常微分方程积分边值问题的正解

【作者】 张强；

【导师】 程建纲；

【作者基本信息】 烟台大学 ， 应用数学， 2009， 硕士

【摘要】 本文考虑积分边值问题和的正解存在性,其中常数λ, a, b和函数f, g0, g1满足下列条件:（H1） 1 -λ- aλ²≥0,并且以下两条件之一满足（H1-1） a > 0;（H1-2） 1 + a + aλ< 0.（H2） 1 +λ+ bλ²≥0,并且以下两条件之一满足（H2-1） b - 1 - bλ> 0;（H2-2） b < 0.（H3） k = 1 + a - b + abλ² + （a + b）λ> 0.（H4） g₀,g₁ : [0,1]→（-∞,+∞）是连续函数,并且其中（H5） f : [0,1]×[0,∞)→[0,∞)是非负连续函数.记其中函数γ0（t）与γ1（t）的定义分别为:当条件（H1-1）和（H2-1）满足时,当条件（H1-1）和（H2-2）满足时,当条件（H1-2）和（H2-1）满足时,当条件（H1-2）和（H2-2）满足时,本文的主要结论是:设条件（H1）–（H5）满足,并且存在L1 > L0 > 0使得（i）如果条件（H1-1）和（H2-1）满足,或者条件（H1-2）和（H2-2）满足,则边值问题（1.1）至少存在一个正解.（ii）如果条件（H1-1）和（H2-2）满足,或者条件（H1-2）和（H2-1）满足,则边值问题（1.2）至少存在一个正解.更多还原

【Abstract】 The existence of positive solutions are considered to the boundary value problemsandwhere constantsλ, a, b and functions f, g0, g1 satisfy the following conditions:(H1) 1 -λ- aλ~2≥0, and one of the following two conditions is satisfied(H1-1) a > 0;(H1-2) 1 + a + aλ< 0.(H2) 1 +λ+ bλ~2≥0, and one of the following two conditions is satisfied(H2-1) b - 1 - bλ> 0;(H2-2) b < 0.(H3) k = 1 + a - b + abλ~2 + (a + b)λ> 0.(H4) g0,g1 : [0,1]→(-∞,+∞) are continuous, andwhere(H5) f : [0,1]×[0,∞)→[0,∞) is continuous and nonnegative.Let whereγ0(t) andγ1(t) are given by :when conditions (H1-1) and (H2-1) are satisfied,when conditions (H1-1) and (H2-2) are satisfied,when conditions (H1-2) and (H2-1) are satisfied,when conditions (H1-2) and (H2-2) are satisfied,The main results of this paper as follows:Assume that (H1)–(H5) hold, and there exist L1 > L0 > 0 such that(i) If conditions (H1-1) and (H2-1) hold, or conditions (H1-2) and (H2-2) hold, thenthe boundary value problems (1.1) has at least one positive solution.(ii) If conditions (H1-1) and (H2-2) hold, or conditions (H1-2) and (H2-1) hold,then the boundary value problems (1.2) has at least one positive solution.更多还原