Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 439176, 22 pages
doi:10.1155/2009/439176
Research Article

Fixed Point Theory for Admissible Type Maps with Applications

Ravi P. Agarwal1 and Donal O'Regan2

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2Department of Mathematics, National University of Ireland, Galway, Ireland

Received 8 December 2008; Accepted 18 June 2009

Academic Editor: Marlene Frigon

Copyright © 2009 Ravi P. Agarwal and Donal O'Regan. 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 present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Fréchet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals.

1. Introduction

In this paper, assuming a natural sequentially compact condition we establish new fixed point theorems for Urysohn type maps between Fréchet spaces. In Section 2 we present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces. Our theory is partly motivated by a variety of authors in the literature (see [1–6] and the references therein).

Existence in Section 2 is based on a Leray-Schauder alternative for Kakutani maps (see [4, 5, 7] for the history of this result) which we state here for the convenience of the reader.

Theorem 1.1. Let 𝐵 be a Banach space, 𝑈 an open subset of 𝐵 , and 0 ∈ 𝑈 . Suppose 𝑇 ∶ 𝑈 → 𝐶 𝐾 ( 𝐵 ) is an upper semicontinuous compact (or countably condensing) map (here 𝐶 𝐾 ( 𝐵 ) denotes the family of nonempty convex compact subsets of 𝐵 ). Then either (A1) 𝑇 has a fixed point in 𝑈 or(A2) there exists 𝑢 ∈ 𝜕 𝑈 (the boundary of 𝑈 in 𝐵 ) and 𝜆 ∈ ( 0 , 1 ) with 𝑢 ∈ 𝜆 𝑇 𝑢 .

Existence in Section 2 will also be based on the topological transversality theorem (see [5, 7] for the history of this result) which we now state here for the convenience of the reader. Let 𝐵 be a Banach space and 𝑈 an open subset of 𝐵 .

Definition 1.2. We let 𝐹 ∈ 𝐾 ( 𝑈 , 𝐵 ) denote the set of all upper semicontinuous compact (or countably condensing) maps 𝐹 ∶ 𝑈 → 𝐶 𝐾 ( 𝐸 ) .

Definition 1.3. We let 𝐹 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) if 𝐹 ∈ 𝐾 ( 𝑈 , 𝐵 ) with 𝑥 ∉ 𝐹 ( 𝑥 ) for 𝑥 ∈ 𝜕 𝑈 .

Definition 1.4. A map 𝐹 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) is essential in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) if for every 𝐺 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) with 𝐺 | 𝜕 𝑈 = 𝐹 | 𝜕 𝑈 there exists 𝑥 ∈ 𝑈 with 𝑥 ∈ 𝐺 ( 𝑥 ) . Otherwise 𝐹 is inessential in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) (i.e., there exists a fixed point free 𝐺 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) with 𝐺 | 𝜕 𝑈 = 𝐹 | 𝜕 𝑈 ).

Definition 1.5. 𝐹 , 𝐺 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) are homotopic in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) , written 𝐹 ≅ 𝐺 in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) , if there exists an upper semicontinuous compact (or countably condensing) map 𝑁 ∶ 𝑈 × [ 0 , 1 ] → 𝐶 𝐾 ( 𝐵 ) such that 𝑁 𝑡 ( 𝑢 ) = 𝑁 ( 𝑢 , 𝑡 ) ∶ 𝑈 → 𝐶 𝐾 ( 𝐵 ) belongs to 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) for each 𝑡 ∈ [ 0 , 1 ] and 𝑁 0 = 𝐹 with 𝑁 1 = 𝐺 .

Theorem 1.6. Let 𝐵 and 𝑈 be as above and let 𝐹 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) . Then the following conditions are equivalent: (i) 𝐹 is inessential in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) ;(ii)there exists a map 𝐺 ∈ 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) with 𝑥 ∉ 𝐺 ( 𝑥 ) for 𝑥 ∈ 𝑈 and 𝐹 ≅ 𝐺 in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) .

Theorem 1.6 immediately yields the topological transversality theorem for Kakutani maps.

Theorem 1.7. Let 𝐵 and 𝑈 be as above. Suppose that 𝐹 and 𝐺 are two maps in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) with 𝐹 ≅ 𝐺 in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) . Then 𝐹 is essential in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) if and only if 𝐺 is essential in 𝐾 𝜕 𝑈 ( 𝑈 , 𝐵 ) .

Also existence in Section 2 will be based on the following result of Petryshyn [8, Theorem  3].

