一类非线性混合分数阶微分方程系统的特征值区间

摘要

研究一类非线性混合分数阶微分方程正解的存在性D0α+u(t) + λf(t,u(t),v(t)) =0,0 ＜ t ＜ 1,D0β+v(t) +μg(t,u(t),v(t)) =0,0 ＜ t ＜ 1,u(0) =u(1) =u'(0) =u'(1) =v(0) =v(1) =v'(0) =v'(1) =0,其中3＜α,β≤4均为实数,D0α+,D0β+是标准的Riemann-Liouville分数阶导数,f,g:[0,1]×[0,+∞)×[0,+∞)-→[0,+∞)是已知的连续函数.利用Krasnoselskii's不动点定理,得到正解存在的几个充分条件,以及使边值问题存在一个正解的特征值区间.%Consider the existence of positive solutions for a system of nonlinear differential equations of mixed fractional orders:{D0α+u(t) + λf(t,u(t),v(t)) =0,0 ＜ t ＜ 1,D0β+v(t) +μg(t,u('t),v(t))'=0,0 ＜ t ＜ 1,u(0) =u(1) =u(0) =u (1) =v(0) =v(1) =v (0) =v'(1) =0,where 3 ＜ α,ββ≤ 4 are real numbers,D0α+,D0β+ are the standard Riemann-Liouville fractional derivatives,and f,g:[0,1] × [0,+ ∞) × [0,+ ∞) → [0,+ ∞) are given continuous functions.By using Krasnoselskii's fixed point theorem,some sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained.