## Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities.(English)Zbl 0977.34061

The authors study the existence of generalized solutions to the antiperiodic boundary value problem $au_{t}(t,x)+Au(t,x)-bu(t,x)+f(u(t,x))\ni h(t,x),\quad u(0,x)=-u(T,x),$ with $$t\in [0,T]$$, $$x\in\Omega$$ (a finite measure space), $$A$$ is a maximal monotone (possibly multivalued) operator in $$L^{2}(\Omega)$$ and a subdifferential in $$L^{2}(\Omega)$$ of a functional $$\varphi:L^{2}(\Omega)\to (-\infty,+\infty], f:\mathbb{R}\to \mathbb{R}$$ and $$h:[0,T]\times\Omega\to \mathbb{R}$$ are given functions, $$a\not =0,$$ and $$b\geq 0$$.
The proof relies on both the approach used in a recent paper by S. Aizicovici and S. Reich [Discrete Contin. Dyn. Syst. 5, No. 1, 35-42 (1999; Zbl 0961.34044)] and the method employed by J. Rauch [Proc. Am. Math. Soc. 64, 277-282 (1977; Zbl 0413.35031)] in studying discontinuous elliptic equations. This paper is a continuation of the same problem considered recently by S. Aizicovici and S. Reich (see Zbl 0961.34044). The same argument is applied to a second-order antiperiodic boundary value problem. Two examples are discussed at the end of the paper illustrating the abstract theory.

### MSC:

 34G25 Evolution inclusions 35D05 Existence of generalized solutions of PDE (MSC2000)

### Citations:

Zbl 0961.34044; Zbl 0413.35031
Full Text:

### References:

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