Critical type of Krasnosel’skii fixed point theorem. (English) Zbl 1223.47061
Let \(K\) be a nonempty bounded closed convex subset of a Banach space \(E\). A unified approach to the classical Schauder’s and Banach-Caccioppoli’s fixed point theorems was given by M. A. Krasnosel’skij [Usp. Mat. Nauk 10, No. 1(63), 123–127 (1955; Zbl 0064.12002)] who proved that the sum \(S+T \) of two operators \(S,T:K \rightarrow E\) has at least one fixed point in \(K\) whenever \(S\) is compact, \(T\) is a contraction on \(K\), and \(Sx+Ty\in K\) for all \(x,y \in K\).
In the paper under review, the authors prove, by means of the technique of measures of noncompactness, some generalizations of this result, where \(S\) is noncompact, \(T\) is not necessarily continuous, and \(I-T\) may not be injective. The results obtained are used to prove the existence of periodic solutions of nonlinear neutral differential equations with delay in the critical case.
In the paper under review, the authors prove, by means of the technique of measures of noncompactness, some generalizations of this result, where \(S\) is noncompact, \(T\) is not necessarily continuous, and \(I-T\) may not be injective. The results obtained are used to prove the existence of periodic solutions of nonlinear neutral differential equations with delay in the critical case.
Reviewer: Giulio Trombetta (Arcavaceta di Rende)
MSC:
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
| 47H10 | Fixed-point theorems |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 47N20 | Applications of operator theory to differential and integral equations |
Citations:
Zbl 0064.12002References:
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