Theorem 1.8. Let 𝐸 be a Banach space and let 𝐶 ⊆ 𝐸 be a closed cone. Let 𝑈 and 𝑉 be bounded open subsets in 𝐸 such that 0 ∈ 𝑈 ⊆ 𝑈 ⊆ 𝑉 and let 𝐹 ∶ 𝑊 → 𝐶 𝐾 ( 𝐶 ) be an upper semicontinuous, 𝑘 -set contractive (countably) map; here 0 ≤ 𝑘 < 1 , 𝑊 = 𝑉 ∩ 𝐶 and 𝑊 denotes the closure of 𝑊 in 𝐶 . Assume that (1) ‖ 𝑦 ‖ ≥ ‖ 𝑥 ‖ ∀ 𝑦 ∈ 𝐹 𝑥 and 𝑥 ∈ 𝜕 Ω and ‖ 𝑦 ‖ ≤ ‖ 𝑥 ‖ ∀ 𝑦 ∈ 𝐹 𝑥 and 𝑥 ∈ 𝜕 𝑊 (here Ω = 𝑈 ∩ 𝐶 and 𝜕 𝑊 denotes the boundary of 𝑊 in 𝐶 ) or(2) ‖ 𝑦 ‖ ≤ ‖ 𝑥 ‖ ∀ 𝑦 ∈ 𝐹 𝑥 and 𝑥 ∈ 𝜕 Ω and ‖ 𝑦 ‖ ≥ ‖ 𝑥 ‖ ∀ 𝑦 ∈ 𝐹 𝑥 and 𝑥 ∈ 𝜕 𝑊 . Then 𝐹 has a fixed point in 𝑊 ⧵ Ω .

Also in Section 2 we consider a class of maps which contain the Kakutani maps.

Suppose that 𝑋 and 𝑌 are Hausdorff topological spaces. Given a class 𝒳 of maps, 𝒳 ( 𝑋 , 𝑌 ) denotes the set of maps 𝐹 ∶ 𝑋 → 2 𝑌 (nonempty subsets of 𝑌 ) belonging to 𝒳 , and 𝒳 𝑐 the set of finite compositions of maps in 𝒳 . A class 𝒰 of maps is defined by the following properties:

(i) 𝒰 contains the class 𝒞 of single-valued continuous functions;(ii)each 𝐹 ∈ 𝒰 𝑐 is upper semicontinuous and compact valued; (iii)for any polytope 𝑃 , 𝐹 ∈ 𝒰 𝑐 ( 𝑃 , 𝑃 ) has a fixed point, where the intermediate spaces of composites are suitably chosen for each 𝒰 .

Definition 1.9. 𝐹 ∈ 𝒰 𝜅 𝑐 ( 𝑋 , 𝑌 ) if for any compact subset 𝐾 of 𝑋 , there is a 𝐺 ∈ 𝒰 𝑐 ( 𝐾 , 𝑌 ) with 𝐺 ( 𝑥 ) ⊆ 𝐹 ( 𝑥 ) for each 𝑥 ∈ 𝐾 .

The class 𝒰 𝜅 𝑐 is due to Park [9] and his papers include many examples in this class. Examples of 𝒰 𝜅 𝑐 maps are the Kakutani maps, the acyclic maps, the approximable maps, and the maps admissible in the sense of Gorniewicz.

Existence in Section 2 is based on a Leray-Schauder alternative [10] which we state here for the convenience of the reader.

Theorem 1.10. Let 𝐸 be a Banach space, 𝑈 an open convex subset of 𝐸 , and 0 ∈ 𝑈 . Suppose 𝐹 ∈ 𝒰 𝜅 𝑐 ( 𝑈 , 𝐸 ) is an upper semicontinuous countably condensing map with 𝑥 ∉ 𝜆 𝐹 𝑥 for 𝑥 ∈ 𝜕 𝑈 and 𝜆 ∈ ( 0 , 1 ) . Then 𝐹 has a fixed point in 𝑈 .

Also existence in Section 2 will be based on some Lefschetz type fixed point theory. Let 𝑋 , 𝑌 , and Γ be Hausdorff topological spaces. A continuous single-valued map 𝑝 ∶ Γ → 𝑋 is called a Vietoris map (written 𝑝 ∶ Γ ⇒ 𝑋 ) if the following two conditions are satisfied:

(i)for each 𝑥 ∈ 𝑋 , the set 𝑝 − 1 ( 𝑥 ) is acyclic,(ii) 𝑝 is a proper map, that is, for every compact 𝐴 ⊆ 𝑋 one has that 𝑝 − 1 ( 𝐴 ) is compact.

Let 𝐷 ( 𝑋 , 𝑌 ) be the set of all pairs 𝑋 𝑝 ⇐ Γ 𝑞 → 𝑌 where 𝑝 is a Vietoris map and 𝑞 is continuous. We will denote every such diagram by ( 𝑝 , 𝑞 ) . Given two diagrams ( 𝑝 , 𝑞 ) and ( 𝑝 , 𝑞 ) , where 𝑋 𝑝 ′ ⇐ Γ 𝑞 ′ → 𝑌 , we write ( 𝑝 , 𝑞 ) ∼ ( 𝑝 , 𝑞 ) if there are maps 𝑓 ∶ Γ → Γ and 𝑔 ∶ Γ → Γ such that 𝑞 ∘ 𝑓 = 𝑞 , 𝑝 ∘ 𝑓 = 𝑝 , 𝑞 ∘ 𝑔 = 𝑞 , and 𝑝 ∘ 𝑔 = 𝑝 . The equivalence class of a diagram ( 𝑝 , 𝑞 ) ∈ 𝐷 ( 𝑋 , 𝑌 ) with respect to ∼ is denoted by or 𝜙 = [ ( 𝑝 , 𝑞 ) ] and is called a morphism from 𝑋 to 𝑌 . We let 𝑀 ( 𝑋 , 𝑌 ) be the set of all such morphisms. For any 𝜙 ∈ 𝑀 ( 𝑋 , 𝑌 ) a set 𝜙 ( 𝑥 ) = 𝑞 𝑝 − 1 ( 𝑥 ) where 𝜙 = [ ( 𝑝 , 𝑞 ) ] is called an image of 𝑥 under a morphism 𝜙 .

