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Existence results for singular three point boundary value problems on time scales. (English) Zbl 1069.34012
Summary: We prove the existence of a positive solution for the three point boundary value problem on time scale $$\mathbb{T}$$ given by $y^{\Delta\Delta}+ f(x, y)=0,\;x\in(0,1]\cap\mathbb{T},\;y(0)=0,\;y(p)=y \bigl(\sigma^2(1)\bigr),$ where $$p \in(0,1)\cup\mathbb{T}$$ is fixed and $$f(x,y)$$ is singular at $$y=0$$ and possibly at $$x=0$$, $$y=\infty$$. We do so by applying a fixed point theorem due to J. A. Gatica, V. Oliker and P. Waltman [J. Differ. Equations 79, 62–78 (1989; Zbl 0685.34017)] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations $$y^{\Delta\Delta}+f(x,y)=0$$, $$y^{\Delta\nabla}+f(x,y)=0$$, and $$y^{\nabla\Delta}+f(x,y)=0$$ satisfying similar three point boundary conditions.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 39A12 Discrete version of topics in analysis
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##### References:
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