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Periodic solutions for nonlinear evolution equations in a Banach space


Author: Ioan I. Vrabie
Journal: Proc. Amer. Math. Soc. 109 (1990), 653-661
MSC: Primary 34G20; Secondary 34C25, 47H15, 58D25
MathSciNet review: 1015686
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Abstract: We prove an existence result for $ T$-periodic mild solutions to nonlinear evolution equations of the form

$\displaystyle u'(t) + Au(t) \mathrel\backepsilon F(t,u(t)),\quad t \in {R_ + }.$

Here $ (X,\vert\vert \cdot \vert\vert)$ is a real Banach space, $ A:D(A) \subset X \to {2^X}$ is an operator with $ A - aI$ $ m$-accretive for some $ a > 0$ and such that $ - A$. generates a compact semigroup, while $ F:{R_ + } \times \overline {D(A)} \to X$ is a Carathéodory mapping which is $ T$-periodic with respect to its first argument and satisfies

$\displaystyle \mathop {\lim }\limits_{r \to + \infty } \tfrac{1}{r}\sup \left\{... ...in {R_ + },v \in \overline {D(A)} ,\vert\vert v\vert\vert \leq r} \right\} < a.$

. As a consequence, we obtain an existence theorem for $ T$-periodic solutions to the porous medium equation.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1015686-4
Keywords: Accretive operator, compact semigroups, periodic solution, porous medium equation
Article copyright: © Copyright 1990 American Mathematical Society