Periodic solutions for nonlinear evolution equations at resonance

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:392

Article

Kokocki, Piotr

关键词: Semigroup;    Evolution equation;    Topological degree;    Periodic solution;    Resonance;   

DOI  :  10.1016/j.jmaa.2012.02.035

来源: Elsevier

【 摘 要 】

We are concerned with periodic problems for nonlinear evolution equations at resonance of the form (u) over dot(t)= -Au (t) + F (t, u (t)), where a densely defined linear operator A: D (A) --> X on a Banach space X is such that -A generates a compact C-0 semigroup and F [0, +infinity) x X --> X is a nonlinear perturbation. Imposing appropriate Landesman-Lazer type conditions on the nonlinear term F we prove a formula expressing the fixed point index of the associated translation along trajectories operator, in the terms of a time averaging of F restricted to Ker A. By the formula, we show that the translation operator has a nonzero fixed point index and, in consequence, we conclude that the equation admits a periodic solution. (C) 2012 Elsevier Inc. All rights reserved.

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