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Anti-periodic solutions for evolution equations associated with maximal monotone mappings. (English) Zbl 1215.34069
The authors consider the existence of anti-periodic solutions for differential inclusions in a real Hilbert space $H$. The first result concerns the problem $$x^{\prime }(t)\in -Ax(t)+f(t)\text{ a.e. on }\Bbb{R},$$ $$x(t)=-x(t+T)\text{ for }t\in \Bbb{R}.$$ $A$ is assumed to be an odd maximal monotone mapping, $D(A)$ is symmetric and convex, $f$ is $L^{2}$ and satisfies $f(t)=-f(t+T)$ for $t\in \Bbb{R}$, $\|g\| \leq M\| x\|$ for all $x\in D(A)$ and $g\in Ax$, and $M>0$ is a constant such that $MT<2$. The other theorem concerns the problem $$x^{\prime }(t)\in -Ax(t)+\partial G(x(t))+f(t)\text{ a.e. on }\Bbb{R},$$ $$x(t)=-x(t+T)\text{ for }t\in \Bbb{R}.$$ In addition to the hypotheses of the first result, it is assumed that $G$ is continuously differentiable and even, $\partial G$ maps bounded sets to bounded sets, and $D(A)$ is compactly embedded in $H$. In the case in which $A$ is the subdifferential of a lower semi-continuous convex function, the authors obtained similar results as in [{\it Y. Q. Chen, D. O’Regan} and {\it J. J. Nieto}, Math. Comput. Modelling 46, No. 9--10, 1183--1190 (2007; Zbl 1142.34313)]. The authors conclude the paper by applying the first of these theorems to a boundary value problem for a partial differential equation.

34G25Evolution inclusions
34C25Periodic solutions of ODE
Full Text: DOI
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