Consider vector spaces over a field 𝐾 . Let 𝐸 be a vector space and 𝑓 ∶ 𝐸 → 𝐸 an endomorphism. Now let 𝑁 ( 𝑓 ) = { 𝑥 ∈ 𝐸 ∶ 𝑓 ( 𝑛 ) ( 𝑥 ) = 0 f o r s o m e 𝑛 } where 𝑓 ( 𝑛 ) is the 𝑛 th iterate of 𝑓 , and let 𝐸 = 𝐸 ⧵ 𝑁 ( 𝑓 ) . Since 𝑓 ( 𝑁 ( 𝑓 ) ) ⊆ 𝑁 ( 𝑓 ) one has the induced endomorphism 𝐸 𝑓 ∶ 𝐸 → . We call 𝑓 admissible if d i m 𝐸 < ∞ ; for such 𝑓 we define the generalized trace T r ( 𝑓 ) of 𝑓 by putting T r ( 𝑓 ) = t r ( 𝑓 ) where tr stands for the ordinary trace.

Let 𝑓 = { 𝑓 𝑞 } ∶ 𝐸 → 𝐸 be an endomorphism of degree zero of a graded vector space 𝐸 = { 𝐸 𝑞 } . We call 𝑓 a Leray endomorphism if (i) all 𝑓 𝑞 are admissible and (ii) almost all 𝐸 𝑞 are trivial. For such 𝑓 we define the generalized Lefschetz number Λ ( 𝑓 ) by

Let 𝐻 be the ̆ C ech homology functor with compact carriers and coefficients in the field of rational numbers 𝐾 from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus 𝐻 ( 𝑋 ) = { 𝐻 𝑞 ( 𝑋 ) } is a graded vector space, with 𝐻 𝑞 ( 𝑋 ) being the 𝑞 -dimensional ̆ C ech homology group with compact carriers of 𝑋 . For a continuous map 𝑓 ∶ 𝑋 → 𝑋 , 𝐻 ( 𝑓 ) is the induced linear map 𝑓 ⋆ = { 𝑓 ⋆ 𝑞 } where 𝑓 ⋆ 𝑞 ∶ 𝐻 𝑞 ( 𝑋 ) → 𝐻 𝑞 ( 𝑋 ) .

The ̆ C ech homology functor can be extended to a category of morphisms (see [11, page 364]) and also note that the homology functor 𝐻 extends over this category, that is, for a morphism we define the induced map by putting 𝜙 ⋆ = 𝑞 ⋆ ∘ 𝑝 ⋆ − 1 .

Let 𝜙 ∶ 𝑋 → 𝑌 be a multivalued map (note for each 𝑥 ∈ 𝑋 we assume 𝜙 ( 𝑥 ) is a nonempty subset of 𝑌 ). A pair ( 𝑝 , 𝑞 ) of single valued continuous maps of the form 𝑋 𝑝 ← Γ 𝑞 → 𝑌 is called a selected pair of 𝜙 (written ( 𝑝 , 𝑞 ) ⊂ 𝜙 ) if the following two conditions hold:

(i) 𝑝 is a Vietoris map, (ii) 𝑞 ( 𝑝 − 1 ( 𝑥 ) ) ⊂ 𝜙 ( 𝑥 ) for any 𝑥 ∈ 𝑋 .

Definition 1.11. An upper semicontinuous compact map 𝜙 ∶ 𝑋 → 𝑌 is said to be admissible (and we write 𝜙 ∈ 𝐴 𝑑 ( 𝑋 , 𝑌 ) ) provided that there exists a selected pair ( 𝑝 , 𝑞 ) of 𝜙 .

Definition 1.12. An upper semicontinuous map 𝜙 ∶ 𝑋 → 𝑌 is said to be admissible in the sense of Gorniewicz (and we write 𝜙 ∈ 𝐴 𝐷 ( 𝑋 , 𝑌 ) ) provided that there exists a selected pair ( 𝑝 , 𝑞 ) of 𝜙 .

Definition 1.13. A map 𝜙 ∈ 𝐴 𝑑 ( 𝑋 , 𝑋 ) is said to be a Lefschetz map if for each selected pair ( 𝑝 , 𝑞 ) ⊂ 𝜙 the linear map 𝑞 ⋆ 𝑝 ⋆ − 1 ∶ 𝐻 ( 𝑋 ) → 𝐻 ( 𝑋 ) (the existence of 𝑝 ⋆ − 1 follows from the Vietoris theorem) is a Leray endomorphism.

