The existence of positive solutions of a class of fourth-order singular boundary value problems.(Chinese. English summary)Zbl 1199.34087

For the singular boundary value problem $\begin{cases} u^{(4)}(t)=f(t, u(t)),\;t\in (0, 1),\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{cases} \tag{1}$ where $$f\in C((0, 1)\times (0, \infty), [0, \infty])$$ and $$f$$ may be singular at $$t=0, 1$$ and $$u=0$$. By using the fixed point index theorem, the authors study the existence of positive solutions of (1). Under appropriate conditions, an existence theorem is proved for (1) to have at least two positive solutions. An example illustrating the results is also given.

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 47N20 Applications of operator theory to differential and integral equations