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Integral inclusions of upper semi-continuous or lower semi-continuous type. (English) Zbl 0860.45007
The author establishes existence results for Volterra inclusions of the form \[ y(t) \in \int^t_0 k(t,s)F \bigl(s,y(s)\bigr) ds+g(t), \quad t \in[0,T], \] and for Hammerstein inclusions of the form \[ y(t)\in \int^T_0k(t,s) F\bigl(s,y(s) \bigr)ds + g(t), \quad t \in[0,T]. \] Here \(F\) is a multivalued function with nonempty compact values and \(F\) satisfies certain other conditions which are shown to imply either lower or upper semi-continuity for certain related mappings. The kernel \(k\) is assumed to be essentially bounded. The proofs rely on fixed-point theorems. Some of the results for superlinear inclusions are new even for integral equations.

MSC:
45G10 Other nonlinear integral equations
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[1] Alberto Bressan and Giovanni Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), no. 1, 69 – 86. · Zbl 0677.54013
[2] A. I. Bulgakov and L. N. Ljapin, Some properties of the set of solutions of a Volterra-Hammerstein integral inclusion, Differentsial\(^{\prime}\)nye Uravneniya 14 (1978), no. 8, 1465 – 1472, 1532 (Russian). · Zbl 0424.45019
[3] C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991. · Zbl 0714.45002
[4] C. Corduneanu, Perturbation of linear abstract Volterra equations, J. Integral Equations Appl. 2 (1990), no. 3, 393 – 401. · Zbl 0824.45013 · doi:10.1216/jiea/1181075570 · doi.org
[5] James Dugundji and Andrzej Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982. · Zbl 0483.47038
[6] Marlène Frigon and Andrzej Granas, Théorèmes d’existence pour des inclusions différentielles sans convexité, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 12, 819 – 822 (French, with English summary). · Zbl 0731.47048
[7] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. · Zbl 0695.45002
[8] Donal O’Regan, Theory of singular boundary value problems, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. · Zbl 0807.34028
[9] ——, Existence results for nonlinear integral equations, J. Math. Anal. Appl. 192 (1995), 705–726. CMP 95:14
[10] ——, Existence theory for nonlinear Volterra and Hammerstein integral equations, Dynamical Systems and Applications , World Scientific Series in Applicable Analysis, Vol. 4 (1995), 601–615. · Zbl 0842.45003
[11] T. Pruszko, Topological degree methods in multivalued boundary value problems, J. Nonlinear Anal. 5 (1981), 953–973. · Zbl 0478.34017
[12] Tadeusz Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. (Rozprawy Mat.) 229 (1984), 48. · Zbl 0543.34008
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