‘Œ ) family.

Proposition 1.9. Let 𝑋 be a nonempty set, let π‘Œ be an almost π’ž -convex subset of an abstract convex uniform space ( 𝐸 , π’ž ) , let 𝑍 and π‘Š be two topological spaces, and let 𝑆 ∢ 𝑋 β†’ 2 π‘Œ be a multifunction. If 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , π‘Œ , 𝑍 ) and if 𝑓 ∢ 𝑍 β†’ π‘Š is continuous, then 𝑓 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , π‘Œ , π‘Š ) .

The Ξ¦ -mappings and the Ξ¦ -spaces, in an abstract convex space setting, were also introduced by Amini et al. [5].

Definition 1.10 (see [5]). Let ( 𝑋 , π’ž ) be an abstract convex space, and π‘Œ a topological space. 𝐴 map 𝑇 ∢ π‘Œ β†’ 2 𝑋 is called a Ξ¦ -mapping if there exists a multifunction 𝐹 ∢ π‘Œ β†’ 2 𝑋 such that(i)for each 𝑦 ∈ π‘Œ , 𝐴 ∈ ⟨ 𝐹 ( 𝑦 ) ⟩ implies 𝑐 π‘œ π’ž ( 𝐴 ) βŠ‚ 𝑇 ( 𝑦 ) , and(ii) π‘Œ = βˆͺ π‘₯ ∈ 𝑋 i n t 𝐹 βˆ’ 1 ( π‘₯ ) . The mapping 𝐹 is called a companion mapping of 𝑇 .
Furthermore, if the abstract convex space ( 𝑋 , π’ž ) which has a uniformity 𝒰 and 𝒰 has a symmetric basis 𝒩 , then 𝑋 is called a Ξ¦ -space if for each entourage 𝑉 ∈ 𝒩 , there exists a Ξ¦ -mapping 𝑇 ∢ 𝑋 β†’ 2 𝑋 such that 𝒒 𝑇 βŠ‚ 𝑉 .

Remark 1.11. (i)If 𝑇 ∢ π‘Œ β†’ 2 𝑋 is a Ξ¦ -mapping, then for each nonempty subset π‘Œ 1 of π‘Œ , 𝑇 | π‘Œ 1 ∢ π‘Œ 1 β†’ 𝑋 is also a Ξ¦ -mapping.(ii)It is easy to see that if 𝑋 1 βŠ‚ 𝑋 and π’ž 1 = { 𝐢 ∩ 𝑋 1 ∢ 𝐢 ∈ π’ž } , then ( 𝑋 1 , π’ž 1 ) is also a Ξ¦ -space.

In order to establish the main result of this paper for the multifunctions with the almost 𝑆 βˆ’ 𝐾 𝐾 𝑀 π’ž property, we need the following definitions concerning the almost Ξ¦ -mappings and the almost Ξ¦ -spaces.

Definition 1.12. Let 𝑋 be an almost π’ž -convex subset of an abstract convex uniform space ( 𝐸 , π’ž ) which has a uniformity 𝒰 and 𝒰 has a symmetric base family 𝒩 , and π‘Œ a topological space. A map 𝑇 ∢ π‘Œ β†’ 2 𝑋 is called an almost Ξ¦ -mapping if there exists a multifunction 𝐹 ∢ π‘Œ β†’ 2 𝑋 such that(i)for each 𝑦 ∈ π‘Œ , 𝐴 ∈ ⟨ 𝐹 ( 𝑦 ) ⟩ and π‘ˆ ∈ 𝒩 , there exists a π’ž -convex-inducing β„Ž 𝐴 , π‘ˆ ∢ 𝐴 β†’ 𝑋 such that 𝑐 π‘œ π’ž ( β„Ž 𝐴 , π‘ˆ ( 𝐴 ) ) βŠ‚ π‘ˆ [ 𝑇 ( 𝑦 ) ] , and(ii) π‘Œ = βˆͺ π‘₯ ∈ 𝑋 i n t 𝐹 βˆ’ 1 ( π‘₯ ) . The mapping 𝐹 is called an almost companion mapping of 𝑇 .
Furthermore, 𝑋 is called an almost Ξ¦ -space, if, for each entourage 𝑉 ∈ 𝒩 , there exists an almost Ξ¦ -mapping 𝑇 ∢ 𝑋 β†’ 2 𝑋 such that 𝒒 𝑇 βŠ‚ 𝑉 .

