This work is concerned with a nonlocal partial neutral differential equation of Sobolev type. Specifically, existence of the solutions to the abstract formulations of such type of problems in a Banach space is established. The results are obtained by using Schauder's fixed point theorem. Finally, an example is provided to illustrate the applications of the abstract results.

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In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii's fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration.The theory of almost periodic functions was first treated and created by Bohr during 1924Bohr during -1926. Later on Bochner extended the Bohr theory to general abstract spaces. The theory has been widely treated by Favard, Levitan [1], Besicovich [2] in monographs. Amerio [3] has extended certain results of Favard and Bochner to differential equations in abstract spaces. Functional differential equations arise as models in several physical phenomena, for example, reaction-diffusion equations, climate models, coupled oscillators, population ecology, neural networks, the propagation of waves etc. A class of functional differential equations called neutral differential equations arises in many phenomena such as in the study of oscillatory systems and also in the modeling of several physical problems [4]. There exists an extensive literature for ordinary neutral functional differential equations; as a reference one can see Hale and Lunel's book [5] and references therein. A partial neutral differential equation with finite delay case arises, for instance, from the transmission line theory. And unbounded delay arises, for instance, from the description of the heat conduction in materials with fading memory (Gurtin and Pipkin [6]).The purpose of this paper is to deal with the existence and uniqueness of an almost periodic solution of the following functional differential equation in a complex Banach space X :where A P(X ) is the set of all almost periodic functions from R to X (cf. Definition 2.1) and the family {A(t) : t ∈ R} of operators in X generates an exponentially stable evolution system {U (t, s), t ≥ s}. We also discuss the differential

In this paper, we prove the existence and uniqueness of mild solutions for the impulsive fractional differential equations for which the impulses are not instantaneous in a Banach space H. The results are obtained by using the analytic semigroup theory and the fixed points theorems.