On second‐order multivalued impulsive functional differential inclusions in Banach spaces

M Benchohra,J Henderson,S K Ntouyas

doi:10.1155/s1085337501000604

Abstract

A fixed point theorem for condensing maps due to Martelli is used to investigate the existence of solutions to second‐order impulsive initial value problem for functional differential inclusions in Banach spaces.

Highlights

Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth
Much has been done under the assumption that the state variables and system parameters change continuously
One may visualize situations in nature where abrupt changes such as shock, harvesting, and disasters may occur. These phenomena are shortterm perturbations whose duration is negligible in comparison with the duration of the whole evolution process

Summary

Introduction

Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. Let C([a, b], E) be the Banach space of continuous functions from [a, b] into E with norm y ∞ = sup |y(t)| : t ∈ [a, b] ∀y ∈ C [a, b], E. A multivalued map G : [a, b] → CC(X) is said to be measurable if for each x ∈ E the function Y : [a, b] → R defined by. F be a multivalued map satisfying the Carathéodory conditions with the set of L1-selections SF is nonempty, and let be a linear continuous mapping from L1(I, X) to C(I, X). We introduce the following hypotheses: (H1) F : J ×C([−r, 0], E) → CC(E); (t, u) → F (t, u) is an L1-Carathéodory multivalued map and for each fixed u ∈ C([−r, 0], E) the set SF,u = g ∈ L1(J, E) : g(t) ∈ F (t, u) for a.e. t ∈ J (2.10).

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