Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 392018, 7 pages http://dx.doi.org/10.1155/2014/392018
Research Article Some New Coincidence Theorems in Product GFC-Spaces with Applications Jianrong Zhao School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China Correspondence should be addressed to Jianrong Zhao;
[email protected] Received 28 February 2014; Accepted 12 April 2014; Published 6 May 2014 Academic Editor: Xie-ping Ding Copyright © 2014 Jianrong Zhao. 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. We first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.
1. Introduction and Preliminaries Recently, Ding [1, 2], X. P. Ding and T. M. Ding [3], Tang et al. [4], Fang and Huang [5] established some collective fixed point theorems, coincidence theorems, and KKM-type theorems for the families of set-valued mappings, which are defined on FC-spaces or product FC-spaces without any convexity structure. Very recently, Khanh and Quan [6], Khanh et al. [7] defined a new notion by GFC-space which is a generalization of FC-space and proved some continuous selections theorems, collectively fixed point theorems, coincidence theorems, and KKM-type theorems in this new GFC-space. Motivated and inspired by the work mentioned above, we will give some new collective fixed point and coincidence theorems in product GFC-spaces in the present paper. For this purpose, we first propose a new concept of GFCsubspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems. For our purpose, first, we present some known definitions and preliminary results. For a nonempty set 𝐴, ⟨𝐴⟩ denotes the family of all nonempty finite subsets and 2𝐴 denotes the family of all subsets of 𝐴. We denote the standard 𝑛simplex with vertices {𝑒𝑖 }𝑛𝑖=0 by Δ 𝑛 . The following notion was introduced by Khanh and Quan [6].
Definition 1 (see [6]). (a) A generalized finitely continuous topological space or a GFC-space (𝑋, 𝑌, {𝜑𝑁}) consists of a topological space 𝑋 and a nonempty set 𝑌 such that for each finite subset 𝑁 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ ⟨𝑌⟩, one has a continuous mapping 𝜑𝑁 : Δ 𝑛 → 𝑋. (b) Let 𝐴, 𝐵 ⊂ 𝑌 and 𝑆 : 𝑌 → 2𝑋 be given. Then 𝐵 is called an 𝑆-subset of 𝑌 with respect to 𝐴 if for any 𝑁 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ ⟨𝑌⟩ and for any {𝑦𝑖0 , . . . , 𝑦𝑖𝑘 } ⊂ 𝐴 ∩ 𝑁, one has 𝜑𝑁(Δ 𝑘 ) ⊂ 𝑆(𝐵), where Δ 𝑘 = co({𝑒𝑖0 , . . . , 𝑒𝑖𝑘 }). Now we give a new concept as follows. Definition 2. Assume that (𝑋, 𝑌, {𝜑𝑁}) is a GFC-space, 𝐶 ⊂ 𝑌, and 𝐷 ⊂ 𝑋. Then 𝐷 is said to be a GFC-subspace of 𝑋 with respect to 𝐶 if for each 𝑁 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ ⟨𝑌⟩ and for any {𝑦𝑖0 , . . . , 𝑦𝑖𝑘 } ⊂ 𝐶 ∩ 𝑁, one has 𝜑𝑁(Δ 𝑘 ) ⊂ 𝐷, where Δ 𝑘 = co({𝑒𝑖0 , . . . , 𝑒𝑖𝑘 }). Remark 3. By the Definition 2, we know that (𝐷, 𝐶, {𝜑𝑁}) is also a GFC-space. Clearly, if 𝐵 is an 𝑆-subset of 𝑌 with respect to 𝐴, then 𝑆(𝐵) is a GFC-subspace of 𝑋 with respect to 𝐴. In addition, if 𝑋 = 𝑌, then a GFC-subspace 𝐷 of 𝑋 with respect to 𝐶 coincides with an FC-subspace 𝐷 of 𝑋 with respect to 𝐶 (see [1]). Assume that {𝐷𝑖 }𝑖∈𝐼 is a family of GFC-subspace of 𝑋 with respect to 𝐶 and ⋂𝑖∈𝐼 𝐷𝑖 ≠ 0, where 𝐼 is an index set. It then follows from Definition 2 that ⋂𝑖∈𝐼 𝐷𝑖 is also a GFC-subspace
2
Abstract and Applied Analysis
of 𝑋 with respect to 𝐶. For any given subset 𝐴 of 𝑋, we define GFC-hull of 𝐴 with respect to 𝐶 by GFC (𝐴, 𝐶) = ∩ {𝐵 ⊂ 𝑋 : 𝐴 ⊂ 𝐵, 𝐵 is a GFC-subspace of 𝑋 with respect to 𝐶} . (1) We simply write GFC(𝑆(𝐶)) instead of GFC(𝑆(𝐶), 𝐶) when 𝐴 = 𝑆(𝐶). We can easily show that GFC(𝐴, 𝐶) is also a GFC-subspace of 𝑋 with respect to 𝐶. Let 𝑋 be a topological space. A set 𝐴 ⊂ 𝑋 is called compactly closed (resp., compactly open) if for each nonempty compact set 𝐾 ⊂ 𝑋 such that 𝐴 ∩ 𝐾 is closed (resp., open) in 𝐾. And the compact interior and the compact closure of 𝐴 (see [8]) are defined by cint (𝐴) = ∪ {𝐵 ⊂ 𝑋 : 𝐵 is compactly open in 𝑋, 𝐵 ⊂ 𝐴} , ccl (𝐴) = ∩ {𝐵 ⊂ 𝑋 : 𝐵 is compactly closed in 𝑋, 𝐵 ⊂ 𝐴} . (2) Clearly if 𝐾 is a nonempty compact subset of 𝑋, then we deriver that cint(𝐴) ∩ 𝐾 = int𝐾 (𝐴 ∩ 𝐾), ccl(𝐴) ∩ 𝐾 = cl𝐾 (𝐴 ∩ 𝐾), and ccl(𝐴 \ 𝐾) = 𝑋 \ cint(𝐴). Assume that 𝑌 and 𝑍 are nonempty sets and 𝑋 is a topological space. Define two set-valued mappings 𝐹 : 𝑋 × 𝑌 → 2𝑍 and 𝐶 : 𝑋 → 2𝑍 . 𝐹(𝑥, 𝑦) is called transfer compactly upper semicontinuous in 𝑥 with respect to 𝐶 (see [9]) if for any nonempty compact subset 𝐾 of 𝑋 and any 𝑥 ∈ 𝐾, the set {𝑦 ∈ 𝑌 : 𝐹(𝑥, 𝑦) ∈ 𝐶(𝑥)} ≠ 0 means that there is a relatively open neighborhood 𝑁(𝑥) of 𝑥 in 𝐾 and a point 𝑦 ∈ 𝑌 such that 𝐹(𝑧, 𝑦 ) ⊂ 𝐶(𝑧) for all 𝑧 ∈ 𝑁(𝑥). A mapping 𝑇 : 𝑋 → 2𝑌 is called transfer compactly open-valued (resp., transfer compactly closed-valued) on 𝑋 (see [8]) if for each 𝑥 ∈ 𝑋 and each nonempty compact subset 𝐾 of 𝑌, 𝑦 ∈ 𝑇(𝑥) ∩ 𝐾 (resp., 𝑦 ∉ 𝑇(𝑥) ∩ 𝐾) means that there is 𝑥 ∈ 𝑋 such that 𝑦 ∈ int𝐾 (𝑇(𝑥 ) ∩ 𝐾) (resp., 𝑦 ∉ cl𝐾 (𝑇(𝑥 ) ∩ 𝐾)). We need the following two results. The first statement was proved by Ding and Park [9]. Lemma 4 (see [9]). Let 𝑌 and 𝑍 be two nonempty sets and 𝑋 a topological space. Let 𝐹 : 𝑋×𝑌 → 2𝑍 and 𝐶 : 𝑋 → 2𝑍 be two set-valued mappings. Then 𝐹(𝑥, 𝑦) is transfer compactly upper semicontinuous in 𝑥 with respect to 𝐶 if and only if the mapping 𝑇 : 𝑌 → 2𝑋 defined by 𝑇(𝑦) = {𝑥 ∈ 𝑋 : 𝐹(𝑥, 𝑦) ⊄ 𝐶(𝑥)} is transfer compactly closed-valued on 𝑌. The second statement is Lemma 1.1 of Ding [8]. Lemma 5. Assume that 𝑋 and 𝑌 are two topological spaces and 𝐹 : 𝑋 → 2𝑌 is a set-valued mapping with nonempty values. Then the following statements are equivalent: (i) 𝐹−1 : 𝑌 → 2𝑋 is transfer compactly open-valued; (ii) for any compact subset 𝐾 of 𝑋 and any 𝑥 ∈ 𝐾, one has 𝑦 ∈ 𝑌 satisfying that 𝑥 ∈ cint 𝐹−1 (𝑦) ∩ 𝐾 and 𝐾 = ⋃𝑦∈𝑌 int𝐾 (𝐹−1 ∩ 𝐾).
