Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

Abstract

MSC: 34A37 34A60 34G25 34K30 34K35 34K45 93B05 Keywords: Nondensely defined operator Impulsive semilinear differential inclusion Fixed point Integral solutions Extremal solution ControllabilityIn this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.

Existence results for differential evolution equations with nonlocal conditions in Banach space

“…In the paper [1], Benchohra et al established sufficient conditions for the existence of mild and extremal mild solutions of first order impulsive functional evolution equations in a separable Banach space (X. |·|) of the form:…”

Approximate controllability of impulsive Hilfer fractional differential inclusions

“…During the last ten years, impulsive differential equations and inclusions with different conditions have been intensely studied by many mathematicians, see [1,10,20]. At present, the foundations of the general theory are already laid, and many of them are formulated in detail in Benchohra et al [6].…”

“…From the mathematical point of view, the problems of exact and approximate controllability are to be distinguished. Exact controllability enables to steer the system to arbitrary final state (see for example [1,5,18,20]) while approximate controllability means that system can be steered an arbitrary small neighborhood of final state. Approximately controllability systems are more prevalent and very often approximate controllability is completely adequate in applications.…”

Existence and controllability for nondensely defined partial neutral functional differential inclusions

“…More recently, it has been shown that the density condition is not necessary for dealing with the existence of integral solutions and the controllability for many classes of both the functional differential equations (see [7], [5], [8]) and the semilinear functional differential inclusions (see [1], [2], [6]). …”