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Positive solutions to a generalized second-order three-point boundary-value problem on time scales. (English) Zbl 1075.34014
Summary: Let \(\mathbf T \) be a time scale with \(0,T \in \mathbf T \). We investigate the existence and multiplicity of positive solutions to the nonlinear second-order three-point boundary value problem \[ \begin{aligned} u^{\Delta\nabla}(t)+a(t)f(u(t))&=0,\quad t\in[0, T]\subset \mathbf T,\\ u(0)=\beta u(\eta),\quad u(T)&=\alpha u(\eta), \end{aligned} \] on time scales \(\mathbf T \), where \(0<\eta<T\), and \(0<\alpha<\frac{T}{\eta}\), \(0<\beta<\frac{T-\alpha\eta}{T-\eta}\) are given constants.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
39A13 Difference equations, scaling (\(q\)-differences)
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