Throughout this paper, we always let 𝐼 and 𝐽 be any given index set. Now we give the following statement, which generalizes Lemma 1.1 of Ding [1]. Lemma 6. Suppose that (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) is a GFC-space for each 𝑖 ∈ 𝐼. If 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 , 𝑌 = ∏𝑖∈𝐼 𝑌𝑖 , and 𝜑𝑁 = ∏𝑖∈𝐼 𝜑𝑁𝑖 , then (𝑋, 𝑌, {𝜑𝑁}) is also a GFC-space. Proof. Let 𝜋𝑖 : 𝑌 → 𝑌𝑖 be the projective mapping from 𝑌 to 𝑌𝑖 for each 𝑖 ∈ 𝐼. For any given 𝑁 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ ⟨𝑌⟩, we denote 𝑁𝑖 = 𝜋𝑖 (𝑁) = {𝜋𝑖 (𝑦0 ), . . . , 𝜋𝑖 (𝑦𝑛 )} ∈ ⟨𝑌𝑖 ⟩. Note that (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) is a GFC-space. Then we have a continuous mapping 𝜑𝑁𝑖 : Δ 𝑛 → 𝑋𝑖 for each 𝑖 ∈ 𝐼. So we may let a mapping 𝜑𝑁 : Δ 𝑛 → 𝑋 by 𝜑𝑁(𝛼) = ∏𝑖∈𝐼 𝜑𝑁𝑖 (𝛼), for any 𝛼 ∈ Δ 𝑛 . It follows that 𝜑𝑁 is a continuous mapping, which means that (𝑋, 𝑌, {𝜑𝑁}) is a GFC-space. So Lemma 6 is proved.
2. Continuous Selection and Collective Fixed Points Theorem 7. Assume that (𝑋, 𝑌, {𝜑𝑁}) is a GFC-space and 𝑍 is a compact topological space. Suppose that 𝐹 : 𝑍 → 2𝑌 and 𝐺 : 𝑍 → 2𝑋 are such that (i) 𝑍 = ⋃𝑦∈𝑌 cint 𝐹−1 (𝑦); (ii) for any given 𝑧 ∈ 𝑍, 𝐺(𝑧) is a GFC-subspace of 𝑋 with respect to 𝐹(𝑧). Then one has a continuous selection 𝑔 : 𝑍 → 𝑋 of 𝐺 satisfying 𝑔 = 𝜑∘𝜓, where 𝜑 : Δ 𝑛 → 𝑋 and 𝜓 : 𝑍 → Δ 𝑛 are continuous for some 𝑛 ∈ Z+ . Proof. By condition (i), we know that there exists 𝑁 = {𝑦0 , 𝑦1 , . . . , 𝑦𝑛 } ∈ ⟨𝑌⟩ such that 𝑍 = ⋃𝑛𝑖=0 cint 𝐹−1 (𝑦𝑖 ) since 𝑍 is compact. Assume that {𝜓𝑖 }𝑛𝑖=0 is the continuous partition of unity subordinated to the open covering {cint 𝐹−1 (𝑦𝑖 )}𝑛𝑖=0 ; then for any 𝑖 ∈ {0, 1, . . . , 𝑛} and 𝑧 ∈ 𝑍, one has 𝜓𝑖 (𝑧) ≠ 0 ⇐⇒ 𝑧 ∈ cint 𝐹−1 (𝑦𝑖 ) ⊂ 𝐹−1 (𝑦𝑖 ) ⇒ 𝑦𝑖 ∈ 𝐹 (𝑧) . (3) Let 𝜓 : 𝑍 → Δ 𝑛 be a mapping with 𝜓(𝑧) = ∑𝑛𝑖=0 𝜓𝑖 (𝑧)𝑒𝑖 . Clearly 𝜓 is continuous and for any 𝑧 ∈ 𝑍, one has 𝜓(𝑧) = ∑𝑖∈𝐽(𝑥) 𝜓𝑗 (𝑧)𝑒𝑗 where 𝐽(𝑥) = {𝑗 ∈ {0, 1, . . . , 𝑛} : 𝜓𝑖 (𝑧) ≠ 0}. So by (3), we get {𝑦𝑗 : 𝑗 ∈ 𝐽(𝑥)} ⊂ 𝐹(𝑧) ∩ 𝑁. It then follows from condition (ii) that for 𝑧 ∈ 𝑍, 𝜑𝑁(Δ 𝐽(𝑥) ) ⊂ 𝐺(𝑧). It is easy to see that 𝑔(𝑧) = 𝜑𝑁 ∘ 𝜓(𝑧) ∈ 𝜑𝑁(Δ 𝐽(𝑥) ) ⊂ 𝐺(𝑧), which implies that 𝑔 = 𝜑𝑁 ∘ 𝜓 is a continuous selection of 𝐺. The proof of Theorem 7 is completed. Remark 8. Applying the definition of GFC-subspace, we extend Theorem 2.2 of Tarafdar [10], Proposition 1 of Browder [11], and Theorem 2.1 of Ding [2] to GFC-spaces without any convexity. Theorem 9. Assume that 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 and (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) is a compact GFC-space for each 𝑖 ∈ 𝐼. Suppose that 𝐹𝑖 : 𝑋 → 2𝑌𝑖
Abstract and Applied Analysis
3
and 𝐺𝑖 : 𝑋 → 2𝑋𝑖 are two set-valued mappings satisfying the following conditions:
𝐵𝑖 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 ) ∩ 𝐵𝑖 ) ⊂ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) .
