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On the existence of solutions for singular boundary value problem of third-order differential equations. (English) Zbl 1240.34088

The authors consider the singular boundary value problem \[ \begin{cases} -u'''(t)= h(t)f(t,u(t)), \quad t\in (0,1), \\ u(0)=u'(0)=0, \quad u'(1)=\alpha u'(\eta), \end{cases} \] where \(0<\eta <1\), \(1<\alpha <\frac {1}{\eta }\), \(h:(0,1)\to [0,+\infty)\) is continuous and is allowed to be singular at \(t=0\) and/or \(t=1\), \(f\) is continuous on \([0,1]\times (-\infty ,+\infty)\) and is, in particular, not necessarily nonnegative. By means of the topological degree theory, under some conditions on \(f(t,u)\) concerning the first eigenvalue corresponding to the relevant linear operator, the existence of nontrivial solutions for the case when \(f\) is bounded below by a nonpositive constant and of one-sign solutions for other the case is obtained.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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