If 𝜙 ∶ 𝑋 → 𝑋 is a Lefschetz map, we define the Lefschetz set Λ ( 𝜙 ) (or Λ 𝑋 ( 𝜙 ) ) by

Definition 1.14. A Hausdorff topological space 𝑋 is said to be a Lefschetz space provided that every 𝜙 ∈ 𝐴 𝑑 ( 𝑋 , 𝑋 ) is a Lefschetz map and Λ ( 𝜙 ) ≠ { 0 } that implies 𝜙 has a fixed point.

Also we present Krasnoselskii compression and expansion theorems in Section 2 in the Fréchet space setting. Let 𝐸 = ( 𝐸 , | ⋅ | ) be a normed linear space and 𝐶 ⊆ 𝐸 a closed cone. For 𝑟 > 0 let 𝐵 𝐶 ( 0 , 𝑟 ) = { 𝑥 ∈ 𝐶 ∶ | 𝑥 | ≤ 𝑟 } and it is well known that 𝐵 𝐶 ( 0 , 𝑟 ) = 𝐵 ( 0 , 𝑅 ) ∩ 𝐶 where 𝐵 ( 0 , 𝑟 ) = { 𝑥 ∈ 𝐸 ∶ | 𝑥 | ≤ 𝑟 } . Our next result, Theorem 1.8, was established in [12] and Theorem 1.10 can be found in [13].

Theorem 1.15. Let 𝐸 = ( 𝐸 , | ⋅ | ) be a normed linear space, 𝐶 ⊆ 𝐸 a closed cone, 𝑟 , 𝑅 constants, and 0 < 𝑟 < 𝑅 . Suppose that 𝐹 ∈ 𝒰 𝜅 𝑐 ( 𝐵 ( 0 , 𝑅 ) ∩ 𝐶 , 𝐶 ) is compact with Then 𝐹 has a fixed point in 𝐵 𝐶 𝑟 , 𝑅 = { 𝑥 ∈ 𝐶 ∶ 𝑟 ≤ ‖ 𝑥 ‖ ≤ 𝑅 } .

Theorem 1.16. Let 𝐸 = ( 𝐸 , | ⋅ | ) be a normed linear space, 𝐶 ⊆ 𝐸 a closed cone, 𝑟 , 𝑅 constants, and 0 < 𝑟 < 𝑅 . Suppose that 𝐹 ∈ 𝐴 𝐷 ( 𝐶 , 𝐶 ) is completely continuous with Then 𝐹 has a fixed point in 𝐵 𝐶 𝑟 , 𝑅 .

Now let 𝐼 be a directed set with order ≤ and let { 𝐸 𝛼 } 𝛼 ∈ 𝐼 be a family of locally convex spaces. For each 𝛼 ∈ 𝐼 , 𝛽 ∈ 𝐼 for which 𝛼 ≤ 𝛽 let 𝜋 𝛼 , 𝛽 ∶ 𝐸 𝛽 → 𝐸 𝛼 be a continuous map. Then the set is a closed subset of ∏ 𝛼 ∈ 𝐼 𝐸 𝛼 and is called the projective limit of { 𝐸 𝛼 } 𝛼 ∈ 𝐼 and is denoted by l i m ← 𝐸 𝛼 (or l i m ← { 𝐸 𝛼 , 𝜋 𝛼 , 𝛽 } or the generalized intersection [14, page 439] ⋂ 𝛼 ∈ 𝐼 𝐸 𝛼 ).

2. Fixed Point Theory in Fréchet Spaces

Let 𝐸 = ( 𝐸 , { | ⋅ | 𝑛 } 𝑛 ∈ 𝑁 ) be a Fréchet space with the topology generated by a family of seminorms { | ⋅ | 𝑛 ∶ 𝑛 ∈ 𝑁 } ; here 𝑁 = { 1 , 2 , … } . We assume that the family of seminorms satisfies A subset 𝑋 of 𝐸 is bounded if for every 𝑛 ∈ 𝑁 there exists 𝑟 𝑛 > 0 such that | 𝑥 | 𝑛 ≤ 𝑟 𝑛 for all 𝑥 ∈ 𝑋 . For 𝑟 > 0 and 𝑥 ∈ 𝐸 we denote 𝐵 ( 𝑥 , 𝑟 ) = { 𝑦 ∈ 𝐸 ∶ | 𝑥 − 𝑦 | 𝑛 ≤ 𝑟 ∀ 𝑛 ∈ 𝑁 } . To 𝐸 we associate a sequence of Banach spaces { ( 𝐄 𝑛 , | ⋅ | 𝑛 ) } described as follows. For every 𝑛 ∈ 𝑁 we consider the equivalence relation ∼ 𝑛 defined by We denote by 𝐄 𝑛 = ( 𝐸 / ∼ 𝑛 , | ⋅ | 𝑛 ) the quotient space, and by ( 𝐄 𝑛 , | ⋅ | 𝑛 ) the completion of 𝐄 𝑛 with respect to | ⋅ | 𝑛 (the norm on 𝐄 𝑛 induced by | ⋅ | 𝑛 and its extension to 𝐄 𝑛 is still denoted by | ⋅ | 𝑛 ). This construction defines a continuous map 𝜇 𝑛 ∶ 𝐸 → 𝐄 𝑛 . Now since (2.1) is satisfied the seminorm | ⋅ | 𝑛 induces a seminorm on 𝐄 𝑚 for every 𝑚 ≥ 𝑛 (again this seminorm is denoted by | ⋅ | 𝑛 ). Also (2.2) defines an equivalence relation on 𝐄 𝑚 from which we obtain a continuous map 𝜇 𝑛 , 𝑚 ∶ 𝐄 𝑚 → 𝐄 𝑛 since 𝐄 𝑚 / ∼ 𝑛 can be regarded as a subset of 𝐄 𝑛 . Now 𝜇 𝑛 , 𝑚 𝜇 𝑚 , 𝑘 = 𝜇 𝑛 , 𝑘 if 𝑛 ≤ 𝑚 ≤ 𝑘 and 𝜇 𝑛 = 𝜇 𝑛 , 𝑚 𝜇 𝑚 if 𝑛 ≤ 𝑚 . We now assume the following condition holds:

Remark 2.1. (i) For convenience the norm on 𝐸 𝑛 is denoted by | ⋅ | 𝑛 .
(ii) In our applications 𝐄 𝑛 = 𝐄 𝑛 for each 𝑛 ∈ 𝑁 .
(iii) Note if 𝑥 ∈ 𝐄 𝑛 (or 𝐄 𝑛 ) then 𝑥 ∈ 𝐸 . However if 𝑥 ∈ 𝐸 𝑛 then 𝑥 is not necessaily in 𝐸 and in fact 𝐸 𝑛 is easier to use in applications (even though 𝐸 𝑛 is isomorphic to 𝐄 𝑛 ). For example if 𝐸 = 𝐶 [ 0 , ∞ ) , then 𝐄 𝑛 consists of the class of functions in 𝐸 which coincide on the interval [ 0 , 𝑛 ] and 𝐸 𝑛 = 𝐶 [ 0 , 𝑛 ] .

Finally we assume (here we use the notation from [14], i.e., decreasing in the generalized sense) Let l i m ← 𝐸 𝑛 (or ⋂ ∞ 1 𝐸 𝑛 where ⋂ ∞ 1 is the generalized intersection [14]) denote the projective limit of { 𝐸 𝑛 } 𝑛 ∈ 𝑁 (note 𝜋 𝑛 , 𝑚 = 𝑗 𝑛 𝜇 𝑛 , 𝑚 𝑗 𝑚 − 1 ∶ 𝐸 𝑚 → 𝐸 𝑛 for 𝑚 ≥ 𝑛 ) and note l i m ← 𝐸 𝑛 ≅ 𝐸 , so for convenience we write 𝐸 = l i m ← 𝐸 𝑛 .

For each 𝑋 ⊆ 𝐸 and each 𝑛 ∈ 𝑁 we set 𝑋 𝑛 = 𝑗 𝑛 𝜇 𝑛 ( 𝑋 ) , and we let 𝑋 𝑛 , i n t 𝑋 𝑛 and 𝜕 𝑋 𝑛 denote, respectively, the closure, the interior, and the boundary of 𝑋 𝑛 with respect to | ⋅ | 𝑛 in 𝐸 𝑛 . Also the pseudointerior of 𝑋 is defined by The set 𝑋 is pseudoopen if 𝑋 = p s e u d o − i n t ( 𝑋 ) . For 𝑟 > 0 and 𝑥 ∈ 𝐸 𝑛 we denote 𝐵 𝑛 ( 𝑥 , 𝑟 ) = { 𝑦 ∈ 𝐸 𝑛 ∶ | 𝑥 − 𝑦 | 𝑛 ≤ 𝑟 } .

We now show how easily one can extend fixed point theory in Banach spaces to applicable fixed point theory in Fréchet spaces. In this case the map 𝐹 𝑛 will be related to 𝐹 by the closure property (2.11).

Theorem 2.2. Let 𝐸 and 𝐸 𝑛 be as described above, 𝑋 a subset of 𝐸 and 𝐹 ∶ 𝑌 → 2 𝐸 where i n t 𝑋 𝑛 ⊆ 𝑌 𝑛 for each 𝑛 ∈ 𝑁 . Also for each 𝑛 ∈ 𝑁 assume that there exists 𝐹 𝑛 ∶ i n t 𝑋 𝑛 → 2 𝐸 𝑛 and suppose the following conditions are satisfied: Then 𝐹 has a fixed point in 𝐸 .

Remark 2.3. Notice that to check (2.10) we need to show that for each 𝑘 ∈ 𝑁 the sequence { 𝑗 𝑘 𝜇 𝑘 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) } 𝑛 ∈ 𝑁 𝑘 − 1 ⊆ i n t 𝑋 𝑘 is sequentially compact.