(i) 𝑋 = ⋃𝑦𝑖 ∈𝑌𝑖 cint 𝐹𝑖−1 (𝑦𝑖 ); (ii) for any given 𝑥 ∈ 𝑋, 𝐺𝑖 (𝑥) is a GFC-subspace of 𝑋𝑖 with respect to 𝐹𝑖 (𝑥). ̂ for Then one has a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 such that 𝑥̂𝑖 ∈ 𝐺𝑖 (𝑥) 𝑖 ∈ 𝐼. Proof. For any given 𝑖 ∈ 𝐼, using Theorem 7, we obtain that there are continuous mappings 𝜑𝑁𝑖 : Δ 𝑛𝑖 → 𝑋𝑖 and 𝜓𝑖 : 𝑋 → Δ 𝑛𝑖 such that 𝑔𝑖 = 𝜑𝑁𝑖 ∘ 𝜓𝑖 is a continuous selection of 𝐺𝑖 for some positive integer 𝑛𝑖 . Assume that 𝐸𝑖 is the linear hull of the set {𝑒0 , 𝑒1 , . . . , 𝑒𝑛𝑖 } for each 𝑖 ∈ 𝐼. Clearly 𝐸𝑖 is a locally convex Hausdorff topological vector space as it is finite dimensional and Δ 𝑛𝑖 is a compact convex subset of 𝐸𝑖 . Moreover, 𝐸 = ∏𝑖∈𝐼 𝐸𝑖 is a locally convex Hausdorff topological vector space and Δ = ∏𝑖∈𝐼 Δ 𝑛𝑖 is a compact convex subset of 𝐸. Now let Φ : Δ → 𝑋 and Ψ : 𝑋 → Δ be two continuous mappings with Φ (𝛿) = ∏𝜑𝑁𝑖 (𝑃𝑖 (𝛿)) , 𝑖∈𝐼
Ψ (𝑥) = ∏𝜓𝑖 (𝑥) , 𝑖∈𝐼
(4)
where 𝑃𝑖 : Δ → Δ 𝑛𝑖 is the projection of Δ onto Δ 𝑛𝑖 for any given 𝑖 ∈ 𝐼. It then follows from the Tychonoff fixed point theorem that the continuous mapping Ψ ∘ Φ : Δ → Δ has a fixed point 𝛿 ∈ Δ; that is, 𝛿 = Ψ ∘ Φ(𝛿). Let 𝑥̂ = Φ(𝛿). It follows that ̂ ̂ = Φ (∏𝜓𝑖 (𝑥)) 𝑥̂ = Ψ ∘ Φ (𝑥) 𝑖∈𝐼
(5)
̂ ̂ . = ∏𝜑𝑁𝑖 ∘ 𝜓𝑖 (𝑥) = ∏𝜑𝑁𝑖 (𝑃𝑖 (∏𝜓𝑖 (𝑥))) 𝑖∈𝐼
𝑖∈𝐼
Proof. For any given 𝑖 ∈ 𝐼, if 𝐵𝑖 is a nonempty compact subset of 𝑍𝑖 , then by (i) one has
𝑖∈𝐼
̂ ∈ 𝐺𝑖 (𝑥) ̂ for each 𝑖 ∈ 𝐼. So This means that 𝑥̂ = 𝜑𝑁𝑖 ∘ 𝜓𝑖 (𝑥) Theorem 9 is proved. Theorem 10. Assume that 𝑍𝑖 is a topological space and (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) is a GFC-space for each 𝑖 ∈ 𝐼. Suppose that 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 , 𝑆𝑖 : 𝑍𝑖 → 2𝑌𝑖 and 𝑇𝑖 : 𝑍𝑖 → 2𝑋𝑖 and 𝑔𝑖 : 𝑋 → 𝑍𝑖 is a continuous mapping such that (i) for any compact subset 𝐷𝑖 of 𝑍𝑖 , 𝐷𝑖 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint 𝑆𝑖−1 (𝑦𝑖 ) ∩ 𝐷𝑖 ); (ii) for any 𝑧𝑖 ∈ 𝑍𝑖 , 𝑇𝑖 (𝑧𝑖 ) is a GFC-subspace of 𝑋𝑖 with respect to 𝑆𝑖 (𝑧𝑖 ); (iii) there is a nonempty subset of 𝑌𝑖0 ⊂ 𝑌𝑖 such that the set 𝐵𝑖 = ⋂𝑦𝑖 ∈𝑌𝑖0 (cint 𝑆𝑖−1 (𝑦𝑖 ))𝑐 is empty or compact in 𝑍𝑖 , and for any 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there exists a compact GFCsubspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 . ̂ Then one has a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 such that 𝑥̂𝑖 ∈ 𝑇𝑖 (𝑔𝑖 (𝑥)) for each 𝑖 ∈ 𝐼.
𝑦𝑖 ∈𝑌𝑖
𝑦𝑖 ∈𝑌𝑖
(6)
Moreover, noticing that 𝐵𝑖 is compact, we can find that there is a finite set 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩ such that 𝑐
𝐵𝑖 = ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) ⊂ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) . 𝑦𝑖 ∈𝑁𝑖
𝑦𝑖 ∈𝑌𝑖0
(7)
It then follows from (7) that 𝑍𝑖 = ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) ∪ ( ⋃ cint 𝑆𝑖−1 (𝑦𝑖 )) .
(8)
𝑦𝑖 ∈𝑁𝑖
𝑦𝑖 ∈𝑌𝑖0
If 𝐵𝑖 is empty in (iii), then we derive that 𝑐
𝑍𝑖 = 𝑍𝑖 \ 𝐵𝑖 = 𝑍𝑖 \ ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) = ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) . 𝑦𝑖 ∈𝑌𝑖0
𝑦𝑖 ∈𝑌𝑖0
(9) From condition (iii), we know that there exists a compact GFC-subspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 . So by (8) and (9), we get 𝑍𝑖 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 )) . 𝑦𝑖 ∈𝐿 𝑁𝑖
(10)
Now let 𝐾𝑁 = ∏𝑖∈𝐼 𝐾𝑁𝑖 , 𝐿 𝑁 = ∏𝑖∈𝐼 𝐿 𝑁𝑖 , and 𝜑𝑁 = ∏𝑖∈𝐼 𝜑𝑁𝑖 . By condition (iii), we deduce that (𝐾𝑁𝑖 , 𝐿 𝑁𝑖 , {𝜑𝑁𝑖 }) is a compact GFC-space. It then follows from Lemma 6 that (𝐾𝑁, 𝐿 𝑁, {𝜑𝑁}) is also a compact GFC-space. Let 𝑃𝑖 : 𝐾𝑁 → 2𝐿 𝑁𝑖 and 𝑄𝑖 : 𝐾𝑁 → 2𝐾𝑁𝑖 be two set-valued mappings with 𝑃𝑖 (𝑥) = 𝑆𝑖 (𝑔𝑖 (𝑥)) ∩ 𝐿 𝑁𝑖 , 𝑄𝑖 (𝑥) = 𝑇𝑖 (𝑔𝑖 (𝑥)) ∩ 𝐾𝑁𝑖
for 𝑥 ∈ 𝐾𝑁.
