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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

POSITIVE SOLUTION FOR SYSTEMS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS ON TIME SCALES

한국수학교육학회지시리즈B:순수및응용수학 / 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 (COLLEGE OF SCIENCE, CHANGCHUN UNIVERSITY)
Ji, Dehong (COLLEGE OF SCIENCE, TIANJIN UNIVERSITY OF TECHNOLOGY)
Zhao, Junfang (DEPARTMENT OF MATHEMATICS, BEIJING INSTITUTE OF TECHNOLOGY)
Ge, Weigao (DEPARTMENT OF MATHEMATICS, BEIJING INSTITUTE OF TECHNOLOGY)
Zhang, Jiani (DEPARTMENT OF MATHEMATICS, BEIJING INSTITUTE OF TECHNOLOGY)
Miao, Chunmei, Ji, Dehong, Zhao, Junfang, Ge, Weigao, & Zhang, Jiani. (2009). POSITIVE SOLUTION FOR SYSTEMS OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS ON TIME SCALES. 한국수학교육학회지시리즈B:순수및응용수학, 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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한국수학교육학회지시리즈B:순수및응용수학

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