On semilinear differential inclusions in Banach spaces with nondensely defined operators

Abstract

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille-Yosida operator whereas Carathèodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.Mathematics Subject Classification (2010). Primary 34G25; Secondary 34A60, 47D62, 47H04, 47H09, 47H10, 49J24.

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

“…(see, for example, [32,33]). Let us recall (see, for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) that a mild solution to problems (15) and 16is a function x ∈ C([0, T], H) of the form…”

“…where P F is the superposition multioperator defined by (17). It is clear that each fixed point x λ ∈ C([0, T]; H) of the multimap F (•, λ), λ ∈ [0, 1] is a mild solution to the problem…”

“…The increasing interest in this subject is explained by its numerous applications in various branches of applied mathematics, physics, engineering, biology, economics, and other sciences (see, e.g., the monographs of [1][2][3]). A number of various approaches to the solving of boundary value problems for differential equations and inclusions in the case of fractional order q ∈ (0, 1) have been widely described in the literature (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein).…”

Generalized Cauchy problems involving nonlocal and impulsive conditions

“…The case when A is nondensely defined has been considered by many authors recently. We are interested in the papers [1,8,19,21,34,33] that are devoted to this situation. Since A is nondensely defined, there is no semigroup generated by A on X .…”

“…2), and then {S (t)} t≥0 is a C 0 -semigroup on D(A). The hypothesis that S (t) (t > 0) is compact was used in [1,8,19,21] to get the existence results whereas in [34] and [33], this condition was weakened by assuming that S is norm-continuous for t > 0. Regarding nonlocal problems, Liu [31] pointed out that one may meet a difficulty in proving the equicontinuity of the solution operator if the Lipschitz or compactness assumption on the nonlocal function is relaxed.…”