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A new characteristic property of Mittag-Leffler functions and fractional cosine functions

Zhan-Dong Mei, Ji-Gen Peng, Jun-Xiong Jia (2014)

Studia Mathematica

A new characteristic property of the Mittag-Leffler function E α ( a t α ) 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.

A stability result for a class of nonlinear integrodifferential equations with L¹ kernels

Piermarco Cannarsa, Daniela Sforza (2008)

Applicationes Mathematicae

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.

Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels

Maurizio Grasselli, Alfredo Lorenzi (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

Approximation of abstract linear integrodifferential equations

Hirokazu Oka, Naoki Tanaka (2000)

Studia Mathematica

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.

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