On the Solvability of Systems of Inclusions Involving Noncompact Operators

Abstract

Abstract. We consider the solvability of a system y e F(x, y), x £ G(x, y) of set-valued maps in two different cases. In the first one, the map (x, y) -o F(x, y) is supposed to be closed graph with convex values and condensing in the second variable and (x, y)-o 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 F is as before with compact, not necessarily convex, values and 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 -o 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.

Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued wλ0-pseudomonotone maps

“…Recent development of the monotonicity method in the theory of differential-operator inclusions and evolution variational inequalities [4,11,12,15,19,20,[23][24][25]43,[62][63][64][65][66][67][68][69][70] ensures resolvability of the given objects under the conditions of −-coercivity, boundedness and the generalized pseudomonotonicity (it is necessary to notice, that the proof is not constructive). With relation to applications it would be actual to weaken some conditions for multi-valued maps in problem (2) replacing −-coercivity by +-coercivity, boundedness by Condition (Π) and pseudomonotonicity in classical sense or generalized pseudomonotonicity by w λ 0 -pseudomonotonicity.…”

“…[1][2][3][4][5][6][19][20][21][22][23][24][25][27][28][29][30][31][32][33][34][35][43][44][45][46][47][48][49][50][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70]73,[79][80][81][82][83] and citations there). Similarly to differential-operator equations at...…”

Systems of Inclusions Involving Fredholm Operators of Nonnegative Index and Nonconvex-Valued Maps

“…when X = X , Y = Y and L 1 and L 2 are the identity operators on X and Y , respectively (see [5,6,21] where compact set-valued maps were considered and [9,26] where condensing maps were studied). Papers [5,21] and [26] use the technique of the so-called weighted maps, while [6] and [9] apply the homological tools rather. In Section 4 we show some applications, while in Appendix we recall basic notions from the cohomotopy theory and provide some technical arguments necessary in Section 3.…”