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ISSN : 1226-0657

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Vol.16 No.4
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POSITIVE SOLUTION FOR SYSTEMS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS ON TIME SCALES

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2009, v.16 no.4, pp.327-344
Miao, Chunmei
Ji, Dehong
Zhao, Junfang
Ge, Weigao
Zhang, Jiani
Miao,, C. , Ji,, D. , Zhao,, J. , Ge,, W. , & Zhang,, J. (2009). POSITIVE SOLUTION FOR SYSTEMS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS ON TIME SCALES. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 16(4), 327-344.
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Abstract

In this paper, we deal with the following system of nonlinear singular boundary value problems(BVPs) on time scale <TEX>$\mathbb{T}$</TEX> <TEX>$$\{{{{{{x^{\bigtriangleup\bigtriangleup}(t)+f(t,\;y(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}\atop{y^{\bigtriangleup\bigtriangleup}(t)+g(t,\;x(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}}\atop{\alpha_1x(a)-\beta_1x^{\bigtriangleup}(a)=\gamma_1x(\sigma(b))+\delta_1x^{\bigtriangleup}(\sigma(b))=0,}}\atop{\alpha_2y(a)-\beta_2y^{\bigtriangleup}(a)=\gamma_2y(\sigma(b))+\delta_2y^{\bigtriangleup}(\sigma(b))=0,}}$$</TEX> where <TEX>$\alpha_i$</TEX>, <TEX>$\beta_i$</TEX>, <TEX>$\gamma_i\;{\geq}\;0$</TEX> and <TEX>$\rho_i=\alpha_i\gamma_i(\sigma(b)-a)+\alpha_i\delta_i+\gamma_i\beta_i$</TEX> > 0(i = 1, 2), f(t, y) may be singular at t = a, y = 0, and g(t, x) may be singular at t = a. The arguments are based upon a fixed-point theorem for mappings that are decreasing with respect to a cone. We also obtain the analogous existence results for the related nonlinear systems <TEX>$x^{\bigtriangledown\bigtriangledown}(t)$</TEX> + f(t, y(t)) = 0, <TEX>$y^{\bigtriangledown\bigtriangledown}(t)$</TEX> + g(t, x(t)) = 0, <TEX>$x^{\bigtriangleup\bigtriangledown}(t)$</TEX> + f(t, y(t)) = 0, <TEX>$y^{\bigtriangleup\bigtriangledown}(t)$</TEX> + g(t, x(t)) = 0, and <TEX>$x^{\bigtriangledown\bigtriangleup}(t)$</TEX> + f(t, y(t)) = 0, <TEX>$y^{\bigtriangledown\bigtriangleup}(t)$</TEX> + g(t, x(t)) = 0 satisfying similar boundary conditions.

keywords
singular boundary value problem, nonlinear system, time scale, fixed point theorem

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