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The paper deals with systems of abstract integrodifferential equations subject to general nonlocal initial conditions. In order to allow the nonlinear terms of the equations to behave independently as much as possible, we use a vector approach based on matrices, vector-valued norms and a vector version of Krasnoselskii's fixed point theorem for a sum of two operators. The assumptions take into account the support of the nonlocal initial conditions and the hybrid character of the system. Two examples are given to illustrate the theory.

The work is concerned with systems of abstract integrodifferential equations with general nonlocal initial conditions. To allow the nonlinear terms of the equations to behave as independently as possible, we employ a vector approach based on matrices, vector-valued norms, and a vector version of Krasnoselskii's fixed point theorem for a sum of two operators. The assumptions take into account the system's hybridity and the support for nonlocal initial conditions. To demonstrate the principle, two examples are given.

In, the recent years, many authors have attracted much attention to the study of existence problems for differential and integrodifferntial equations. In this work, we investigate the existence and regularity of solutions for some partial functional integrodifferential equations with finite delay. The continuous dependence upen initial values and asymptotic stability are also studied. Firstly, we show the existence of the mild solutions. Secondly, we give suffcient conditions ensuring the existence of the strict solutions. The method used treats the equations in the domain of A with the graph norm employing results from linear semigroup theory. To illustrate our abstract result, we conclude this work with an application.

In this paper, we study a class of neutral partial functional integrodifferential equations with finite delay in Banach spaces. We are interested in the global existence, uniqueness and regularity of solutions with values in the subspace $D(A).$ The method used are based on Banach's fixed point theorem and on the technique of the graph norm. In our work the nonlinear term is treated as a perturbation of the linear equation. As an application, we consider a diffusive neutral partial functional integrodifferential equation.

In this work, we study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Frechet spaces. We assume that the linear part has a resolvent operator in the sense given by Grimmer (Trans Am Math Soc 273: 333–349, 1982). The nonlinear part is a sum of a Lipschitzian function and another satisfies the Caratheodory’s conditions. Our approach is based on a nonlinear alternative of Avramescu type and the resolvent operators theory. An application is provided to a reaction-diffusion equation with delay.