Proof. From Theorem 1.1 for each 𝑛 ∈ 𝑁 there exists 𝑦 𝑛 ∈ i n t 𝑋 𝑛 with 𝑦 𝑛 ∈ 𝐹 𝑛 𝑦 𝑛 (we apply Theorem 1.1 with 𝑈 = i n t 𝑋 𝑛 and note 𝑗 𝑛 𝜇 𝑛 ( 0 ) ∈ 𝑋 𝑛 ⧵ 𝜕 𝑋 𝑛 = i n t 𝑋 𝑛 ). Let us look at { 𝑦 𝑛 } 𝑛 ∈ 𝑁 . Notice 𝑦 1 ∈ i n t 𝑋 1 and 𝑗 1 𝜇 1 , 𝑘 𝑗 𝑘 − 1 ( 𝑦 𝑘 ) ∈ i n t 𝑋 1 for 𝑘 ∈ { 2 , 3 , … } from (2.7). Now (2.10) with 𝑘 = 1 guarantees that there exists a subsequence 𝑁 1 ⊆ { 2 , 3 , … } and a 𝑧 1 ∈ i n t 𝑋 1 with 𝑗 1 𝜇 1 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) → 𝑧 1 in 𝐸 1 as 𝑛 → ∞ in 𝑁 1 . Look at { 𝑦 𝑛 } 𝑛 ∈ 𝑁 1 . Now 𝑗 2 𝜇 2 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) ∈ i n t 𝑋 2 for 𝑘 ∈ 𝑁 1 . Now (2.10) with 𝑘 = 2 guarantees that there exists a subsequence 𝑁 2 ⊆ { 3 , 4 , … } of 𝑁 1 and a 𝑧 2 ∈ i n t 𝑋 2 with 𝑗 2 𝜇 2 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) → 𝑧 2 in 𝐸 2 as 𝑛 → ∞ in 𝑁 2 . Note from (2.4) and the uniqueness of limits that 𝑗 1 𝜇 1 , 2 𝑗 2 − 1 𝑧 2 = 𝑧 1 in 𝐸 1 since 𝑁 2 ⊆ 𝑁 1 (note 𝑗 1 𝜇 1 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) = 𝑗 1 𝜇 1 , 2 𝑗 2 − 1 𝑗 2 𝜇 2 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) for 𝑛 ∈ 𝑁 2 ). Proceed inductively to obtain subsequences of integers and 𝑧 𝑘 ∈ i n t 𝑋 𝑘 with 𝑗 𝑘 𝜇 𝑘 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) → 𝑧 𝑘 in 𝐸 𝑘 as 𝑛 → ∞ in 𝑁 𝑘 . Note 𝑗 𝑘 𝜇 𝑘 , 𝑘 + 1 𝑗 − 1 𝑘 + 1 𝑧 𝑘 + 1 = 𝑧 𝑘 in 𝐸 𝑘 for 𝑘 ∈ { 1 , 2 , … } .
Fix 𝑘 ∈ 𝑁 . Note for every 𝑚 ≥ 𝑘 . We can do this for each 𝑘 ∈ 𝑁 . As a result 𝑦 = ( 𝑧 𝑘 ) ∈ l i m ← 𝐸 𝑛 = 𝐸 and also note 𝑦 ∈ 𝑌 since 𝑧 𝑘 ∈ i n t 𝑋 𝑘 ⊆ 𝑌 𝑘 for each 𝑘 ∈ 𝑁 . Also since 𝑦 𝑛 ∈ 𝐹 𝑛 𝑦 𝑛 in 𝐸 𝑛 for 𝑛 ∈ 𝑁 𝑘 and 𝑗 𝑘 𝜇 𝑘 , 𝑛 𝑗 𝑛 − 1 ( 𝑦 𝑛 ) → 𝑧 𝑘 = 𝑦 in 𝐸 𝑘 as 𝑛 → ∞ in 𝑁 𝑘 one has from (2.11) that 𝑦 ∈ 𝐹 𝑦 in 𝐸 .

Remark 2.4. From the proof we see that condition (2.7) can be removed from the statement of Theorem 2.2. We include it only to explain condition (2.10) (see Remark 2.3).

Remark 2.5. Note that we could replace i n t 𝑋 𝑛 ⊆ 𝑌 𝑛 above with i n t 𝑋 𝑛 a subset of the closure of 𝑌 𝑛 in 𝐸 𝑛 if 𝑌 is a closed subset of 𝐸 (so in this case we can take 𝑌 = 𝑋 if 𝑋 is a closed subset of 𝐸 ). To see this note 𝑧 𝑘 ∈ i n t 𝑋 𝑘 , 𝑦 = ( 𝑧 𝑘 ) ∈ l i m ← 𝐸 𝑛 = 𝐸 and 𝜋 𝑘 , 𝑚 ( 𝑦 𝑚 ) → 𝑧 𝑘 in 𝐸 𝑘 as 𝑚 → ∞ and we can conclude that 𝑦 ∈ 𝑌 = 𝑌 (note that 𝑞 ∈ 𝑌 if and only if for every 𝑘 ∈ 𝑁 there exists ( 𝑥 𝑘 , 𝑚 ) ∈ 𝑌 , 𝑥 𝑘 , 𝑚 = 𝜋 𝑘 , 𝑛 ( 𝑥 𝑛 , 𝑚 ) for 𝑛 ≥ 𝑘 with 𝑥 𝑘 , 𝑚 → 𝑗 𝑘 𝜇 𝑘 ( 𝑞 ) in 𝐸 𝑘 as 𝑚 → ∞ ).