(11)
In order to show that the conditions (i) and (ii) of Theorem 9 hold, we only need to show that 𝑄𝑖 (𝑥) is a compact GFC-subspace of 𝐾𝑁𝑖 with respect to 𝑃𝑖 (𝑥), and 𝐾𝑁 = ⋃𝑦𝑖 ∈𝐿 𝑁 cint 𝑃𝑖−1 (𝑦𝑖 ). By conditions (ii) and (iii), it is easy to see 𝑖 that 𝑄𝑖 (𝑥) is a compact GFC-subspace of 𝐾𝑁𝑖 with respect to 𝑃𝑖 (𝑥). It remains to show that 𝐾𝑁 = ⋃𝑦𝑖 ∈𝐿 𝑁 cint 𝑃𝑖−1 (𝑦𝑖 ). On 𝑖 one hand, for each 𝑦𝑖 ∈ 𝐿 𝑁𝑖 , we deduce that 𝑃𝑖−1 (𝑦𝑖 ) = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑃𝑖 (𝑥)} = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑆𝑖 (𝑔𝑖 (𝑥)) ∩ 𝐿 𝑁𝑖 } = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑆𝑖 (𝑔𝑖 (𝑥))} = {𝑥 ∈ 𝐾𝑁 : 𝑥 ∈ 𝑔𝑖−1 (𝑆𝑖−1 (𝑦𝑖 ))} = 𝑔𝑖−1 (𝑆𝑖−1 (𝑦𝑖 )) .
(12)
4
Abstract and Applied Analysis On the other hand, by (10), we obtain 𝑔𝑖 (𝐾𝑁) ⊂ 𝑍𝑖 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 )) . 𝑦𝑖 ∈𝐿 𝑁𝑖
(13)
It then follows form (12), (13), and the continuity of 𝑔𝑖 that 𝐾𝑁 ⊂ 𝑔𝑖−1 ( ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ))
3. Coincidence Theorems for Two Families of Set-Valued Mappings
𝑦𝑖 ∈𝐿 𝑁𝑖
= ⋃ 𝑔𝑖−1 (cint 𝑆𝑖−1 (𝑦𝑖 ))
(14)
𝑦𝑖 ∈𝐿 𝑁𝑖
= ⋃ (cint 𝑃𝑖−1 (𝑦𝑖 )) ⊂ 𝐾𝑁. 𝑦𝑖 ∈𝐿 𝑁𝑖
Hence 𝐾𝑁 = ⋃𝑦𝑖 ∈𝐿 𝑁 cint 𝑃𝑖−1 (𝑦𝑖 ). Then Theorem 9 tells us 𝑖 ̂ = that there exists a point 𝑥̂ ∈ 𝐾𝑁 ⊂ 𝑋 such that 𝑥̂𝑖 ∈ 𝑄𝑖 (𝑥) ̂ ∩ 𝐾𝑁𝑖 ⊂ 𝑇𝑖 (𝑔𝑖 (𝑥)) ̂ for each 𝑖 ∈ 𝐼. This completes the 𝑇𝑖 (𝑔𝑖 (𝑥)) proof of Theorem 10. Theorem 11. Let 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 , 𝑍𝑖 be a topological space and (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) a GFC-space for each 𝑖 ∈ 𝐼. Assume that 𝑆𝑖 : 𝑍𝑖 → 2𝑌𝑖 , 𝑇𝑖 : 𝑍𝑖 → 2𝑋𝑖 , and 𝑅𝑖 : 𝑌𝑖 → 2𝑋𝑖 are set-valued mappings and 𝑔𝑖 : 𝑋 → 𝑍𝑖 is a continuous mapping such that (i) for any compact subset 𝐷𝑖 of 𝑍𝑖 , 𝐷𝑖 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint 𝑆𝑖−1 (𝑦𝑖 ) ∩ 𝐷𝑖 ); (ii) for any 𝑧𝑖 ∈ 𝑍𝑖 , GFC(𝑅𝑖 (𝑆𝑖 (𝑧𝑖 ))) ⊂ 𝑇𝑖 (𝑧𝑖 ); (iii) if 𝑍𝑖 is not compact, then there is a nonempty subset of 𝑌𝑖0 ⊂ 𝑌𝑖 such that a compact subset 𝐷𝑖 of 𝑍𝑖 satisfying 𝑍𝑖 \ 𝐷𝑖 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint 𝑆𝑖−1 (𝑦𝑖 ), and for any 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there exists a compact GFC-subspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 . ̂ Then we have a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 such that 𝑥̂𝑖 ∈ 𝑇𝑖 (𝑔𝑖 (𝑥)) for each 𝑖 ∈ 𝐼. Proof. Using the similar argument of Theorem 10, we only need to show that the conditions (ii) and (iii) of Theorem 10 are satisfied. By the definition of GFC(𝑅𝑖 (𝑆𝑖 (𝑧𝑖 ))), we know that GFC(𝑅𝑖 (𝑆𝑖 (𝑧𝑖 ))) is a GFC-subspace of 𝑋𝑖 with respect to 𝑆𝑖 (𝑧𝑖 ). It then follows from condition (ii) that 𝑇𝑖 (𝑧𝑖 ) is a GFC-subspace of 𝑋𝑖 with respect to 𝑆𝑖 (𝑧𝑖 ), which means that the condition (i) of Theorem 10 holds. It remains to deal with condition (iii). If 𝑍𝑖 is noncompact, by (iii), we get 𝑐
𝐵𝑖 = ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) = 𝑌𝑖 \ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) ⊂ 𝐷𝑖 . (15) 0 0 𝑦𝑖 ∈𝑌𝑖
assumption (i) in Theorems 9 and 10, then the statements of Theorems 9 and 10 are still true. (b) The results of Theorems 10 and 11 generalize Theorem 2.1 of Lan and Webb [12], Theorem 1 of Ansari and Yao [13], Theorem 2.2 of Ding and Park [14], Theorem 3.1 of Ding and Park [15], and Theorem 3.1 of Lin and Ansari [16] and Ding [2] to GFC-spaces.
𝑦𝑖 ∈𝑌𝑖
Clearly 𝐵𝑖 is a closed subset of compact set 𝐷𝑖 . If 𝐵𝑖 is nonempty, then 𝐵𝑖 is compact in 𝑍𝑖 . This means that the condition (iii) of Theorem 10 is satisfied. So the statement of Theorem 11 follows immediately from Theorem 10. Remark 12. (a) Let 𝑉𝑗 be a topological spaces for each 𝑗 ∈ 𝐽. By Lemma 5, if using assumption (i) in Lemma 5 for
Theorem 13. Let (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) and (𝑈𝑗 , 𝑉𝑗 , {𝜑𝑁𝑗 }) be GFCspaces for any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. Assume that 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 and 𝑈 = ∏𝑖∈𝐽 𝑈𝑗 . Let 𝑆𝑖 : 𝑈 → 2𝑌𝑖 , 𝑇𝑖 : 𝑈 → 2𝑋𝑖 , 𝐹𝑗 : 𝑋 → 2𝑉𝑗 , and 𝐺𝑗 : 𝑋 → 2𝑈𝑗 be set-valued mappings satisfying the following conditions: (i) for any compact subset 𝐴 of 𝑋, 𝐴 = ⋃V𝑗 ∈𝑉𝑗 (cint 𝐹𝑗−1 (V𝑗 ) ∩ 𝐴); (ii) for any 𝑥 ∈ 𝑋, 𝐺𝑗 (𝑥) is a GFC-subspace of 𝑈𝑗 with respect to 𝐹𝑗 (𝑥); (iii) there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 such that the set 𝐷𝑖 = ⋂𝑦𝑖 ∈𝑌𝑖0 (cint 𝑆𝑖−1 (𝑦𝑖 ))𝑐 is empty or compact in 𝑈 and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFCsubspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 ;
(iv) for any compact subset 𝐵 of 𝑈, 𝐵 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint 𝑆𝑖−1 (𝑦𝑖 )∩ 𝐵); (v) for any 𝑢 ∈ 𝑈, 𝑇𝑖 (𝑢) is a GFC-subspace of 𝑋𝑖 with respect to 𝑆𝑖 (𝑢).
