## 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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