一类分数阶微分方程多点边值问题的多解性  

Multiple Solutions of Multiple-points Boundary Value Problem for a Class of Fractional Differential Equation

作　　者：黄燕萍 韦煜明 HUANG Yanping;WEI Yuming(College of Mathematics and Statistics,Guangxi Normal University,Guilin Guangxi 541006,China)

机构地区：[1]广西师范大学数学与统计学院,广西桂林541006

出　　处：《广西师范大学学报（自然科学版）》2018年第3期41-49,共9页Journal of Guangxi Normal University:Natural Science Edition

基　　金：国家自然科学基金(11361010);广西自然科学基金(2014GXNSFAA118002)

摘　　要：本文主要研究一类Riemann-Liouville分数阶微分方程多点边值问题:{D_(0+)~αu(t)+f(t,u(t),u′(t))=0,u(0)=u′(0)=u″(0)=…=u^(n-2)(0)=0,u′(1)=∑m-2i=1β_iu′(ξ_i),其中0≤t≤1,n-1<α≤n,n≥2,0<β_i<1,0<ξ_i<1,i=1,2,…,m-2。a_i>0,∑m-2i=1β_iξ_i^(α-2)<1。先利用Schauder不动点定理得到边值问题解的存在性,再由Leggett-Williams不动点定理证明边值问题至少存在3个正解的存在性,所得结论更为丰富,推广了已有文献的结果,最后举例子说明本文结论的正确性。In this paper,a class of Riemann-Liouville fractional differential equation with multiple-points boundary conditions are studied:Dα0+u(t)+f(t,u(t),u′(t))=0,u(0)=u′(0)=u″(0)=…=u(n-2)(0)=0,u′(1)=∑m-2[]i=1βiu′(ξi),其中0≤t≤1,n-1<α≤n,n≥2,0<βi<1,0<ξi<1,i=1,2,…,m-2.a i>0,∑m-2[]i=1βiξα-2 i<1.The existence of solution for the boundary value problem is proved by using the Schauder fixed point theorem.At least three positive solutions of the above fractional differential equation are acquired by using the Leggett-Williams fixed point theorem.An example is given to illustrate the validity of the conclusion obtained in this paper.

关 键 词：分数阶微分方程 多点边值问题 SCHAUDER不动点定理 LEGGETT-WILLIAMS不动点定理 多解性 

分 类 号：O175.1[理学—数学]