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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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