Then one has 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 and 𝑢̂ = (̂ 𝑢𝑗 )𝑗∈𝐽 ∈ 𝑈 such that ̂ for any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. 𝑢) and 𝑢̂𝑗 ∈ 𝐺𝑗 (𝑥) 𝑥̂𝑖 ∈ 𝑇𝑖 (̂ Proof. For any given 𝑖 ∈ 𝐼, if 𝐷𝑖 is empty in (iii), it is easy to see that 𝑐
𝑈 = 𝑈 \ 𝐷𝑖 = 𝑈 \ ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) = ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) . 𝑦𝑖 ∈𝑌𝑖0
𝑦𝑖 ∈𝑌𝑖0
(16) If 𝐷𝑖 is nonempty compact set in 𝑈, by (iv), we know that 𝐷𝑖 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 ) ∩ 𝐷𝑖 ) ⊂ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) . 𝑦𝑖 ∈𝑌𝑖
𝑦𝑖 ∈𝑌𝑖
(17)
Note that 𝐷𝑖 is compact. Then we can find a finite set 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩ such that 𝑐
𝐷𝑖 = ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) ⊂ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) . 𝑦𝑖 ∈𝑌𝑖0
𝑦𝑖 ∈𝑁𝑖
(18)
It then follows from (16) and (18) that if either 𝐷𝑖 is empty or compact in 𝑈, we get 𝑈 = ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) ∪ ( ⋃ cint 𝑆𝑖−1 (𝑦𝑖 )) . 𝑦𝑖 ∈𝑌𝑖0
𝑦𝑖 ∈𝑁𝑖
(19)
Abstract and Applied Analysis
5
Furthermore, by (iii), there exists a compact GFCsubspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 . So by (19), we get 𝑈 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 )) . 𝑦𝑖 ∈𝐿 𝑁𝑖
(20)
Assume that 𝐾𝑁 = ∏𝑖∈𝐼 𝐾𝑁𝑖 , 𝐿 𝑁 = ∏𝑖∈𝐼 𝐿 𝑁𝑖 , and 𝜑𝑁 = ∏𝑖∈𝐼 𝜑𝑁𝑖 . It then follows from Lemma 6 that 𝐾𝑁 is a compact GFC-subspace of 𝑋 with respect to 𝐿 𝑁. So (𝐾𝑁, 𝐿 𝑁, {𝜑𝑁}) is a compact GFC-space. We consider the restrictions 𝐹𝑗 |𝐾𝑁 and 𝐺𝑗 |𝐾𝑁 of 𝐹𝑗 and 𝐺𝑗 on 𝐾𝑁, respectively, for each 𝑗 ∈ 𝐽. Then by condition (ii), we have 𝐺𝑗 |𝐾𝑁 (𝑥) as a GFC-subspace of 𝑈𝑗 with respect to 𝐹𝑗 |𝐾𝑁 (𝑥) for any 𝑥 ∈ 𝑋. From condition (i), we obtain 𝐾𝑁 = ⋃ (cint 𝐹𝑗−1 (V𝑗 ) ∩ 𝐾𝑁) = ⋃ cint 𝐹𝑗−1 |𝐾𝑁 (V𝑗 ) . V𝑗 ∈𝑉𝑗
V𝑗 ∈𝑉𝑗
(21) Moreover, by Theorem 7, there exists a continuous selection 𝑔𝑗 : 𝐾𝑁 → 𝑈𝑗 of 𝐺𝑗 |𝐾𝑁 for any 𝑗 ∈ 𝐽. Let 𝑔 : 𝐾𝑁 → 𝑈 be a mapping with 𝑔(𝑥) = ∏𝑗∈𝐽 𝑔𝑗 (𝑥) for any 𝑥 ∈ 𝐿 𝑁. Clearly 𝑔 is a continuous mapping. Define two set-valued mappings 𝑃𝑖 : 𝐾𝑁 → 2𝐿 𝑁𝑖 and 𝑄𝑖 : 𝐾𝑁 → 2𝐾𝑁𝑖 by 𝑃𝑖 (𝑥) = 𝑆𝑖 (𝑔 (𝑥)) ∩ 𝐿 𝑁𝑖 ,
𝑄𝑖 (𝑥) = 𝑇𝑖 (𝑔 (𝑥)) ∩ 𝐾𝑁𝑖 . (22)
Since 𝐾𝑁𝑖 is a compact GFC-subspace of 𝑋𝑖 relative to 𝐿 𝑁𝑖 , by (v), we know that 𝑄𝑖 (𝑥) is also a compact GFCsubspace of 𝑋𝑖 with respect to 𝑃𝑖 (𝑥). Now we claim that 𝐾𝑁 = ⋃𝑦𝑖 ∈𝐿 𝑁 cint 𝑃𝑖−1 (𝑦𝑖 ). For any 𝑖 given 𝑦𝑖 ∈ 𝐿 𝑁𝑖 , we deduce that 𝑃𝑖−1 (𝑦𝑖 ) = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑃𝑖 (𝑥)} = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑆𝑖 (𝑔 (𝑥)) ∩ 𝐿 𝑁𝑖 } = {𝑥 ∈ 𝐾𝑁 : 𝑦𝑖 ∈ 𝑆𝑖 (𝑔 (𝑥))}
(23)
= {𝑥 ∈ 𝐾𝑁 : 𝑥 ∈ 𝑔−1 (𝑆𝑖−1 (𝑦𝑖 ))} =𝑔
−1
(𝑆𝑖−1
(i) for any compact subset 𝐴 of 𝑋, 𝐴 = ⋃V𝑗 ∈𝑉𝑗 (cint 𝐹𝑗−1 (V𝑗 ) ∩ 𝐴), (ii) for any 𝑥 ∈ 𝑋, GFC(𝐻𝑗 (𝐹𝑗 (𝑥))) ⊂ 𝐺𝑗 (𝑥), (iii) if 𝑈 is not compact, then there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 as well as a compact subset 𝐾 of 𝑈 such that 𝑈 \ 𝐾 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint 𝑆𝑖−1 (𝑦𝑖 ) and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFC-subspace 𝐾𝑁𝑖 of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 , (iv) for any compact subset 𝐵 of 𝑈, 𝐵 = ⋃V𝑗 ∈𝑉𝑗 (cint 𝐹𝑗−1 (V𝑗 ) ∩ 𝐵), (v) for any 𝑢 ∈ 𝑈, GFC(𝑅𝑖 (𝑆𝑖 (𝑢))) ⊂ 𝑇𝑖 (𝑢), 𝑢𝑗 )𝑗∈𝐽 ∈ 𝑈 such that then there exist 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 and 𝑢̂ = (̂ ̂ for any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. 𝑢) and 𝑢̂𝑗 ∈ 𝐺𝑗 (𝑥) 𝑥̂𝑖 ∈ 𝑇𝑖 (̂ Proof. By the definition of the GFC-hull with respect to 𝐹𝑗 (𝑥) and condition (ii), it is easy to see that 𝐺𝑗 (𝑥) is a GFCsubspace of 𝑈𝑗 with respect to 𝐹𝑗 (𝑥) for any 𝑥 ∈ 𝑋 and 𝑗 ∈ 𝐽. Similarly, by (v), we obtain that 𝑇𝑖 (𝑢) is a GFC-subspace of 𝑋𝑖 with respect to 𝑆𝑖 (𝑢) for any 𝑢 ∈ 𝑈 and 𝑖 ∈ 𝐼. Then conditions (ii) and (v) of Theorem 13 hold. On the other hand, by (iii), if 𝑈 is not compact, we know that 𝑐
(𝑦𝑖 )) .
