Multiple positive solutions of singular third-order periodic boundary value problem. (English) Zbl 1068.34020

Sun, Jingxian; Liu, Yansheng

Acta Math. Sci., Ser. B, Engl. Ed. 25, No. 1, 81-88 (2005).

Summary: This paper deals with the singular nonlinear third-order periodic boundary value problem \[ u'''+ \rho^3 u= f(t,u),\quad 0\leq t\leq 2\pi, \] with \(u^{(i)}(0)= u^{(i)}(2\pi)\), \(i= 0,1,2\), where \(\rho\in (0,{1\over\sqrt{3}})\) and \(f\) is singular at \(t= 0\), \(t= 1\) and \(u= 0\). Under suitable conditions, it is proved by constructing a special cone in \(C[0,2\pi]\) and employing the fixed-point index theory, that the problem has at least one or at least two positive solutions.

MSC:

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

Keywords:

singular boundary value problem; third-order differential system; positive solution