# Properties of the solution set of nonlinear evolution inclusions

Properties of the solution set of nonlinear evolution inclusions In this paper we examine nonlinear, nonautonomous evolution inclusions defined on a Gelfand triple of spaces. First we show that the problem with a convex-valued, h *-usc in x orientor field F ( t, x ) has a solution set which is an R δ -set in C ( T, H ). Then for the problem with a nonconvex-valued F ( t, x ) which is h -Lipschitz in x , we show that the solution set is path-connected in C ( T, H ). Subsequently we prove a strong invariance result and a continuity result for the solution multifunction. Combining these two results we establish the existence of periodic solutions. Some examples of parabolic partial differential equations with multivalued terms are also included. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Properties of the solution set of nonlinear evolution inclusions

, Volume 36 (1) – Jul 1, 1997
20 pages

/lp/springer_journal/properties-of-the-solution-set-of-nonlinear-evolution-inclusions-KnN92clhUS
Publisher
Springer-Verlag
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/BF02683335
Publisher site
See Article on Publisher Site

### Abstract

In this paper we examine nonlinear, nonautonomous evolution inclusions defined on a Gelfand triple of spaces. First we show that the problem with a convex-valued, h *-usc in x orientor field F ( t, x ) has a solution set which is an R δ -set in C ( T, H ). Then for the problem with a nonconvex-valued F ( t, x ) which is h -Lipschitz in x , we show that the solution set is path-connected in C ( T, H ). Subsequently we prove a strong invariance result and a continuity result for the solution multifunction. Combining these two results we establish the existence of periodic solutions. Some examples of parabolic partial differential equations with multivalued terms are also included.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jul 1, 1997

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