EIGENVALUES OF COUNTABLY CONDENSING MAPS

Abstract

Abstract. Using an index theory for countably condensing maps, weshow the existence of eigenvalues for countably k-set contractive maps andcountably condensing maps in an infinite dimensional Banach space X,under certain condition that depends on the quantitative characteristic,that is, the infimum of all k≥ 1 for which there is a countably k-set-contractive retraction of the closed unit ball of Xonto its boundary. 1. IntroductionA starting point of our investigation is a result of Guo [5] which states asfollows:Let Ω be a bounded open subset of an infinite dimensional Banach space X. IfF : Ω → X is a compact map such that inf x∈∂Ω ||Fx|| > 0 and Fx 6= λx for allx ∈ ∂Ω and all λ ∈ (0,1], then ind(F,Ω) = 0.It has been attempted to extend the result to strict-set contractions; see [2,9, 12]. In a recent work [4], some generalizations to these contractions andcondensing maps are obtained, under suitable condition that depends on thequantitative characteristic R γ , where γ is a measure of noncompactness onX. This means the infimum of all k ≥ 1 for which there is a k-γ-contractiveretraction of the closed unit ball of X onto its boundary. See [3, 4, 6, 11].It is known in [5] that the above result is closely related to the problem offinding solutions for nonlinear equations. It is natural to consider this problemfor a large class of countably condensing maps, roughly speaking, condensingon countable subsets of the space. The use of such countable sets to solvenonlinear equations can be found in [7].Motivated by the work [4], our goal in the present paper is to study non-linear eigenvalue problem for countably k-set contractive maps and countablycondensing maps. To this end, we introduce an index theory for countably con-densing maps due to Vath [10] and the corresponding characteristic R