## Positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces.(English)Zbl 1105.34306

Summary: This paper is devoted to study the existence of positive solutions to the second-order semipositone periodic boundary value problem $x''+ a(t)x=f(t,x),\;x(0)=x(1),\quad x'(0)=x'(1).$ Here, $$f(t,x)$$ may be singular at $$x=0$$ and may be superlinear at $$x=+\infty$$. Our analysis relies on a fixed-point theorem in cones.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, (), 109-114 · Zbl 0616.34033 [2] Majer, P.; Terracini, S., Periodic solutions to some problems of n-body type, Arch. rational mech. anal., 124, 381-404, (1993) · Zbl 0782.70010 [3] del Pino, M.; Mangsevich, R.; Montero, A., T-periodic solutions for some second-order differential equations with singularities, (), 231-243 · Zbl 0761.34031 [4] Yujun, D., Invariance of homotopy and an extension of a theorem by habets-metzen on periodic solutions of Duffing equations, Nonlinear anal., 46, 1123-1132, (2001) · Zbl 1005.34011 [5] Zhang, M.R., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, (), 1099-1114 · Zbl 0918.34025 [6] Zhang, M.R., Periodic solutions of liönard equations with singular forces of repulsive type, J. math. anal. appl., 203, 254-269, (1996) · Zbl 0863.34039 [7] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, (), 1-78 · Zbl 0889.34018 [8] Fonda, A.; Manasevich, R.; Zanolin, F., Subharmonic solutions for some second order differential equations with singularities, SIAM J. math. anal., 24, 1294-1311, (1993) · Zbl 0787.34035 [9] del Pino, M.A.; Manasevich, R.F., Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. differential equations, 103, 260-277, (1993) · Zbl 0781.34032 [10] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025 [11] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, (), 743-748 · Zbl 0802.34018 [12] Erbe, L.H.; Mathsen, R.M., Positive solutions for singular nonlinear boundary value problems, Nonlinear anal., 46, 979-986, (2001) · Zbl 1007.34020 [13] Rachunkovâ, I.; Tvrdy, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [14] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040 [15] P.J. Torres, M.R. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nach. (to appear). · Zbl 1024.34030 [16] Deimling, K., Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.