Definition 1.13. Let 𝑋 be an almost Ξ¦ -space, and let 𝑇 ∢ 𝑋 β†’ 2 𝑋 . We say that 𝑇 has the approximate fixed point property if, for each π‘ˆ ∈ 𝒩 , there exists π‘₯ ∈ 𝑋 such that π‘ˆ [ π‘₯ ] ∩ 𝑇 ( π‘₯ ) β‰  πœ™ .

2. Main Results

Using the above introduced concepts and definitions, we now state our main theorem.Theorem 2.1. Let 𝑋 be an almost Ξ¦ -space, and let 𝑠 ∢ 𝑋 β†’ 𝑋 be a surjective single-valued function. If 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , 𝑋 , 𝑋 ) is compact, then 𝑇 has the approximate fixed point property.

Proof. Let 𝒩 be a symmetric basis of the uniform structure, and let π‘ˆ ∈ 𝒩 . Take 𝑉 ∈ 𝒩 such that 𝑉 ∘ 𝑉 βŠ‚ π‘ˆ . Then, by the definition of the almost Ξ¦ -space, there exists an almsot Ξ¦ -mapping 𝐹 ∢ 𝑋 β†’ 2 𝑋 such that 𝒒 𝐹 βŠ‚ 𝑉 . Since 𝐹 is an almsot Ξ¦ -mapping, there exists an almost companion mapping 𝐺 ∢ 𝑋 β†’ 2 𝑋 such that 𝑋 = βˆͺ π‘₯ ∈ 𝑋 i n t 𝐺 βˆ’ 1 ( π‘₯ ) .
Let 𝐾 = 𝑇 ( 𝑋 ) . Then 𝐾 is compact, since 𝑇 is compact. Hence there exists 𝐴 ∈ ⟨ 𝑋 ⟩ such that 𝐾 βŠ‚ βˆͺ π‘₯ ∈ 𝐴 i n t 𝐺 βˆ’ 1 ( π‘₯ ) . Since 𝑠 is surjective, there exists a finite subset 𝐡 of 𝑋 such that 𝐾 βŠ‚ βˆͺ 𝑧 ∈ 𝐡 i n t 𝐺 βˆ’ 1 ( 𝑠 ( 𝑧 ) ) .
Now, we define 𝑃 ∢ 𝑋 β†’ 2 𝑋 by
𝑃 ( 𝑧 ) = 𝐾 ⧡ i n t 𝐺 βˆ’ 1 ( 𝑠 ( 𝑧 ) ) , f o r e a c h 𝑧 ∈ 𝑋 . ( 2 . 1 ) By the definition of 𝑃 , we obtain that 𝑃 is not an almost 𝑠 βˆ’ 𝐾 𝐾 𝑀 π’ž mapping with respect to 𝑇 . Hence, there exist 𝑁 = { 𝑧 1 , 𝑧 2 , … , 𝑧 π‘˜ } βŠ‚ 𝑋 and 𝐷 ∈ ⟨ 𝑠 ( 𝑁 ) ⟩ such that for any π’ž -convex-inducing β„Ž 𝐷 , 𝑉 ∢ 𝐷 β†’ π‘Š ∞ , we have 𝑇 ξ€· 𝑐 π‘œ π’ž ξ€· β„Ž 𝐷 , 𝑉 ( ΜΈ 𝐷 ) ξ€Έ ξ€Έ βŠ† βˆͺ π‘˜ 𝑖 = 1 𝑃 ξ€· 𝑧 𝑖 ξ€Έ . ( 2 . 2 )
So, for any π’ž -convex-inducing β„Ž 𝐷 , 𝑉 ∢ 𝐷 β†’ 𝑋 , there exist π‘₯ π‘ˆ ∈ 𝑐 π‘œ π’ž ( β„Ž 𝐷 , 𝑉 ( 𝐷 ) ) and 𝑦 π‘ˆ ∈ 𝑇 ( π‘₯ π‘ˆ ) such that 𝑦 π‘ˆ βˆ‰ βˆͺ π‘˜ 𝑖 = 1 𝑃 ( 𝑧 𝑖 ) . Consequently, 𝑦 π‘ˆ ∈ ∩ π‘˜ 𝑖 = 1 i n t 𝐺 βˆ’ 1 ( 𝑠 ( 𝑧 𝑖 ) ) , and so 𝑠 ( 𝑧 𝑖 ) ∈ 𝐺 ( 𝑦 π‘ˆ ) for all 𝑖 = 1 , 2 , … , π‘˜ . Since 𝐹 is an almost Ξ¦ -mapping, there exists a π’ž -convex-inducing β„Ž βˆ— 𝐷 , 𝑉 ∢ 𝐷 β†’ 𝑋 such that 𝑐 π‘œ π’ž ( β„Ž βˆ— 𝐷 , 𝑉 ( 𝐷 ) ) βŠ‚ 𝑉 [ 𝐹 ( 𝑦 π‘ˆ ) ] . So π‘₯ π‘ˆ ∈ π‘Ž 𝑑 π’ž ( β„Ž βˆ— 𝐷 , 𝑉 ( 𝐷 ) ) and π‘₯ π‘ˆ ∈ 𝑉 [ 𝐹 ( 𝑦 π‘ˆ ) ] . Thus, there exists 𝑧 π‘ˆ ∈ 𝐹 ( 𝑦 π‘ˆ ) such that π‘₯ π‘ˆ ∈ 𝑉 [ 𝑧 π‘ˆ ] . Since 𝑋 is an almost Ξ¦ -space, we have ( 𝑦 π‘ˆ , 𝑧 π‘ˆ ) ∈ 𝒒 𝐹 βŠ‚ 𝑉 , and so ( 𝑦 π‘ˆ , π‘₯ π‘ˆ ) = ( 𝑦 π‘ˆ , 𝑧 π‘ˆ ) ∘ ( 𝑧 π‘ˆ , π‘₯ π‘ˆ ) ∈ 𝑉 ∘ 𝑉 βŠ‚ π‘ˆ , that is, 𝑦 π‘ˆ ∈ π‘ˆ [ π‘₯ π‘ˆ ] .
Therefore, 𝑦 π‘ˆ ∈ π‘ˆ [ π‘₯ π‘ˆ ] ∩ 𝑇 ( π‘₯ π‘ˆ ) . The proof is finished.

