An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

Abstract

The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of “impulses.” In fact, there are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Impulsive differential equations take an important place in some area such that physics, chemical technology, population dynamics, biotechnology, and economics. The study of such equations is relatively less developed due to the difficulties created by the state-dependent impulses. In the case of impulses at variable times, a “beating phenomenon” may occur, that is to say, a solution of the differential equation may hit a given barrier several times (including infinitely many times). In this work, we study the existence of solutions for some partial impulsive functional differential equations with variable times in Banach spaces by using the fractional power of closed operators theory. We suppose that the undelayed part admits an analytic semigroup. The delayed part is assumed to be Lipschitz. We use Schaefer fixed-point Theorem to prove the existence of solutions for this first order equation with impulse in <i>α</i>-norm.