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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) 0 t u x (t,s)ds,

where f:R + ×R n R n is a single valued map, u x S U,x ,S U,x is the set of selections of the multivalued map U:H×R n 2 R n , and H={(t,s)R + ×R + :st}. 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; 1023.34056)] for condensing multivalued maps on ordered Banach spaces.

45G10Nonsingular nonlinear integral equations
47H09Mappings defined by “shrinking” properties