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

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
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