𝑦𝑖 ∈𝑌𝑖
𝑔 (𝐾𝑁) ⊂ 𝑈 = ⋃ (cint 𝑆𝑖−1 (𝑦𝑖 )) . 𝑦𝑖 ∈𝐿 𝑁𝑖
(24)
It then follows form (23), (24), and the continuity of 𝑔 that
𝑦𝑖 ∈𝐿 𝑁𝑖
= ⋃ 𝑔−1 (cint 𝑆𝑖−1 (𝑦𝑖 )) 𝑦𝑖 ∈𝐿 𝑁𝑖
𝑦𝑖 ∈𝑌𝑖
If 𝐷 is nonempty, then 𝐷 is a closed subset of compact set 𝐾. This implies that 𝐷 is compact in 𝑈. So condition (iii) tells us condition (iii) of Theorem 13 holds. Then the statement of Theorem 14 follows immediately from Theorem 13. Remark 15. Theorems 13 and 14 improve Theorem 9 of Yu and Lin [17], Theorem 3.3 of Lin and Ansari [16], and Theorems 3.1 and 3.2 of Ding [2] to GFC-spaces without any convexity structure.
𝐾𝑁 ⊂ 𝑔−1 ( ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ))
𝑦𝑖 ∈𝐿 𝑁𝑖
Theorem 14. Let (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) and (𝑈𝑗 , 𝑉𝑗 , {𝜑𝑁𝑗 }) be GFCspaces for any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. Assume that 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 and 𝑈 = ∏𝑖∈𝐽 𝑈𝑗 . Let 𝑆𝑖 : 𝑈 → 2𝑌𝑖 , 𝑇𝑖 : 𝑈 → 2𝑋𝑖 , 𝑅𝑖 : 𝑌𝑖 → 2𝑋𝑖 , 𝐹𝑗 : 𝑋 → 2𝑉𝑗 , 𝐺𝑗 : 𝑋 → 2𝑈𝑗 , and 𝐻𝑗 : 𝑉𝑗 → 2𝑈𝑗 be set-valued mappings for each 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. If the following conditions hold:
𝐷 = ⋂ (cint 𝑆𝑖−1 (𝑦𝑖 )) = 𝑈 \ ⋃ cint 𝑆𝑖−1 (𝑦𝑖 ) ⊂ 𝐾. (26) 0 0
Then by (20) we obtain
= ⋃
Hence 𝐾𝑁 = ⋃𝑦𝑖 ∈𝐿 𝑁 cint 𝑃𝑖−1 (𝑦𝑖 ). So the claim is proved. 𝑖 Note that 𝑄𝑖 (𝑥) is a compact GFC-subspace of 𝑋𝑖 relative to 𝑃𝑖 (𝑥). So by the above claim and Theorem 9, we know that ̂ = there exists a point 𝑥̂ ∈ 𝐾𝑁 ⊂ 𝑋 such that 𝑥̂𝑖 ∈ 𝑄𝑖 (𝑥) ̂ ∩ 𝐾𝑁𝑖 ⊂ 𝑇𝑖 (𝑔(𝑥)) ̂ for any given 𝑖 ∈ 𝐼. Assume that 𝑇𝑖 (𝑔(𝑥)) ̂ Thus we derive that there exist 𝑥̂ ∈ 𝑋 ̂ = ∏𝑗∈𝐽 𝑔𝑗 (𝑥). 𝑢̂ = 𝑔(𝑥) ̂ for any 𝑖 ∈ 𝐼 and 𝑢̂ ∈ 𝑈 such that 𝑥̂𝑖 ∈ 𝑇𝑖 (̂ 𝑢) and 𝑢̂𝑗 ∈ 𝐺𝑗 (𝑥) and 𝑗 ∈ 𝐽, which implies that Theorem 13 is true.
(25)
4. Applications (cint 𝑃𝑖−1
(𝑦𝑖 )) ⊂ 𝐾𝑁.
In the current section, we will give some applications of our theorems.
6
Abstract and Applied Analysis
Theorem 16. Let (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) be a GFC-space and 𝑍𝑖 a topological space for each 𝑖 ∈ 𝐼. Assume that 𝐴 𝑖 and 𝐵𝑖 are a subset of 𝑌𝑖 × 𝑍𝑖 and 𝑋𝑖 × 𝑍𝑖 , respectively. Let 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 and for each 𝑖 ∈ 𝐼, let 𝑔𝑖 : 𝑋 → 𝑍𝑖 be a continuous mapping and 𝑅𝑖 : 𝑌𝑖 → 2𝑋𝑖 a set-valued mapping satisfying the following conditions: (i) for any compact subset 𝐷𝑖 of 𝑍𝑖 , 𝐷𝑖 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint{𝑧𝑖 ∈ 𝑍𝑖 : (𝑦𝑖 , 𝑧𝑖 ) ∈ 𝐴 𝑖 } ∩ 𝐷𝑖 ); (ii) for any 𝑧𝑖 ∈ 𝑍𝑖 , GFC(𝑅𝑖 ({𝑦𝑖 ∈ 𝑌𝑖 : (𝑦𝑖 , 𝑧𝑖 ) ∈ 𝐴 𝑖 })) ⊂ {𝑥𝑖 ∈ 𝑋𝑖 : (𝑥𝑖 , 𝑧𝑖 ) ∈ 𝐵𝑖 }; (iii) if 𝑍𝑖 is not compact, then there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 such that a compact subset 𝐷𝑖 of 𝑍𝑖 such that 𝑍𝑖 \ 𝐷𝑖 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint{𝑧𝑖 ∈ 𝑍𝑖 : (𝑦𝑖 , 𝑧𝑖 ) ∈ 𝐴 𝑖 } and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFC-subspace of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 . ̂ ∈ 𝐵𝑖 Then we have a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 such that (𝑥̂𝑖 , 𝑔𝑖 (𝑥)) for each 𝑖 ∈ 𝐼. Proof. For any given 𝑖 ∈ 𝐼, we define two set-valued mappings 𝑆𝑖 : 𝑍𝑖 → 2𝑌𝑖 and 𝑇𝑖 : 𝑍𝑖 → 2𝑋𝑖 by 𝑆𝑖 (𝑧𝑖 ) = {𝑦𝑖 ∈ 𝑌𝑖 : (𝑦𝑖 , 𝑧𝑖 ) ∈ 𝐴 𝑖 } ,
In the rest of this section, for each 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽, we always let 𝑍𝑖 and 𝑊𝑖 be two nonempty set and (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) and (𝑈𝑗 , 𝑉𝑗 , {𝜑𝑁𝑗 }) be two GFC-spaces. Assume that 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 and 𝑈 = ∏𝑖∈𝐽 𝑈𝑗 . For any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽, let 𝑅𝑖 : 𝑌𝑖 → 2𝑋𝑖 , 𝐻𝑗 : 𝑉𝑗 → 2𝑈𝑗 , 𝐶𝑗 : 𝑋 → 2𝑍𝑗 , 𝐷𝑖 : 𝑈 → 2𝑊𝑖 , 𝐵𝑗 : 𝑋 × 𝑉𝑗 → 2𝑍𝑗 , 𝐴 𝑗 : 𝑋 × 𝑈𝑗 → 2𝑍𝑗 , 𝑄𝑖 : 𝑌𝑖 × 𝑈 → 2𝑊𝑖 , and 𝑃𝑖 : 𝑋𝑖 × 𝑈 → 2𝑊𝑖 be set-valued mappings. Then we have the follows results. Theorem 18. Suppose that the following conditions hold: (i) for any compact subset 𝐴 of 𝑋, 𝐴 = ⋃V𝑗 ∈𝑉𝑗 (cint{𝑥 ∈ 𝑋 : 𝐵𝑗 (𝑥, V𝑗 ) ⊂ 𝐶𝑗 (𝑥)} ∩ 𝐴); (ii) for any 𝑥 ∈ 𝑋, GFC(𝐻𝑗 ({V𝑗 ∈ 𝑉𝑗 : 𝐵𝑗 (𝑥, V𝑗 ) ⊂ 𝐶𝑗 (𝑥)})) ⊂ {𝑢𝑗 ∈ 𝑈𝑗 : 𝐴 𝑗 (𝑥, 𝑢𝑗 ) ⊂ 𝐶𝑗 (𝑥)}; (iii) if 𝑈 is not compact, then there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 such that a compact subset 𝐾 of 𝑈 such that 𝑈 \ 𝐾 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint{𝑢 ∈ 𝑈 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)} and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFC-subspace of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 ; (iv) for any compact subset 𝐵 of 𝑈, 𝐵 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint{𝑦𝑖 ∈ 𝑌𝑖 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)} ∩ 𝐵);
(27)
(v) for any 𝑢 ∈ 𝑈, GFC(𝑅𝑖 ({𝑦𝑖 ∈ 𝑌𝑖 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)})) ⊂ {𝑥𝑖 ∈ 𝑋𝑖 : 𝑃𝑖 (𝑥𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)}.
