Fixed Point Theory for Admissible Type Maps with Applications

Fixed Point Theory and Applications, Jul 2009

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.

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Fixed Point Theory for Admissible Type Maps with Applications

Fixed Point Theory and Applications Hindawi Publishing Corporation Fixed Point Theory for Admissible Type Maps with Applications Ravi P. Agarwal 1 Donal O'Regan 0 0 Department of Mathematics, National University of Ireland , Galway , Ireland 1 Department of Mathematical Sciences, Florida Institute of Technology , Melbourne, FL 32901 , USA We present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Fre´chet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals. Correspondence should be addressed to Ravi P; Agarwal; agarwal@fit; edu 1. Introduction In this paper, assuming a natural sequentially compact condition we establish new fixed point theorems for Urysohn type maps between Fre´chet spaces. In Section 2 we present new LeraySchauder alternatives, Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Fre´chet 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 → CK B is an upper semicontinuous compact (or countably condensing) map (here CK B denotes the family of nonempty convex compact subsets of B). Then either A1 T has a fixed point in U or 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 → CK E . 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 → CK B such that Nt u N u, t : U → CK B belongs to K∂U U, B for each t ∈ 0, 1 and N0 F with N1 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 ; Theorem 1.6 immediately yields the topological transversality theorem for Kakutani 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 → CK 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 y ≥ x ∀y ∈ Fx and x ∈ ∂Ω and y ≤ x ∀y ∈ Fx and x ∈ ∂W (here Ω and ∂W denotes the boundary of W in C) or U ∩ C y ≤ x ∀y ∈ F x and x ∈ ∂Ω and y ≥ x ∀y ∈ Fx and x ∈ ∂W. 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 → 2Y nonempty subsets of Y belonging to X, and Xc 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 ∈ Uc is upper semicontinuous and compact valued; iii for any polytope P , F ∈ Uc P, P has a fixed point, where the intermediate spaces of composites are suitably chosen for each U. Definition 1.9. F ∈ Ucκ X, Y if for any compact subset K of X, there is a G ∈ Uc K, Y with G x ⊆ F x for each x ∈ K. Theorem 1.10. Let E be a Banach space, U an open convex subset of E, and 0 ∈ U. Suppose F ∈ Ucκ U, E is an upper semicontinuous countably condensing map with x /∈ λFx 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, 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 qp−1 x where φ p, q is called an image of x under a morphism φ. Consider (...truncated)


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Ravi P. Agarwal, Donal O'Regan. Fixed Point Theory for Admissible Type Maps with Applications, Fixed Point Theory and Applications, 2009, pp. 439176, Volume 2009, Issue 1, DOI: 10.1155/2009/439176