On integro‐differential inclusions with operator‐valued kernels

Abstract

We study integro-differential inclusions in Hilbert spaces with operator-valued kernels and give sufficient conditions for the well-posedness. We show that several types of integro-differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well-posedness within this framework. As an example, we apply our results to the equations of visco-elasticity and to a class of nonlinear integro-differential inclusions describing phase transition phenomena in materials with memory.

Well-posedness for a general class of differential inclusions

“…To find the right conditions on k to ensure (b) is more delicate and will be postponed to future research. In case of a kernel k(t, s) = k(t−s), this was done in [29,30] even for operator-valued kernels.…”

Exponential stability of a second‐order integro‐differential equation with delay

“…We state the following exponential stability result for material laws of the above form. This proposition can be used to study the exponential stability of integro-differential equations (for the treatment of integrodifferential equations within the framework of evolutionary equations we refer to [4]).…”

Complete Volterra Integrodifferential Equations of the Second Order Unsolved with Respect to the Higher Derivative

“…Among the recent publications, we should mention [3] (one operator coefficient and the operation of integration from −1 to t), [4] (general nonlinear problems for the functions from R n ), [5] ✓ special equations of the first and second orders with [6] (Gurtin-Pipkin-type equation; one operator coefficient), and [7] (two comparable operator coefficients with integration from −1 to t). Moreover, all these papers deal with a version of equations solvable with respect to the higher derivative.…”