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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A method of lines for a nonlinear abstract functional evolution equation

Authors: A. G. Kartsatos and M. E. Parrott
Journal: Trans. Amer. Math. Soc. 286 (1984), 73-89
MSC: Primary 34K30
MathSciNet review: 756032
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Abstract: Let $ X$ be a real Banach space with $ {X^\ast}$ uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E)

$\displaystyle u\prime(t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].$

The operators $ A(t):D \subset X \to X$ are $ m$-accretive and $ G(t,\phi )$ is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators $ \hat A(t)$ and the corresponding use of nonlinear evolution operator theory.

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Keywords: Functional evolution equation, uniformly convex dual, method of lines, $ m$-accretive operator
Article copyright: © Copyright 1984 American Mathematical Society

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