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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 $\bbfT$ given by $$y^{\Delta\Delta}+ f(x, y)=0,\ x\in(0,1]\cap\bbfT,\ y(0)=0,\ y(p)=y \bigl(\sigma^2(1)\bigr),$$ where $p \in(0,1)\cup\bbfT$ 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 {\it J. A. Gatica, V. Oliker} and {\it 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 ODE 34B16 Singular nonlinear boundary value problems for ODE 39A12 Discrete version of topics in analysis
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##### References:
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