Eigenvectors and fixed point of non-linear operators

Abstract

Let X be a real infinite-dimensional Banach space and ψ a measure of noncompactness on X. Let Ω be a bounded open subset of X and A : Ω → X a ψ-condensing operator, which has no fixed points on ∂Ω.
Then the fixed point index, ind(A,Ω), of A on Ω is defined (see, for example, ([1] and [18]). In particular, if A is a compact operator ind(A,Ω) agrees with the classical Leray-Schauder degree of I −A on Ω relative to the point 0, deg(I −A,Ω,0). The main aim of this note is to investigate boundary conditions, under which the fixed point index of strict- ψ-contractive or ψ-condensing operators A : Ω → X is equal to zero. Correspondingly, results on eigenvectors and nonzero fixed points of k-ψ-contractive and ψ-condensing operators are obtained. In particular we generalize the Birkhoff-Kellog theorem [4] and Guo’s domain compression and expansion theorem [17]. The note is based mainly on the results contained in [7] and [8].