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Transactions of the American Mathematical Society

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On the solvability of systems of inclusions involving noncompact operators

Authors: P. Nistri, V. V. Obukhovskiĭ and P. Zecca
Journal: Trans. Amer. Math. Soc. 342 (1994), 543-562
MSC: Primary 47H15; Secondary 34G20, 34K30, 47H04, 47N20, 47N70, 49J45
MathSciNet review: 1232189
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Abstract: We consider the solvability of a system

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end{array} } \right.$

of set-valued maps in two different cases. In the first one, the map $ (x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $ (x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $ \bar F$ is as before with compact, not necessarily convex, values and $ \bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $ x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

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Additional Information

Keywords: Condensing multivalued map, degree theory, Borsuk-Ulam condition, control problem, periodic solution
Article copyright: © Copyright 1994 American Mathematical Society