Remark 2.6. Suppose in Theorem 2.2 we replace (2.10) with In addition we assume 𝐹 ∶ 𝑌 → 2 𝐸 with i n t 𝑋 𝑛 ⊆ 𝑌 𝑛 for each 𝑛 ∈ 𝑁 is replaced by 𝐹 ∶ 𝑋 → 2 𝐸 and suppose (2.11) is true with 𝑤 ∈ 𝑌 replaced by 𝑤 ∈ 𝑋 . Then the result in Theorem 2.2 is again true.

The proof follows the reasoning in Theorem 2.2 except in this case 𝑧 𝑘 ∈ i n t 𝑋 𝑘 and 𝑦 ∈ 𝑋 .

Remark 2.7. In fact we could replace (in fact we can remove it as mentioned in Remark 2.4) (2.7) in Theorem 2.2 with and the result above is again true.

Remark 2.8. Usually in our applications one has 𝜕 𝑋 𝑛 = 𝜕 i n t 𝑋 𝑛 (so 𝑋 𝑛 = i n t 𝑋 𝑛 ). If 𝑋 is a pseudoopen subset of 𝐸 then for each 𝑛 ∈ 𝑁 one has (see [15]) that 𝑋 𝑛 is a open subset of 𝐸 𝑛 so i n t 𝑋 𝑛 = 𝑋 𝑛 .

Essentially the same reasoning as in Theorem 2.2 (now using Theorem 1.7) establishes the following result. We will need the following definitions.

Let 𝐸 and 𝐸 𝑛 be as described in Section 2. For the definitions below 𝑋 ⊆ 𝐸 and 𝐹 ∶ 𝑌 → 2 𝐸 with i n t 𝑋 𝑛 ⊆ 𝑌 𝑛 for each 𝑛 ∈ 𝑁 (or i n t 𝑋 𝑛 a subset of the closure of 𝑌 𝑛 in 𝐸 𝑛 if 𝑌 is a closed subset of 𝐸 ). In addition assume for each 𝑛 ∈ 𝑁 that 𝐹 𝑛 ∶ i n t 𝑋 𝑛 → 2 𝐸 𝑛 .

Definition 2.9. We say 𝐹 ∈ 𝐾 ( 𝑌 , 𝐸 ) if for each 𝑛 ∈ 𝑁 one has 𝐹 𝑛 ∈ 𝐾 ( i n t 𝑋 𝑛 , 𝐸 𝑛 ) (i.e., for each 𝑛 ∈ 𝑁 , 𝐹 𝑛 ∶ i n t 𝑋 𝑛 → 𝐶 𝐾 ( 𝐸 𝑛 ) is an upper semicontinuous countably condensing map).

Definition 2.10. 𝐹 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) if 𝐹 ∈ 𝐾 ( 𝑌 , 𝐸 ) and for each 𝑛 ∈ 𝑁 one has 𝑥 ∉ 𝐹 𝑛 ( 𝑥 ) for 𝑥 ∈ 𝜕 i n t 𝑋 𝑛 .

Definition 2.11. 𝐹 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) is essential in 𝐾 𝜕 ( 𝑌 , 𝐸 ) if for each 𝑛 ∈ 𝑁 one has that 𝐹 𝑛 ∈ 𝐾 𝜕 i n t 𝑋 𝑛 ( i n t 𝑋 𝑛 , 𝐸 𝑛 ) is essential in 𝐾 𝜕 i n t 𝑋 𝑛 ( i n t 𝑋 𝑛 , 𝐸 𝑛 ) (i.e., for each 𝑛 ∈ 𝑁 , every map 𝐺 𝑛 ∈ 𝐾 𝜕 i n t 𝑋 𝑛 ( i n t 𝑋 𝑛 , 𝐸 𝑛 ) with 𝐺 𝑛 | 𝜕 i n t 𝑋 𝑛 = 𝐹 𝑛 | 𝜕 i n t 𝑋 𝑛 has a fixed point in i n t 𝑋 𝑛 ).

Remark 2.12. Note that if 𝑗 𝑛 𝜇 𝑛 ( 0 ) ∈ 𝑈 𝑛 for each 𝑛 ∈ 𝑁 then 0 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) is essential in 𝐾 𝜕 ( 𝑌 , 𝐸 ) (see [7]).

Definition 2.13. (We assume 𝑗 𝑛 𝜇 𝑛 ( 0 ) ∈ i n t 𝑋 𝑛 for 𝑛 ∈ 𝑁 .) 𝐹 , 0 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) are homotopic in 𝐾 𝜕 ( 𝑌 , 𝐸 ) , written 𝐹 ≅ 𝐺 in 𝐾 𝜕 ( 𝑌 , 𝐸 ) , if for each 𝑛 ∈ 𝑁 one has 𝐹 𝑛 ≅ 𝑗 𝑛 𝜇 𝑛 ( 0 ) in 𝐾 𝜕 i n t 𝑋 𝑛 ( i n t 𝑋 𝑛 , 𝐸 𝑛 ) .

