Multiple positive solutions for singular semipositone periodic boundary value problems with derivative dependence. (English) Zbl 1255.34026

Lu, Huiqin

J. Appl. Math. 2012, Article ID 295209, 12 p. (2012).

Summary: By constructing a special cone in \(C^1[0, 2\pi]\) and applying a fixed point theorem, this paper investigates second-order singular semipositone periodic boundary value problems with dependence on the first-order derivative and obtains the existence of multiple positive solutions. Further, an example is given to demonstrate the applicability of our main results.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

References:

[1]	W. B. Gordon, “Conservative dynamical systems involving strong forces,” Transactions of the American Mathematical Society, vol. 204, pp. 113-135, 1975. · Zbl 0276.58005
[2]	P. Majer and S. Terracini, “Periodic solutions to some problems of n-body type,” Archive for Rational Mechanics and Analysis, vol. 124, no. 4, pp. 381-404, 1993. · Zbl 0782.70010
[3]	S. Zhang, “Multiple closed orbits of fixed energy for n-body-type problems with gravitational potentials,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 462-475, 1997. · Zbl 0880.70008
[4]	D. Jiang, J. Chu, D. O’Regan, and R. P. Agarwal, “Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 563-576, 2003. · Zbl 1042.34047
[5]	D. Jiang, “On the existence of positive solutions to second order periodic BVPS,” Acta Mathematica Sinica, vol. 18, pp. 31-35, 1998.
[6]	Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99-107, 2003. · Zbl 1030.34024
[7]	X. Lin, X. Li, and D. Jiang, “Positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces,” Computers & Mathematics with Applications, vol. 51, no. 3-4, pp. 507-514, 2006. · Zbl 1105.34306
[8]	J. Sun and Y. Liu, “Multiple positive solutions of singular third-order periodic boundary value problem,” Acta Mathematica Scientia. Series B, vol. 25, no. 1, pp. 81-88, 2005. · Zbl 1068.34020
[9]	X. Hao, L. Liu, and Y. Wu, “Existence and multiplicity results for nonlinear periodic boundary value problems,” Nonlinear Analysis, vol. 72, no. 9-10, pp. 3635-3642, 2010. · Zbl 1195.34033
[10]	B. Liu, L. Liu, and Y. Wu, “Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3337-3345, 2010. · Zbl 1190.34049
[11]	R. Ma, J. Xu, and X. Han, “Global structure of positive solutions for superlinear second-order periodic boundary value problems,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5982-5988, 2012. · Zbl 1254.34039
[12]	D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988. · Zbl 0661.47045