Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. (English) Zbl 1004.34008

Rachůnková, Irena; Tvrdý, Milan; Vrkoč, Ivo

J. Differ. Equations 176, No. 2, 445-469 (2001).

The authors are interested in nonnegative and nonpositive solutions to the boundary value problem \[ u''=f(t,u),\quad u(0)=u(1), u'(0)=u'(1), \tag{1} \] where \(f\) satisfies Carathéodory conditions. They provide sets of assumptions that imply existence of lower and upper solutions \(\alpha\) and \(\beta\) which can be well-ordered \(\alpha\leq\beta\) or ordered in the reversed side \(\beta\leq\alpha\). Using such techniques together with degree arguments, the authors generalize previous results on (1). In particular, they consider Duffing equations with repulsive or attracting singularity at the origin and extend existence results by A.C. Lazer and S. Solimini and other authors. Multiple solutions are obtained and weak singularities are allowed in the repulsive case. Results are illustrated by examples.

Reviewer: P.Habets (Louvain-La-Neuve)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34A40	Differential inequalities involving functions of a single real variable
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

second-order nonlinear ordinary differential equation; periodic solution; lower and upper solutions; differential inequalities; nonnegative solution; nonpositive solution; attractive and repulsive singularity; Duffing equation

References:

[1]	del Pino, M.; Manásevich, R.; Montero, A., T-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect., 120, 231-243, (1992) · Zbl 0761.34031
[2]	Fonda, A.; Manásevich, R.; Zanolin, F., Subharmonic solutions for some second-order differential equations with singularities, SIAM J. math. anal., 24, 1294-1311, (1993) · Zbl 0787.34035
[3]	Habets, P.; Sanchez, L., Periodic solutions of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1035-1044, (1990) · Zbl 0695.34036
[4]	Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. amer. math. soc., 99, 109-114, (1987) · Zbl 0616.34033
[5]	J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in Topological Methods for Ordinary Differential Equations (M. Furiet al. , Eds.), Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 24-July 2, 1991, Lecture Notes in Mathematics, Vol. 1537, pp. 74-142, Springer-Verlag, Berlin, 1993.
[6]	Nkashama, M.N.; Santanilla, J., Existence of multiple solutions for some nonlinear boundary value problems, J. differential equations, 84, 148-164, (1990) · Zbl 0693.34011
[7]	Omari, P.; Ye, W., Necessary and sufficient conditions for the existence of periodic solutions of second order ordinary differential equations with singular nonlinearities, Differential integral equations, 8, 1843-1858, (1995) · Zbl 0831.34048
[8]	Rachůnková, I.; Tvrdý, M., Nonlinear systems of differential inequalities and solvability of certain nonlinear second order boundary value problems, J. inequal. appl., 6, 199-226, (2001) · Zbl 1039.34013
[9]	Sanchez, L., Positive solutions for a class of semilinear two-point boundary value problems, Bull. austral. math. soc., 45, 439-451, (1992) · Zbl 0745.34017
[10]	Vrkoč, I., Comparison of two definitions of lower and upper functions of nonlinear second order differential equations, J. inequal. appl., 6, 191-198, (2001) · Zbl 1039.34014
[11]	Zhang, M., A relationship between the periodic and the Dirichlet BVP’s of singular differential equations, Proc. roy. soc. Edinburgh sect. A, 128, 1099-1114, (1998) · Zbl 0918.34025