Positive solutions of fourth-order periodic boundary value problems. (English) Zbl 1030.34025

Li, Yongxiang

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 54, No. 6, 1069-1078 (2003).

Summary: Existence results for positive solutions are obtained on the fourth-order periodic boundary value problem \[ u^{(4)}-\beta u''+\alpha u=f(t,u),\;0\leq t\leq 1,\quad u^{(i)}(0)=u^{(i)}(1),\;i=0,1,2,3, \] where \(f:[0,1] \times\mathbb{R}^+ \to\mathbb{R}^+\) is continuous, \(\alpha,\beta \in\mathbb{R}\) and satisfy \(0< \alpha <(\beta/2+ 2\pi^2)^2\), \(\beta>-2\pi^2\), \(\alpha/ \pi^4+\beta/ \pi^2+1>0\). The discussion is based on a new maximum principle for the operator \(L_4u= u^{(4)} -\beta u''+ \alpha u\) and on fixed-point index theory in cones.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10	Fixed-point theorems

Keywords:

fourth-order periodic boundary value problem; maximum principle; positive solution; cone; fixed-point index

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