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.
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.
Reviewer: Mouffak Benchohra (Sidi Bel Abbes)
MSC:
| 34G25 | Evolution inclusions |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
Keywords:
antiperiodic solution; evolution equation; maximal monotone operator; discontinuous nonlinearityReferences:
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