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This paper is concerned with double families of evolution operators employed in the study of dynamical systems in which cause and effect are represented in different Banach spaces. The main tool is the Laplace transform of vector-valued functions. It is used to define the generator of the double family which is a pair of unbounded linear operators and relates to implicit evolution equations in a direct manner. The characterization of generators for a special class of evolutions is presented.
Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell’equazione di evoluzione non autonoma , con il dato iniziale , in spazi di Banach. I dominii degli operatori variano in e non sono necessariamente densi in . Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità holderiana della soluzione e della sua derivata.
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 of bounded operators.
This paper presents existence results for initial and boundary value problems for nonlinear differential equations in Banach spaces.
Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation are studied for the general case, even when the algebraic equation is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.
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