On a class of second-order anti-periodic boundary value problems. (English) Zbl 0767.34047

The paper deals with second-order boundary value problems like (i) \(- u''(t)+au'(t)+Au(t)\ni f(t)\), \(0<t<T\); (ii) \(u(0)=-u(T)\), \(u'(0)=-u'(T)\), with an odd \(m\)-accretive multivalued operator \(A\). Existence and uniqueness are discussed under the assumption that the Banach space \(X\) and its dual are sufficiently smooth. Then, (ii) is replaced by \(u(0)=- u(T)\), \(u'(0)+u'(T)\in\alpha(u(0))\) or \(u(0)+u(T)\in\beta(u'(0))\), \(u'(0)=-u'(T)\), where \(\alpha\) and \(\beta\) are maximal monotone operators acting in \(X\). The case of time-dependent coefficients in \(u'(t)\) and \(u''(t)\) is also studied. Finally, the authors consider the problem (i)– (ii) in a Hilbert space where the nonlinear operator \(A\) depends on \(t\). The investigations rely on the theory of \(m\)-accretive operators. Some examples of boundary value problems for ordinary and partial differential equations are also given.


34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
47J05 Equations involving nonlinear operators (general)
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