Abstract
In this paper we investigate the existence of mild solutions for rst and second order impulsive semilinear evolution inclusions in real separable Banach spaces. By using suitable xed point theorems, we study the case when the multivalued map has convex and nonconvex values.
Highlights
We shall be concerned with the existence of mild solutions for first and second order impulsive semilinear damped differential inclusions in a real Banach space
Where F : J × E → P (E) is a multivalued map (P (E) is the family of all nonempty subsets of E), A is the infinitesimal generator of a family of semigroup {T (t) : t ≥ 0}, B is a bounded linear operator from E into E, y0 ∈ E, 0 < t1 < . . . < tm < tm+1 =
A fixed point theorem for contraction multivalued maps due to Covitz and Nadler [10] is applied in the second one
Summary
We shall be concerned with the existence of mild solutions for first and second order impulsive semilinear damped differential inclusions in a real Banach space. We consider the following first order impulsive semilinear differential inclusions of the form:. We study the second order impulsive semilinear evolution inclusions of the form:. A fixed point theorem for contraction multivalued maps due to Covitz and Nadler [10] is applied in the second one. The special case (for B=0) of the problem (1)–(3) was studied by Benchohra et al in [5] by using the concept of upper and lower mild solutions combined with the semigroup theory and by Benchohra and Ntouyas in [7] with the aid to a fixed point theorem due to Martelli for condensing multivalued maps [24]. The results of the present paper can be seen as an extension of the problems considered in [6], [5] and [7]
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