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EIGENVECTORS AND FIXED POINTS OF NON-LINEAR OPERATORS

Authors
Publisher
Dipartimento di Matematica e Informatica
Publication Date
Keywords
  • Measure Of Noncompactness
  • K-ψ-Contraction
  • Retraction
  • Fixed Point Index
  • 47H09
  • 47H11
  • 45G10
Disciplines
  • Mathematics

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].

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