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Positive periodic solutions of second-order differential equations. (English) Zbl 1127.34325

The authors consider the second-order nonlinear differential equation \[ x''(t)+a(t)x'(t)+f(t,x(t))=0, \] where \(a(t)\in C(\mathbb{R},\mathbb{R})\), \(a(t+\omega)\equiv a(t)\), \(f(t,u)\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\), \(f(t+\omega,u)\equiv f(t,u)\) and investigate existence of positive periodic solutions using a fixed point theorem for cones [see e.g. D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones. Boston, MA: Academic Press (1988; Zbl 0661.47045) or K. Deimling, Nonlinear Functional Analysis. Berlin etc.: Springer (1985; Zbl 0559.47040)]. They extend some of results achieved by Y. Li [Chin. Ann. Math., Ser. B 25, No. 3, 413–420 (2004; Zbl 1073.34041)], where criteria for existence of positive periodic solutions of the equation \[ x''(t)+f(t,x(t))=0, \] are proved.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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