一类三阶积分边值问题的奇数个正解  

Existence of Odd Number of Positive Solutions to a Class of Third-order Integral Boundary Value Problem

作　　者：李丝雨 杨赟瑞 宋雪 LI Siyu;YANG Yunrui;SONG Xue(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,甘肃兰州730070

出　　处：《山西大学学报（自然科学版）》2024年第3期555-563,共9页Journal of Shanxi University(Natural Science Edition)

基　　金：国家自然科学基金(11761046);甘肃省自然科学基金(20JR5RA411);兰州交通大学百名青年优秀人才培养计划基金。

摘　　要：为了发展并完善常微分方程非局部问题的基本理论,本文利用Guo-Krasnoselskill不动点定理建立了一类三阶积分边值问题奇数个正解的存在性。首先,通过研究相应线性积分边值问题的格林函数得到解的形式,并讨论格林函数的性质和解的非负性、单调性等性质。其次,将三阶积分边值问题解的存在性转化为锥上算子的不动点问题,并验证该算子的全连续性。接下来,借助Guo-Krasnoselskill不动点定理证明非线性项满足特定增长性条件下该算子存在奇数个不动点,从而得到三阶积分边值问题奇数个正解的存在性。最后,给出具体的例子说明了研究结果的合理性。基于此,本文在方法上将常用于建立正解以及至少两个正解存在性的Guo-Krasnoselskill不动点定理运用到无穷多(奇数)个正解的存在性研究,推广并完善了三阶边值问题正解的研究结果,丰富了常微分方程边值问题的研究内容,为常微分方程非局部问题在应用数学、物理学领域的广泛应用提供了理论依据。In order to develop and perfect the basic theory for nonlocal problems of ordinary differential equations,we establish the existence of odd number of positive solutions to a class of third-order integral boundary value problems by using Guo-Krasnoselskill fixed point theorem in this paper.Firstly,the form of solution is obtained by investigating the corresponding Green's function of lin-ear integral boundary value problem,and the properties of Green's function and the nonnegativity,monotonicity and other properties of solutions are discussed at the same time.Secondly,the existence of solutions to the third-order integral boundary value problem is transformed into a fixed point problem to an operator defined on a cone,and the complete continuity of the operator is examined.Next,with the help of Guo-Krasnoselskill fixed point theorem,it is proven that the operator has an odd number of fixed points when the nonlinear term satisfies specific growth conditions,and thus the existence of odd number of positive solutions for the third-order integral boundary value problem is obtained.Finally,a concrete example is given to illustrate the rationality of our results.Based on the facts above,Guo-Krasnoselskill fixed point theorem,that is often used to establish the existence of one positive solution and two positive solutions at least,is applied to investigate the existenceof infinite(odd)positive solutions in this paper,which extends and perfects the research results of positive solutions to third-order boundary value problems,enriches the research contents of boundary value problems for ordinary differential equations,and provides theoretical basis for wide applications of nonlocal problems of ordinary differential equations in applied mathematics and physics.

关 键 词：积分边值问题 不动点定理 正解 

分 类 号：O175.8[理学—数学]