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Positive solutions for nonlinear singular third order boundary value problem. (English) Zbl 1221.34059
Summary: We investigate the problem of existence of positive solutions for the nonlinear third order boundary value problem $u'''(t)+\lambda a(t)f(u(t))=0,\quad t\in(0,1),$ $u(0)=u'(0)=0,\quad \alpha u'(1)+\beta u''(1)=0,$ where $$\lambda$$ is a positive parameter. By using Krasnoselskii’s fixed-point theorem in cones, we establish various results on the existence of positive solutions of the boundary value problem. Under various assumptions on $$a(t)$$ and $$f(u(t))$$, we give the intervals of the parameter $$\lambda$$ which yield the existence of the positive solutions. An example is also given to illustrate the main results.

##### MSC:
 34B16 Singular 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
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