### A Construction of Monotonically Convergent Sequences from Successive Approximations in Certain Banach Spaces.

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A Cauchy problem for an abstract nonlinear Volterra integrodifferential equation is considered. Existence and uniqueness results are shown for any given time interval under weak time regularity assumptions on the kernel. Some applications to the heat flow with memory are presented.

In this paper we get an algebraic derivative relative to the convolution $(f*g)\left(t\right)={\int}_{0}^{t}if(t-\psi )g\left(\psi \right)d\psi $ associated to the operator ${D}^{\delta}$, which is used, together with the corresponding operational calculus, to solve an integral-differential equation. Moreover we show a certain convolution property for the solution of that equation

We prove exponential decay for the solution of an abstract integrodifferential equation. This equation involves coefficients of polynomial type, weakly singular kernels as well as different powers of the unknown in some norms.

The objective of this work is the application of Krasnosel'skii's fixed point technique to prove the existence of periodic solutions of a system of coupled nonlinear integro-differential equations with variable delays. An example is given to illustrate this work.