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Existence of solutions for integral inclusions. (English) Zbl 1125.45006

This paper presents sufficient conditions for the existence of positive solutions to a class of nonlinear integral inclusion of the form
\[ x(t)=f(t,x)\int_{0}^{t}u_{x}(t,s)\,ds, \]
where \(f:R_{+}\times R^{n}\to R^{n}\) is a single valued map, \(u_{x}\in S_{U,x}, \;S_{U,x}\) is the set of selections of the multivalued map \(U: H\times R^{n}\to 2^{R^{n}}, \) and \(H=\{(t,s)\in R_{+}\times R_{+} : s\leq t\}\). These results are obtained via a fixed point theorem due to M. Martelli [Boll. Unione Mat. Ital., IV. Ser. 11, Suppl. Fasc. 3, 70–76 (1975; Zbl 0314.47035)] or the author [S. Hong, Electron. J. Differ. Equ. 2003, Paper No. 32 (2003; Zbl 1023.34056)] for condensing multivalued maps on ordered Banach spaces.

MSC:

45G10 Other nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

[1] Papageorgiou, N. S., Boundary value problems for evolution, Comment. Math. Univ. Carolin., 29, 355-363 (1988) · Zbl 0696.35074
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