Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space. (English) Zbl 0743.34067

The boundary value problem \(-au''(t)+bu'(t)+\partial\varphi(u(t))- \partial\psi(u(t))\ni f(t)\), a.e. \(f\in R\), \(u(t+T)=-u(t)\), in a real Hilbert space \(H\) is considered. Here \(a\geq 0\), \(b\neq 0\), \(T>0\), \(f\in L^ 2_{loc}(R;H)\) satisfies \(f(t+T)=-f(t)\) a.e. \(t\in R\), \(\varphi\) and \(\psi\) are convex, even functionals on \(H\) and “\(\partial\)” stands for subdifferential. The existence of solution of this problem is studied by the methods of the theory of multi-valued monotone operators. The applications of the achieved results to nonlinear partial differential equations are discussed.


34G20 Nonlinear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
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