Family of mappings with an equicontractive-type condition
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Journal of Fixed Point Theory and Applications
Family of mappings with an equicontractivetype condition Dariusz Wardowski Abstract. In a real Banach space X and a complete metric space M , we consider a compact mapping C defined on a closed and bounded subset A of X with values in M and the operator T : A × C(A) → X. Using a new type of equicontractive condition for a certain family of mappings and β-condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator x → T (x, C(x)) has a fixed point. The obtained results are applied to the initial value problem. Mathematics Subject Classification. Primary: 47H10, 47N20. Keywords. Compact operator, mappings of equicontractive type, operator equation, fixed point, hausdorff measure of noncompactness.
1. Introduction and preliminaries The investigations concerning compact operators together with contractive mappings have their origin in the famous Krasnosel’ski˘ı’s result [7]. This known theorem states that if M is a nonempty closed convex and bounded subset of the given Banach space X and there are given two mappings: a contraction A : M → X and a compact operator B : M → X satisfying A(M ) + B(M ) ⊂ M then A + B has a fixed point. In the literature, one can find many contributions, where the authors extend this idea. In [3], Burton replaced the Banach contractive condition with the more general so-called large contraction. In [4], the authors merged the concepts due to Krasnosel’ski˘ı with the Schaefer’s result [11]. In addition, in [9], Reich considered condensing mappings with bounded ranges and applied them to obtain the Schaefer’s alternative and a Krasnosel’ski˘ı type fixed point theorem. Using Krasnosel’ski˘ı-Schaefer type method, Vetro and Wardowski [12] have recently proved an existence theorem producing a periodic solution of a nonlinear integral equation. Przeradzki in his work [8], using a concept of Hausdorff measure of noncompactness, relaxed a strong condition: A(M ) + B(M ) ⊂ M, 0123456789().: V,-vol
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by considering a weaker one (A + B)(M ) ⊂ M, where a contractive type operator is a generalized contraction. In addition, in [14], there was investigated a wide class of (ϕ, F )-contractions. The author proved that a certain subclass of these mappings is β-condensing. Applying the Sadovskii’s result, the fixed point result for the sum of compact mapping with (ϕ, F )-contraction was obtained. On the other hand in [6], Karakostas gave an extension of Krasnosel’ski˘ı’s theorem by involving both operators (contractive and compact) in the resulting one given in an implicit form. In this way, the author was interested in finding a solution of the equation given by the formula: x = T (x, C(x)), (1) where C : A → Y , T : A × C(A) → X, A is a subset of a real Banach space X and Y is a metric space. In the present paper, we prove two theorems which improve the results in [6]. One of the derived tools will be applied to some nonlinear problem which, according to the author’s knowledge, cannot be solved using th
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