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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:

Abstract. The objective of the present paper is to study the local existence, global existence, uniqueness, continuous dependence, asymptotic stability and other properties of solutions of a nonlinear Volterra integrodifferential equation in Banach spaces of more general type. The technique employed in our analysis is based on treating the equation in the domain of the infinitesimal generator of semigroups of linear operators in a Banach space with graph norm and using results from linear semigroup theory. (2000): 45N05, 45J05. Mathematics subject classification

In this paper, we study the existence and uniqueness of mild solutions for second order initial value problems, with nonlocal conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family.

In the present paper, we investigate the global existence of solutions to initial value problem for nonlinear mixed Volterra–Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray–Schauder alternative and rely on a priori bounds of solution.