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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.

MSC:

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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[1] Aftabizadeh, A. R.; Pavel, N. H., Boundary value problems for second-order differential equations and a convex problem of Bolza, Differential Integral Equations, 2, 495-509 (1989) · Zbl 0723.34045
[2] Aftabizadeh, A. R.; Aizicovici, S.; Pavel, N. H., Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces, Nonlinear Anal., 18, 253-267 (1992) · Zbl 0779.34054
[3] Aizicovici, S.; Pavel, N. H., Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space, J. Funct. Anal., 99, 387-408 (1991) · Zbl 0743.34067
[4] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces (1976), Noordhoff: Noordhoff Leyden
[5] Brézis, H., Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, (Math. Studies, Vol. 5 (1973), North-Holland: North-Holland Amsterdam) · Zbl 0252.47055
[6] Haraux, A., Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math., 63, 479-505 (1989) · Zbl 0684.35010
[7] Okochi, H., On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan, 40, 541-553 (1988) · Zbl 0679.35046
[8] Okochi, H., On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains, Nonlinear Anal., 14, 771-783 (1990) · Zbl 0715.35091
[9] Okochi, H., On the existence of anti-periodic solutions to nonlinear evolution equationsv associated with odd subdifferential operators, J. Funct. Anal., 91, 246-258 (1990) · Zbl 0735.35071
[10] Poffald, E. I.; Reich, S., An incomplete Cauchy problem, J. Math. Anal. Appl., 113, 514-543 (1986) · Zbl 0599.34078
[11] Cai, Z.; Pavel, N. H., Generalized periodic and anti-periodic solutions for the heat equation in \(R^1\), Libertas Math., 10, 109-121 (1990) · Zbl 0731.47051
[12] Cai, Z., Boundary Value Problems of Periodic and Anti-periodic Type, (Ph.D. Dissertation (1991), Ohio University)
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