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A singular third-order 3-point boundary-value problem with nonpositive Green’s function. (English) Zbl 1138.34308

Summary: We find a Green’s function for the singular third-order three-point BVP \[ u'''(t)=-a(t)f(t,u(t)),\quad u(0)=u'(1)= u''(\eta )=0 \] where \(0\leq \eta <1/2\). Then we apply the classical Krasnosel’skii’s fixed point theorem for finding solutions in a cone. Although the Green’s function is not positive, the obtained solution is still positive and increasing. Our techniques rely on a combination of a fixed point theorem and the properties of the corresponding vector field.

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
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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