for all 𝑧𝑖 ∈ 𝑍𝑖 . It then follows from assumptions (i)–(iii) and Theorem 11 that for any 𝑖 ∈ 𝐼, there exists a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ ̂ which implies that (𝑥̂𝑖 , 𝑔𝑖 (𝑥)) ̂ ∈ 𝐵𝑖 . 𝑋 such that 𝑥̂𝑖 ∈ 𝑇𝑖 (𝑔𝑖 (𝑥)), So Theorem 16 is true.
̂ 𝑢̂) ∈ 𝑋 × 𝑈 and 𝑢̂ = (̂ Then there exist (𝑥, 𝑢𝑗 )𝑗∈𝐽 ∈ 𝑈 such that ̂ 𝑢̂𝑗 ) ⊂ 𝐶𝑗 (𝑥) ̂ and 𝑃𝑖 (𝑥̂𝑖 , 𝑢̂) ⊂ 𝐷𝑖 (̂ 𝑢) for each 𝑖 ∈ 𝐼 and 𝐴 𝑗 (𝑥, 𝑗 ∈ 𝐽.
𝑇𝑖 (𝑧𝑖 ) = {𝑥𝑖 ∈ 𝑋𝑖 : (𝑥𝑖 , 𝑧𝑖 ) ∈ 𝐵𝑖 }
Corollary 17. For each 𝑖 ∈ 𝐼, let (𝑋𝑖 , 𝑌𝑖 , {𝜑𝑁𝑖 }) be a GFC-space, 𝑅𝑖 : 𝑌𝑖 → 2𝑋𝑖 a set-valued mapping, 𝑋 = ∏𝑖∈𝐼 𝑋𝑖 , and 𝑋𝑖 = ∏𝑗∈𝐼,𝑗 ≠ 𝑖 𝑋𝑗 . Assume that {𝐴 𝑖 }𝑖∈𝐼 and {𝐵𝑖 }𝑖∈𝐼 are the families of subsets of 𝑌𝑖 × 𝑋𝑖 and 𝑋𝑖 × 𝑋𝑖 , respectively. If the following conditions hold: (i) for any compact subset 𝐷𝑖 of 𝑋𝑖 , 𝐷𝑖 = ⋃𝑦𝑖 ∈𝑌𝑖 (cint{𝑧𝑖 ∈ 𝑍𝑖 : (𝑦𝑖 , 𝑥𝑖 ) ∈ 𝐴 𝑖 } ∩ 𝐷𝑖 ), (ii) for all 𝑥𝑖 ∈ 𝑋𝑖 , GFC(𝑅𝑖 ({𝑦𝑖 ∈ 𝑌𝑖 : (𝑦𝑖 , 𝑥𝑖 ) ∈ 𝐴 𝑖 })) ⊂ {𝑥𝑖 ∈ 𝑋𝑖 : (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵𝑖 }, (iii) if 𝑋𝑖 is not compact, then there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 such that a compact subset 𝐷𝑖 of 𝑋𝑖 such that 𝑋𝑖 \ 𝐷𝑖 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint{𝑥𝑖 ∈ 𝑋𝑖 : (𝑦𝑖 , 𝑧𝑖 ) ∈ 𝐴 𝑖 } and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFC-subspace of 𝑋𝑖 relative to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 , then ⋂𝑖∈𝐼 𝐵𝑖 ≠ 0. 𝑖
Proof. Define a mapping 𝑔𝑖 : 𝑋 → 𝑋 as the projection of 𝑋 onto 𝑋𝑖 = ∏𝑗∈𝐼,𝑗 ≠ 𝑖 𝑋𝑗 . Clearly 𝑔𝑖 is a continuous mapping. Using Theorem 16 with 𝑍𝑖 = 𝑋𝑖 and 𝐷𝑖 = 𝐷𝑖 , we find that ̂ ∈ 𝐵𝑖 there exists a point 𝑥̂ = (𝑥̂𝑖 )𝑖∈𝐼 ∈ 𝑋 such that (𝑥̂𝑖 , 𝑔𝑖 (𝑥)) ̂ ∈ ⋂𝑖∈𝐼 𝐵𝑖 . for any 𝑖 ∈ 𝐼, which implies that 𝑥̂ = (𝑥̂𝑖 , 𝑔𝑖 (𝑥)) Hence ⋂𝑖∈𝐼 𝐵𝑖 ≠ 0.
Proof. Assume that 𝑆𝑖 : 𝑈 → 2𝑌𝑖 , 𝑇𝑖 : 𝑈 → 2𝑋𝑖 , 𝐹𝑗 : 𝑋 → 2𝑉𝑗 , and 𝐺𝑗 : 𝑋 → 2𝑈𝑗 are set-valued mappings as follows: 𝑆𝑖 (𝑢) = {𝑦𝑖 ∈ 𝑌𝑖 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)} , 𝑇𝑖 (𝑢) = {𝑥𝑖 ∈ 𝑋𝑖 : 𝑃𝑖 (𝑥𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)}
∀𝑢 ∈ 𝑈,
(28)
𝐹𝑗 (𝑥) = {V𝑗 ∈ 𝑉𝑗 : 𝐵𝑗 (𝑥, V𝑗 ) ⊂ 𝐶𝑗 (𝑥)} , 𝐺𝑗 (𝑥) = {𝑢𝑗 ∈ 𝑈𝑗 : 𝐴 𝑗 (𝑥, 𝑢𝑗 ) ⊂ 𝐶𝑗 (𝑥)}
∀𝑥 ∈ 𝑋.
