On Some Qualitative Properties of Mild Solutions of Nonlocal Semilinear Functional Differential Equations

Abstract

In the present paper, we investigate the qualitative properties such as existence, uniqueness and continuous dependence on initial data of mild solutions of first and second order nonlocal semilinear functional differential equations with delay in Banach spaces. Our analysis is based on semigroup theory and modified version of Banach contraction theorem. IntroductionThe problems of existence, uniqueness and other qualitative properties of solutions for semilinear differential equations in Banach spaces has been studied extensively in the literature for last many years, see [1]-[4], [7]-[12], [18]. On the other hand, as nonlocal condition is more precise to describe natural phenomena than classical initial condition, the Cauchy problem with nonlocal condition also received much attention in recent years, see [1]-[3], [8], [10]. These type of problems were first studied by L. Byszewski. Also, the problems of qualitative properties of solutions of second order functional differential equations have been studied by many authors, see [5]-[7], [9], [12], [14]-[16]. It is advantageous to treat second order abstract differential equations directly rather than to convert into first order differential system. For direct applications of second order differential system, one may refer Fitzgibbon [6].In the present paper, we consider semilinear functional differential problem of first order of the type:

Approximate controllability of semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions

“…according to [4,11,15]. Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions (1.1).…”

Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay

“…Also, Klamka [21] gave necessary and sufficient conditions for different kinds of stochastic relative controllability in a given time interval which are proved for stochastic finite-dimensional linear systems with multiple delays in control. e existence of solutions for impulsive evolution equations with delays has been studied by Hernandez, Sakthivel, and Tamaka [22]; Abada, Benchohra, and Hammouche [23]; Shikharchan and Baburao [24] and Chang [25,26]. Besides, impulsive and stochastic effects appear in real-life systems.…”

Existence of a mild solution for an impulsive nonlocal non-autonomous neutral functional differential equation

“…In [17], authors have concerned with the existence of α-mild solutions for a fractional stochastic integro-differential equations with nonlocal initial conditions in a real separable Hilbert space by using an approximation technique. For more details, we refer to papers [11,[14][15][16][17][18][19][20]24,26,28] and references given therein.…”

“…In [8], author has shown the controllability of a system of impulsive semilinear non-autonomous differential equations via Rothe's type fixed-point theorem. For more details and study on such differential equation, we refer to the monographs [9,10] and papers [11][12][13][14][15][16][17][18][19][20] and reference cited therein.…”