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ISSN : 1226-0657

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Vol.17 No.1

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SOLVABILITY FOR SECOND-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON AN UNBOUNDED DOMAIN AT RESONANCE

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
2010, v.17 no.1, pp.39-49
Yang, Ai-Jun
Wang, Lisheng
Ge, Weigao
Yang,, A. , Wang,, L. , & Ge,, W. (2010). SOLVABILITY FOR SECOND-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON AN UNBOUNDED DOMAIN AT RESONANCE. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 17(1), 39-49.

Abstract

This paper deals with the second-order differential equation (p(t)x'(t))' + g(t)f(t, x(t), x'(t)) = 0, a.e. in (0, <TEX>$\infty$</TEX>) with the boundary conditions <TEX>$$x(0)={\int}^{\infty}_0g(s)x(s)ds,\;{lim}\limits_{t{\rightarrow}{\infty}}p(t)x'(t)=0,$$</TEX> where <TEX>$g\;{\in}\;L^1[0,{\infty})$</TEX> with g(t) > 0 on [0, <TEX>$\infty$</TEX>) and <TEX>${\int}^{\infty}_0g(s)ds\;=\;1$</TEX>, f is a g-Carath<TEX>$\acute{e}$</TEX>odory function. By applying the coincidence degree theory, the existence of at least one solution is obtained.

keywords
integral boundary condition, unbounded domain, g-Carath<tex> $\acute{e}$</tex>odory function, resonance

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