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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 ' (t)-Ax(t)+f(t)a.e.on,
x(t)=-x(t+T)fort·

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, gMx for all xD(A) and gAx, and M>0 is a constant such that MT<2.

The other theorem concerns the problem

x ' (t)-Ax(t)+G(x(t))+f(t)a.e.on,
x(t)=-x(t+T)fort·

In addition to the hypotheses of the first result, it is assumed that G is continuously differentiable and even, 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 [Y. Q. Chen, D. O’Regan and 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.

MSC:
34G25Evolution inclusions
34C25Periodic solutions of ODE
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