Theorem 2.14. Let 𝐸 and 𝐸 𝑛 be as described above, 𝑋 a subset of 𝐸 and 𝐹 ∶ 𝑌 → 2 𝐸 where i n t 𝑋 𝑛 ⊆ 𝑌 𝑛 for each 𝑛 ∈ 𝑁 or i n t 𝑋 𝑛 a subset of the closure of 𝑌 𝑛 in 𝐸 𝑛 (if 𝑌 is a closed subset of 𝐸 ). Also for each 𝑛 ∈ 𝑁 assume that there exists 𝐹 𝑛 ∶ i n t 𝑋 𝑛 → 2 𝐸 𝑛 and suppose 𝐹 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) , (2.6), (2.7), and the following condition holds: Also assume (2.10) and (2.11) hold. Then 𝐹 has a fixed point in 𝐸 .

Proof. Fix 𝑛 ∈ 𝑁 . Now Remark 2.12 guarantees that the zero map (i.e., 𝐺 ( 𝑥 ) = 𝑗 𝑛 𝜇 𝑛 ( 0 ) ) is essential in 𝐾 𝜕 𝑈 𝑛 ( 𝑈 𝑛 , 𝐸 𝑛 ) for each 𝑛 ∈ 𝑁 . Now Theorem 1.7 guarantees that 𝐹 𝑛 is essential in 𝐾 𝜕 𝑈 𝑛 ( 𝑈 𝑛 , 𝐸 𝑛 ) so in particular there exists 𝑦 𝑛 ∈ 𝑈 𝑛 with 𝑦 𝑛 ∈ 𝐹 𝑛 𝑦 𝑛 . Essentially the same reasoning as in Theorem 2.2 (with Remark 2.5) establishes the result.

Remark 2.15. Notice that (2.6) and (2.17) could be replaced by 𝐹 ≅ 𝐺 in 𝐾 𝜕 ( 𝑌 , 𝐸 ) (of course we assume 𝐺 ∈ 𝐾 𝜕 ( 𝑌 , 𝐸 ) and we must specify 𝐺 𝑛 for 𝑛 ∈ 𝑁 here).

Remark 2.16. Condition (2.7) can be removed from the statement of Theorem 2.14.

Remark 2.17. Note that Remark 2.6 holds in this situation also.

As an application of Theorem 2.2 we discuss the integral equation

Theorem 2.18. Let 1 ≤ 𝑝 < ∞ be a constant and 1 < 𝑞 ≤ ∞ the conjugate to 𝑝 . Suppose the following conditions are satisfied: Then (2.17) has at least one solution in 𝐄 𝑘 .

Remark 2.19. One could also obtain a multivalued version of Theorem 2.18 by using the ideas in the proof below with the ideas in [16].

Proof. Here 𝐸 , [ 0 , 𝑘 ] consists of the class of functions in 𝐸 𝑘 = 𝐶 [ 0 , 𝑘 ] which coincide on the interval 𝜋 𝑛 , 𝑚 = 𝑗 𝑛 𝜇 𝑛 , 𝑚 𝑗 𝑚 − 1 ∶ 𝐸 𝑚 → 𝐸 𝑛 , 𝜋 𝑛 , 𝑚 ( 𝑥 ) = 𝑥 | [ 0 , 𝑛 ] with of course [ 𝑋 = 𝑢 ∈ 𝐶 0 , ∞ ) ∶ | 𝑢 | 𝑛 ≤ 𝑟 f o r e a c h 𝑛 ∈ 𝑁 , ( 2 . 2 4 ) defined by | 𝑢 | 𝑛 = s u p 𝑡 ∈ [ 0 , 𝑛 ] | 𝑢 ( 𝑡 ) | . We will apply Theorem 2.2 with here 𝑋 𝑛 = 𝑋 𝑛 = [ ] 𝑢 ∈ 𝐶 0 , 𝑛 ∶ | 𝑢 | 𝑛 ≤ 𝑟 ( 2 . 2 5 ) . Fix i n t 𝑋 𝑛 = [ ] ∶ 𝑢 ∈ 𝐶 0 , 𝑛 | 𝑢 | 𝑛 < 𝑟 . ( 2 . 2 6 ) and note with Let 𝑌 = 𝑋 be given by Also let 𝐹 𝑦 ( 𝑡 ) = ∞ 0 𝐾 ( 𝑡 , 𝑠 ) 𝑓 ( 𝑠 , 𝑦 ( 𝑠 ) ) 𝑑 𝑠 . ( 2 . 2 8 ) (we will use Remark 2.5) and let 𝐹 ∶ i n t 𝑋 𝑛 → 𝐸 𝑛 be given by Clearly (2.6) and (2.7) hold, and a standard argument in the literature guarantees that 𝑥 ∈ 𝜕 i n t 𝑋 𝑛 is continuous and compact so (2.8) holds. To show (2.9) fix | 𝑥 | 𝑛 = 𝑟 and suppose that there exists 𝜆 ∈ ( 0 , 1 ] (so 𝑥 = 𝜆 𝐹 𝑛 𝑥 ) and 𝑡 ∈ [ 0 , 𝑛 ] with | | | | 𝑥 ( 𝑡 ) ≤ 𝜓 | 𝑥 | 𝑛 𝑛 0 | | | | 𝐾 ( 𝑡 , 𝑠 ) 𝜙 ( 𝑠 ) 𝑑 𝑠 ≤ 𝜓