Remark 2.2. In the case, if 𝑋 is a Ξ¦ -space and 𝑇 ∈ 𝑠 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , 𝑋 , 𝑋 ) , then the above theorem reduces to Amini et al. [5, Theorem 2.5]

From Theorem 2.1 above, we obtain immediately the following fixed point theorem.

Theorem 2.3. Suppose that all of the assumptions of Theorem 2.1 hold. If 𝑇 is closed, then 𝑇 has a fixed point in 𝑋 .

Proof. By Theorem 2.1, for each π‘ˆ ∈ 𝒩 , there exist π‘₯ π‘ˆ , 𝑦 π‘ˆ ∈ 𝑋 such that 𝑦 π‘ˆ ∈ π‘ˆ [ π‘₯ π‘ˆ ] ∩ 𝑇 ( π‘₯ π‘ˆ ) . Since 𝑇 is compact, without loss of generality, we may assume that 𝑦 π‘ˆ converges to some 𝑦 in 𝑋 ; then π‘₯ π‘ˆ also converges to 𝑦 since 𝑋 is a Hausdorff uniform space and ( π‘₯ π‘ˆ , 𝑦 π‘ˆ ) ∈ π‘ˆ for each π‘ˆ ∈ 𝒩 . By the closedness of 𝑇 , we have that 𝑦 ∈ 𝑇 ( 𝑦 ) .

Corollary 2.4. Let 𝑋 be an almost Ξ¦ -space, and let 𝑠 ∢ 𝑋 β†’ 𝑋 be a surjective single-valued function. Suppose 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , 𝑋 , 𝑋 ) such that 𝑇 ( 𝑋 ) is totally bounded. Then 𝑇 has the approximate fixed point property.

Corollary 2.5. Suppose that all of the assumptions of the above Corollary 2.5 hold. If 𝑇 is closed, then 𝑇 has a fixed point in 𝑋 .

In case 𝑋 is an almost convex subset of Hausdorff topological vector spaces and for each 𝐴 βŠ‚ 𝑋 , we have

(i) 𝑐 π‘œ π’ž ( 𝐴 ) = 𝑐 π‘œ ( 𝐴 ) , and(ii) π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 π’ž ( 𝑋 , 𝑋 , 𝑋 ) = π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 ( 𝑋 , 𝑋 , 𝑋 ) .

This allows us to state the following results.

Theorem 2.6. Let 𝐸 be a Hausdorff locally convex space, let 𝑋 be an almost convex subset of 𝐸 , and let 𝑠 ∢ 𝑋 β†’ 𝑋 be a surjective function. Assume that 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 ( 𝑋 , 𝑋 , 𝑋 ) is compact and closed, then 𝑇 has a fixed point in 𝑋 .

