### A duality approach for solving identification problems related to integrodifferential Maxwell's equations

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A new characteristic property of the Mittag-Leffler function ${E}_{\alpha}\left(a{t}^{\alpha}\right)$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.

We study second order nonlinear integro-differential equations in Hilbert spaces with weakly singular convolution kernels obtaining energy estimates for the solutions, uniform in t. Then we show that the solutions decay exponentially at ∞ in the energy norm. Finally, we apply these results to a problem in viscoelasticity.

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.

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.

One parabolic integrodifferential problem in the abstract real Hilbert spaces is studied in this paper. The semidiscrete and full discrete approximate solution is defined and the error estimate of Rothe's function in some function spaces is established.

This paper is devoted to the approximation of abstract linear integrodifferential equations by finite difference equations. The result obtained here is applied to the problem of convergence of the backward Euler type discrete scheme.