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Zbl 1125.45006
Hong, Shihuang; Wang, Li
Existence of solutions for integral inclusions.
(English)
[J] J. Math. Anal. Appl. 317, No. 2, 429-441 (2006). ISSN 0022-247X

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 {\it M. Martelli} [Boll. Unione Mat. Ital., IV. Ser. 11, Suppl. Fasc. 3, 70--76 (1975; Zbl 0314.47035)] or the author [{\it S. Hong}, Electron. J. Differ. Equ. 2003, Paper No. 32 (2003; 1023.34056)] for condensing multivalued maps on ordered Banach spaces.
[Mouffak Benchohra (Sidi Bel Abbes)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
47H09 Mappings defined by "shrinking" properties

Keywords: upper semicontinuous multivalued map; cone; fixed point; positive solutions; nonlinear integral inclusion; ordered Banach spaces

Citations: Zbl 0314.47035; Zbl 1023.34056

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