Proof. Let π’ž be the family of all convex subsets of 𝐸 , and let ℬ 0 = { 𝑉 𝛼 ∢ 𝛼 ∈ Ξ› } be a local basis of 𝐸 such that each 𝑉 𝛼 ∈ ℬ 0 is symmetric and convex for each 𝛼 ∈ Ξ› . For each π‘₯ ∈ 𝑋 , we set 𝑉 𝛼 [ π‘₯ ] = π‘₯ + 𝑉 𝛼 . Noting that π‘₯ ∈ 𝑉 𝛼 [ π‘₯ ] . Set ξ€½ 𝑉 𝒩 = 𝛼 ∣ 𝑉 𝛼 = βˆͺ π‘₯ ∈ 𝑋 ξ€½ ( π‘₯ , 𝑦 ) ∢ 𝑦 ∈ 𝑉 𝛼 [ π‘₯ ] ξ€Ύ ξ€Ύ . , 𝛼 ∈ Ξ› ( 2 . 3 ) Then 𝒩 is a basis of a uniformity of 𝐸 . For each 𝑉 𝛽 ∈ 𝒩 , 𝛽 ∈ Ξ› , we define the two set-valued mappings 𝐺 , 𝐹 ∢ 𝑋 β†’ 2 𝑋 by 𝐺 ( π‘₯ ) = 𝐹 ( π‘₯ ) = 𝑉 𝛽 [ π‘₯ ] for each π‘₯ ∈ 𝑋 . Then we have (i) for each π‘₯ ∈ 𝑋 , 𝑐 π‘œ ( 𝐺 ( π‘₯ ) ) = 𝑐 π‘œ ( 𝑉 𝛽 [ π‘₯ ] ) βŠ‚ 𝑉 𝛽 [ 𝑉 𝛽 [ π‘₯ ] ] = 𝑉 𝛽 [ 𝐹 ( π‘₯ ) ] , and (ii) 𝑋 = βˆͺ π‘₯ ∈ 𝑋 i n t 𝐺 βˆ’ 1 ( π‘₯ ) . So, 𝐺 is an almost companion mapping of 𝐹 . This implies that 𝐹 is an almost Ξ¦ -mapping such that 𝒒 𝐹 βŠ‚ 𝑉 𝛽 . Therefore, 𝑋 is an almost Ξ¦ -space.
All conditions of Theorems 2.1 and 2.3 are therefore fulfilled; the result follows from an argument similar to those in the proofs of Theorems 2.1 and 2.3.

Theorem 2.7. Let 𝐸 be a topological vector space, let 𝑋 be an almost convex subset of 𝐸 , and let 𝑠 ∢ 𝑋 β†’ 𝑋 be a surjective function. Suppose that 𝑇 ∈ π‘Ž 𝑙 βˆ’ 𝑠 βˆ’ 𝐾 𝐾 𝑀 ( 𝑋 , 𝑋 , 𝑋 ) is compact, then for any symmetric convex neighborhood 𝑉 of 0 in 𝐸 , there is π‘₯ 𝑉 ∈ 𝑋 such that ( π‘₯ 𝑉 + 𝑉 ) ∩ 𝑇 ( π‘₯ 𝑉 ) β‰  πœ™ .

Proof. Let π’ž be the family of all convex subsets of 𝐸 , and let ℬ 0 = { π‘Ž 𝑉 ∢ π‘Ž > 0 } be a new local basis of 𝐸 . We will use ℬ 0 to construct a weaker topology on 𝐸 such that 𝐸 becomes a new topological vector space. For each π‘₯ ∈ 𝑋 , we set 𝑉 π‘Ž [ π‘₯ ] = π‘₯ + π‘Ž 𝑉 . Noting that π‘₯ ∈ 𝑉 π‘Ž [ π‘₯ ] . Set ξ€½ 𝑉 𝒩 = π‘Ž ∣ 𝑉 π‘Ž = βˆͺ π‘₯ ∈ 𝑋 ξ€½ ( π‘₯ , 𝑦 ) ∢ 𝑦 ∈ 𝑉 π‘Ž [ π‘₯ ] ξ€Ύ ξ€Ύ . , π‘Ž > 0 ( 2 . 4 ) Then 𝒩 is a basis of a uniformity of 𝐸 . In vein of the reasonings similar to those of Theorems 2.1 and 2.6, we complete the proof.

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