# zbMATH — the first resource for mathematics

On nonconvex perturbations of maximal monotone differential inclusions. (English) Zbl 0820.47072
Summary: The paper is concerned with the evolution inclusion $$x'\in Ax+ F(t, x)$$, where $$A$$ generates a contractive semigroup and $$F$$ is a lower semicontinuous multifunction. Constructing a suitable directionally continuous selection from $$F$$, we prove the existence of solutions on a closed domain and the connectedness of the set of trajectories.

##### MSC:
 47H20 Semigroups of nonlinear operators 34A60 Ordinary differential inclusions 54C65 Selections in general topology
Full Text:
##### References:
 [1] Aubin, J. P. and Cellina, A.:Differential Inclusions, Springer, Berlin, 1984. · Zbl 0538.34007 [2] Barbu, V.:Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff, 1976. · Zbl 0328.47035 [3] Benilan, P.: Solutions integrales d’equation d’evolution dans un espace de Banach,C. R. Acad. Sci. Paris 274 (1972), 47-50. · Zbl 0246.47068 [4] Bressan, A.: Upper and lower semicontinuous differential inclusions. A unified approach, in H. Sussmann (ed.),Nonlinear Controllability and Optimal Control, M. Dekker, 1990, pp. 21-31. · Zbl 0704.49011 [5] Bressan, A. and Colombo, G.: Selections and representations of multifunctions in paracompact spaces,Studia Math. 102 (1992), 209-216. · Zbl 0807.54020 [6] Bressan, A. and Cortesi, A.: Directionally continuous selections in Banach spaces,Nonlinear Analysis 13 (1989), 987-992. · Zbl 0687.34013 [7] Deimling, K.:Multivalued Differential Equations, Walter De Gruyter, Berlin, 1992. · Zbl 0760.34002 [8] Cellina, A. and Marchi, M. V.: Nonconvex perturbations of maximal monotone differential inclusions,Israel J. Math. 46 (1983), 1-11. · Zbl 0542.47036 [9] Colombo, G., Fonda, A., and Ornelas, A.: Lower semicontinuous perturbations of maximal monotone differential inclusions,Israel J. Math. 61 (1988), 211-218. · Zbl 0661.47038 [10] Crandall, M. G. and Liggett, T.: Generations of semigroups of nonlinear transformations in general Banach spaces,Amer. J. Math. 93 (1971), 265-298. · Zbl 0226.47038 [11] Dugundji, J.:Topology, Allyn and Bacon, Boston, 1966. [12] Fryszkowski, A.: Continuous selections for a class of non-convex multivalued maps,Studia Math. 76 (1983), 163-174. · Zbl 0534.28003 [13] Mitidieri, E. and Vrabie, I. I.: Differential inclusions governed by non-convex perturbations of m-accretive operators,Differential Integral Equations 2 (1989), 525-531. · Zbl 0736.34014 [14] Pavel, N. H. and Vrabie, I. I.: Semi-linear evolution equations with multivalued right hand side in Banach spaces,Anal. Stiint. Univ. A.I. Cuza, Iasi 25 (1979), 137-157. · Zbl 0421.34065 [15] Saks, H.:Theory of the Integral, Dover, New York, 1964. · Zbl 1196.28001 [16] Tolstonogov, A. A.:Differential Inclusions in Banach Spaces, Nauka Publishing House, Siberian Division, Novosibirsk, 1986 (in Russian). · Zbl 0689.34014 [17] Tolstonogov, A. A.: Extreme continuous selectors of multivalued maps and the ?bang-bang? principle for evolution inclusions,Soviet Math. Dokl. 317 (1991). · Zbl 0784.54024 [18] Vrabie, I. I.: Compactness methods and flow-invariance for perturbed nonlinear semigroups,Anal. Stiint. Univ. A.I. Cuza, Iasi 27 (1981), 117-125. · Zbl 0463.34054 [19] Vrabie, I. I.:Compactness Methods for Nonlinear Evolutions, Longman Scientific & Technical, Harlow, 1987. · Zbl 0721.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.