(29)
It then follows from Theorem 14 that there exist 𝑥̂ ∈ 𝑋 and ̂ for any 𝑖 ∈ 𝐼 𝑢) and 𝑢̂𝑗 ∈ 𝐺𝑗 (𝑥) 𝑢̂ ∈ 𝑈 such that 𝑥̂𝑖 ∈ 𝑇𝑖 (̂ ̂ 𝑢̂𝑗 ) ⊂ and 𝑗 ∈ 𝐽. By the above definition, we obtain that 𝐴 𝑗 (𝑥, ̂ and 𝑃𝑖 (𝑥̂𝑖 , 𝑢̂) ⊂ 𝐷𝑖 (̂ 𝑢) for any 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. This 𝐶𝑗 (𝑥) completes the proof. Theorem 19. For each 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽, let 𝑌𝑖 and 𝑉𝑗 be topological spaces. Suppose that the following conditions hold: (i) for each 𝑥 ∈ 𝑋, the set {V𝑗 ∈ 𝑉𝑗 : 𝐵𝑗 (𝑥, V𝑗 ) ⊂ 𝐶𝑗 (𝑥)} is nonempty and 𝐵𝑗 (𝑥, V𝑗 ) is transfer compactly upper semicontinuous in 𝑥 with respect to 𝐶𝑗 ; (ii) for any 𝑥 ∈ 𝑋, GFC(𝐻𝑗 ({V𝑗 ∈ 𝑉𝑗 : 𝐵𝑗 (𝑥, V𝑗 ) ⊂ 𝐶𝑗 (𝑥)})) ⊂ {𝑢𝑗 ∈ 𝑈𝑗 : 𝐴 𝑗 (𝑥, 𝑢𝑗 ) ⊂ 𝐶𝑗 (𝑥)}; (iii) if 𝑈 is not compact, then there exists a nonempty subset of 𝑌𝑖0 of 𝑌𝑖 as well as a compact subset 𝐾 of 𝑈 such that
Abstract and Applied Analysis 𝑈 \ 𝐾 ⊂ ⋃𝑦𝑖 ∈𝑌𝑖0 cint{𝑢 ∈ 𝑈 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)} and for each 𝑁𝑖 ∈ ⟨𝑌𝑖 ⟩, there is a compact GFC-subspace of 𝑋𝑖 with respect to 𝐿 𝑁𝑖 containing 𝑌𝑖0 ∪ 𝑁𝑖 ; (iv) for each 𝑢 ∈ 𝑈, the set {𝑦𝑖 ∈ 𝑌𝑖 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)} is nonempty and 𝑄𝑖 (𝑦𝑖 , 𝑢) is transfer compactly upper semicontinuous in 𝑢 with respect to 𝐷𝑖 ; (v) for any 𝑢 ∈ 𝑈, GFC(𝑅𝑖 ({𝑦𝑖 ∈ 𝑌𝑖 : 𝑄𝑖 (𝑦𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)})) ⊂ {𝑥𝑖 ∈ 𝑋𝑖 : 𝑃𝑖 (𝑥𝑖 , 𝑢) ⊂ 𝐷𝑖 (𝑢)}. ̂ 𝑢̂) ∈ 𝑋 × 𝑈 and 𝑢̂ = (̂ Then there exist (𝑥, 𝑢𝑗 )𝑗∈𝐽 ∈ 𝑈 such that ̂ 𝑢̂𝑗 ) ⊂ 𝐶𝑗 (𝑥) ̂ and 𝑃𝑖 (𝑥̂𝑖 , 𝑢̂) ⊂ 𝐷𝑖 (̂ 𝑢) for all 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽. 𝐴 𝑗 (𝑥, Proof. Let 𝑆𝑖 , 𝑇𝑖 , 𝐹𝑗 , and 𝐺𝑗 be set-valued mappings as defined in the proof of Theorem 18. Using the same argument of Theorem 18, we only need to show that the assumptions (i) and (iv) of Theorem 18 hold. From assumption (i) and Lemma 4, we know that for any 𝑥 ∈ 𝑋, 𝐹𝑗−1 is transfer compactly open-valued. Thus Lemma 5 tells us that the assumption (i) of Theorem 18 holds. In a similar way, from (iv) and Lemma 4, we obtain that the assumption (iv) of Theorem 18 is satisfied. So the statement of Theorem 19 follows immediately from Theorem 18.
Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment The author would like to thank the anonymous referees for careful reading of the paper and for helpful comments and suggestions which improved its presentation.
References [1] X. P. Ding, “Maximal element theorems in product FC-spaces and generalized games,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 29–42, 2005. [2] X. P. Ding, “Collective fixed points and a system of coincidence theorems in product FC-spaces and applications,” Nonlinear Analysis: Theory, Methods and Applications, vol. 66, no. 11, pp. 2604–2617, 2007. [3] X. P. Ding and T. M. Ding, “KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1230–1245, 2007. [4] G. Tang, Q. Zhang, and C. Cheng, “W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1481–1491, 2007. [5] M. Fang and N. Huang, “KKM type theorems with applications to generalized vector equilibrium problems in FC-spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 3, pp. 809–817, 2007. [6] P. Q. Khanh and N. H. Quan, “Intersection theorems coincidence theorems and maximal-element theorems in GFCspaces,” Optimization, vol. 59, pp. 29–42, 2010.
7 [7] P. Q. Khanh, N. H. Quan, and J. Yao, “Generalized KKM-type theorems in GFC-spaces and applications,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 3-4, pp. 1227–1234, 2009. [8] X. P. Ding, “Coincidence theorems in topological spaces and their applications,” Applied Mathematics Letters, vol. 12, no. 7, pp. 99–105, 1999. [9] X. P. Ding and J. Y. Park, “Generalized vector equilibrium problems in generalized convex spaces,” Journal of Optimization Theory and Applications, vol. 120, no. 2, pp. 327–353, 2004. [10] E. Tarafdar, “Fixed point theorems in H-spaces and equilibrium points of abstract economies,” Journal of the Australian Mathematical Society. Series A, vol. 53, no. 1, pp. 252–260, 1992. [11] F. E. Browder, “Coincidence theorems, minimax theorems, and variational inequalities,” Contemporary Mathematics, vol. 26, pp. 67–80, 1984. [12] K. Lan and J. Webb, “New fixed point theorems for a family of mappings and applications to problems on sets with convex sections,” Proceedings of the American Mathematical Society, vol. 126, no. 4, pp. 1127–1132, 1998. [13] Q. H. Ansari and J. Yao, “A fixed point theorem and its applications to a system of variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 433–442, 1999. [14] X. Ding and J. Park, “New collectively fixed point theorems and applications in G-convex spaces,” Applied Mathematics and Mechanics. English Edition, vol. 23, no. 11, pp. 1237–1249, 2002. [15] X. P. Ding and J. Y. Park, “Collectively fixed-point theorems in noncompact G-convex spaces,” Applied Mathematics Letters, vol. 16, no. 3, pp. 137–142, 2003. [16] L. Lin and Q. H. Ansari, “Collective fixed points and maximal elements with applications to abstract economies,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 455– 472, 2004. [17] Z. Yu and L. Lin, “Continuous selection and fixed point theorems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 52, no. 2, pp. 445–455, 2003.
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
