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Zbl 1069.34012
DaCunha, Jeffrey J.; Davis, John M.; Singh, Parmjeet K.
Existence results for singular three point boundary value problems on time scales. (English)
[J] J. Math. Anal. Appl. 295, No. 2, 378-391 (2004). ISSN 0022-247X

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 2000:: *34B10 Multipoint boundary value problems
34B16 Singular nonlinear boundary value problems
39A12 Discrete version of topics in analysis

Keywords: Singular boundary value problem; Time scale; Fixed point theorem

Citations: Zbl 0685.34017

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