摘要
设(i)f(t,u):(0,1)×(0,+∞)→[0,+∞)连续,关于u单调增加;(ii)存在函数g:[1,+∞)→(0,+∞),g(b)<b且g(b)/b2在(1,+∞)上可积,使得对任何(t,u)∈(0,1)×(0,∞)有f(t,bu)≤g(b)f(t,u).则奇异边值问题u″(t)+f(t,u(t))=0,0<t<1,au(0)-βu′(0)=0,γu(1)+δu′(1)=0.有C[0,1]正解的充分必要条件为0<∫10G(s,s)f(s,1)ds<∞,有C1[0,1]正解的充分必要条件为0<∫10f(s,G(s,s))ds<∞,也得到正解的唯一性及其迭代方法.其中α,β,δ,γ≥0,αγ+αδ+βγ>0,G(t,s)是相应问题的Green函数.
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,bu)≤g(b)f(t,u),(t,u)∈(0,1)×(0,∞). Consider the singular problemu″(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 (*) having C positive solutions is that 0<∫~1 -0G(s,s)f(s,1)ds<∞, a necessary and sufficient condition for the equation (*) having C^1 positive solutions is that 0<∫~1 -0f(s,G(s,s))ds<∞, and obtain the uniqueness, iterative 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 (*).
出处
《数学物理学报（A辑）》
CSCD
北大核心
2005年第3期393-403,共11页
Acta Mathematica Scientia
基金
国家自然科学基金(10471075)
山东省自然科学基金(Y2001A03)
山东省优秀中青年科学家科研奖励基金(02BS119)资助
关键词
奇异边值问题
正解
充分必要条件
Singular boundary value problem
Positive solution
Necessary and sufficient condition.
