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)