Eigenvectors and fixed point of non-linear operators
AbstractLet 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, ( and ). 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  and Guo’s domain compression and expansion theorem . The note is based mainly on the results contained in  and .
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