Abstract
We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results.
Highlights
The aim of this paper is to establish a result of the existence of mild solution for a class of the non-autonomous second order nonlinear differential equation with non-instantaneous impulses described in the form
We discuss the existence of mild solutions for system (1)
Let us propose the definition of the mild solution of system (1)
Summary
Useful for the study of abstract second order equations is the existence of an evolution system S(t, s) for the homogenous equation y00 (t) = A(t)y(t), for t ≥ 0. For this purpose there are many techniques to show the existence of S(t, s) which has been developed by Kozak [15]. In many problems, such as the transverse motion of an extensible beam, the vibration of hinged bars and many other physical phenomena, we deal with the second-order abstract differential equations in the infinite dimensional spaces.
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