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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 second-order impulsive Cauchy problem.

Hernández Morales, Eduardo (2002)

International Journal of Mathematics and Mathematical Sciences

A spectrality condition for infinitesimal generators of cosine operator functions

Amina Šahović, Fikret Vajzović (2008)

Matematički Vesnik

Almost-distribution cosine functions and integrated cosine functions

Pedro J. Miana (2005)

Studia Mathematica

We introduce the notion of almost-distribution cosine functions in a setting similar to that of distribution semigroups defined by Lions. We prove general results on equivalence between almost-distribution cosine functions and α-times integrated cosine functions.

Analyticity of the propagators of second order linear differential equations in Banach spaces.

T. Xio, J. Liang (1992)

Semigroup forum

Commutativity of set-valued cosine families

Andrzej Smajdor, Wilhelmina Smajdor (2014)

Open Mathematics

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If F t: t ≥ 0 is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_{t} \circ F_{s} (x) = F_{s} \circ F_{t} (x) f o r s, t ⩾ 0 a n d x \in K$ .

Complex Powers of Operators

Marko Kostić (2008)

Publications de l'Institut Mathématique

Continuous selections of solution sets to second order evolution equations.

Lupulescu, Vasile (2004)

Acta Universitatis Apulensis. Mathematics - Informatics

Convoluted C-operator families and abstract Cauchy problems

Marko Kostić, Stevan Pilipović (2007)

Kragujevac Journal of Mathematics

Cosine families of unbounded normal operators

Jan Niechwiej (1989)

Annales Polonici Mathematici

Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions.

Mugnolo, Delio, Romanelli, Silvia (2006)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Entire solutions of the abstract Cauchy problem.

R. de Laubenfels (1991)

Semigroup forum

Equicontinuous families of operators generating mean periodic maps

Valentina Casarino (1999)

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

The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group $U$ or an equicontinuous cosine function $C$ forces the spectral structure of the infinitesimal generator of $U$ or $C$ . In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.

Existence and uniqueness of mild solutions of second order Volterra integrodifferential equations with nonlocal conditions.

Tidke, Haribhau Laxman, Dhakne, Machindra Baburao (2009)

Applied Mathematics E-Notes [electronic only]

Existence of mild solutions of second-order neutral functional differential inclusions with nonlocal conditions in Banach spaces.

Marshal Anthoni, S., Kim, J.-H., Dauer, J.P. (2004)

International Journal of Mathematics and Mathematical Sciences

Existence of solutions for a second order abstract functional differential equation with state-dependent delay.

Hernández, Eduardo M. (2007)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of solutions for an abstract second-order differential equation with nonlocal conditions.

Hernández, Eduardo (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of solutions of a second order abstract functional Cauchy problem with nonlocal conditions

Hernán R. Henríquez, Eduardo Hernández M. (2006)

Annales Polonici Mathematici

We establish the existence of mild, strong, classical solutions for a class of second order abstract functional differential equations with nonlocal conditions.

Existence of solutions to a second order partial differential equation with nonlocal conditions.

Hernández M., Eduardo (2003)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of the fundamental solution of a second order evolution equation

Jan Bochenek (1997)

Annales Polonici Mathematici

We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence $A_{n} (t)$ of bounded operators.

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