A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems. (English) Zbl 1119.34020

Graef, John R.; Kong, Lingju

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 11, 2389-2412 (2007).

Summary: We study the nonlinear boundary value problem \[ (\varphi(u''))''= f(t,u,u',u''),\;t\in(0,1), \]
\[ u^{(2i)}(0)=u^{(2i)}(1)=0,\;i=0,1, \] and obtain a necessary and sufficient condition for the existence of symmetric positive solutions. We also discuss the application of our result to the special case where \(f\) is a power function of \(u\) and its derivatives. Moreover, similar conclusions for a more general higher-order boundary value problem are established. Our analysis mainly relies on the lower and upper solution method.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C14	Symmetries, invariants of ordinary differential equations

Keywords:

boundary value problems; existence; symmetric positive solutions; symmetric lower and upper solutions; Schauder fixed point theorem

References:

[1]	Avery, R. I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett., 13, 1-7 (2000) · Zbl 0961.34014
[2]	Cabada, A.; Grossinho, M. R.; Minhós, F., Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order, Nonlinear Anal., 62, 1109-1121 (2005) · Zbl 1084.34013
[3]	Cabada, A.; Pouso, R. L., Existence results for the problem \((\phi(u^\prime))^\prime = f(t, u, u^\prime)\) with nonlinear boundary conditions, Nonlinear Anal., 35, 221-231 (1999) · Zbl 0920.34029
[4]	Chen, S.; Zhang, Y., Singular boundary value problems on a half-line, J. Math. Anal. Appl., 195, 449-468 (1995) · Zbl 0852.34019
[5]	Davis, J. M.; Eloe, P. W.; Henderson, J., Triple positive solutions and dependence on higher order derivatives, J. Math. Anal. Appl., 237, 710-720 (1999) · Zbl 0935.34020
[6]	Davis, J. M.; Erbe, L. H.; Henderson, J., Multiplicity of positive solutions for higher order Sturm-Liouville problems, Rocky Mountain J. Math., 31, 169-184 (2001) · Zbl 0989.34012
[7]	Ehme, J.; Eloe, P. W.; Henderson, J., Upper and lower solution methods for fully nonlinear boundary value problems, J. Differential Equations, 180, 51-64 (2002) · Zbl 1019.34015
[8]	Franco, D.; O’Regan, D.; Perán, J., Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174, 315-327 (2005) · Zbl 1068.34013
[10]	Graef, J. R.; Qian, C.; Yang, B., Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations, Proc. Amer. Math. Soc., 131, 577-585 (2003) · Zbl 1046.34037
[11]	Henderson, J.; Thompson, H. B., Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc., 128, 2373-2379 (2000) · Zbl 0949.34016
[12]	Kong, L.; Kong, Q., Positive solutions of higher-order boundary-value problems, Proc. Edinburgh Math. Soc., 48, 445-464 (2005) · Zbl 1084.34023
[13]	Shi, G.; Chen, S., Positive solutions of even higher-order singular superlinear boundary value problems, Comput. Math. Appl., 45, 593-603 (2003) · Zbl 1054.34039
[14]	Wang, H., On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl., 281, 287-306 (2003) · Zbl 1036.34032
[15]	Wang, H., On the structure of positive radial solutions for quasilinear equations in annular domain, Adv. Differential Equations, 8, 111-128 (2003) · Zbl 1042.34052
[16]	Wei, Z., Positive solutions of singular boundary value problems of fourth order differential equations, Acta Math. Sin., 42, 715-722 (1999) · Zbl 1022.34018
[17]	Wei, Z., Existence of positive solutions for 2nth-order singular sublinear boundary value problems, J. Math. Anal. Appl., 306, 619-636 (2005) · Zbl 1078.34010
[18]	Xu, Y.; Li, L.; Debnath, L., A necessary and sufficient condition for the existence of positive solutions of singular boundary value problems, Appl. Math. Lett., 18, 881-889 (2005) · Zbl 1095.34013
[19]	Zhao, Z., Positive solutions of boundary value problems for nonlinear singular differential equations, Acta Math. Sin., 43, 179-188 (2000) · Zbl 1018.34018