一类非线性奇异微分方程正解的存在性定理

Abstract

Abstract:

Suppose (i) f(t,u):(0,1)×(0,+∞)→[0,+∞) is continuous and is increasing on u; (ii) there exists a function g:[1,+∞)→ (0,+∞),g(b)<b and g(b)b^2 is integrable on (1,+∞) such that f(t,b u)≤g(b)f(t,u),(t,u)∈(0,1)×(0,∞).Consider the singular problem{u″(t)+f(t,u(t))=0, 0<t<1,αu(0)-βu′(0)=0,γu(1)+δu′(1)=0.)(*)Then a necessary and sufficient condition for the equation (*) havingC[0,1] positive solutions is that 0<∫^1_0G(s,s)f(s,1)ds<∞, a necessary and sufficient condition for the equation (*) having C^1[0,1] positive solutions is that 0<∫^1_0f(s,G(s,s))ds<∞, and obtain the uniqueness, iterativ e method of the positive solutions. Where α,β,δ,γ≥0, αγ+αδ+δγ>0,G(t,s) is the Green function of the problem (*). 

Key words: Singular boundary value problem, Positive solution, Necessary and sufficient condition.

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DIAO Ceng-Qi. On Existence Theorems of Positive Solutions for Nonlinear Singular Differential Equations[J].Acta mathematica scientia,Series A, 2005, 25(3): 393-403.

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References

[1]Fink A M, Gatica J A, Hernandez G E, Waltman P. Approximation of solutions of singular second order boundary value problems. SIAM J Math Anal, 1991,22(2): 440-462

[2]Gatica J A, Oliker V, Waltman P. Singular nonlinear boundary value prob lems for second order ordinary differential equations.J Differential Equations, 1989,79: 62-78

[3]柴国庆。奇异边值问题的正解存在性。数学物理学报，2001，21( 4)：521-526

[4]Donal O′Regan. Singular Dirichlet boundary value problems I，superlin ear and nonresonant case. Nonlinear Analysis (TMA),1997， 29(2)： 221-245

[5]Zhao Z Q. Uniqueness of positive solutions for singular nonlinear second order boundaryvalue problems. Nonlinear Analysis(TMA), 1994,23(6): 755-765

[6]Luning C D, Perry W L. Positive solutions of negative exponent generali zed EmdenFowler boundary value problems.SIAM J Math Anal, 1981, 12(6):874-879

[7]Zhang Y. Positive solutions of singular sublinear EmdenFowler bounda ry value problems. J Math Anal Appl,1994, 185(1): 215-222

[8]Wei Z L. Positive solutions of singular sublinear second order bound ary v alue problems. Systems Science and Mathematical Sciences， 1998, 11(1): 82-88

[9]赵增勤. 一类奇异次线性边值问题正解存在的充分必要条件. 数学学报, 1998,41(5): 1025-1034

[10]Hartman P. Ordinary Differential Eqnations, 2nd Ed. Boston: Birk hauser, 1982

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On Existence Theorems of Positive Solutions for Nonlinear Singular Differential Equations

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