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Positive solutions of third order boundary value problems. (English) Zbl 1140.34011

The author considers the boundary value problem of the third order \[ \begin{aligned} &u'''(t) = f(t,u(t),u'(t),u''(t)), \quad 0 < t < 1,\\ &u(0) = u'(a) = u(1) = 0, \end{aligned} \] where \(f\) is a Carathéodory function on \([0,1]\times{\mathbb R}^3\), \(0< a < 1\). Sufficient conditions for the existence of at least one positive solution are given. The arguments are based on a priori bounds of solutions and the topological transversality theory. A multiplicity result is obtained, too.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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