具有共振的2n阶m点边值问题的可解性  

Solvability of 2n-Order m-Point Boundary Value Problem at Resonance

作　　者：江卫华[1] 郭彦平[2] 仇计清[2] 

机构地区：[1]河北师范大学数学与信息科学学院,石家庄050016 [2]河北科技大学理学院,石家庄050018

出　　处：《应用数学和力学》2007年第9期1087-1094,共8页Applied Mathematics and Mechanics

基　　金：河北省自然科学基金资助项目(A2006000298);河北省博士基金资助项目(B2004204);河北省科技攻关资助项目(07217141)

摘　　要：对具有共振的高阶多点边值问题进行研究.首先在具有2n-1阶连续导数的函数全体所成的空间X的子集上定义了指数为0的Fredholm算子L,并在X上定义了投影算子P,使得算子L在其定义域和P的核的交集上是可逆的.然后,在Lebesgue可积函数全体所成的空间Y上定义了投影算子Q,使得L的逆与I-Q及非线性项f的复合是紧算子,其中,I是Y上的恒同算子.最后通过赋予f一定的增长条件,利用Mawhin的重合度理论,证明了具有共振的2n阶m点边值问题至少存在一个解,并给出一个例子验证这一结果.在这里不要求f具有连续性.The higher order multiple point boundary value problem at resonance is studied. Firstly, a Fredholm operator L with index zero and a projector operator P are defined in the subset of X and in X, respectively, such that L is inverdble in the intersection of the domain of L and the kernel of P, where X is the space of functions whose （2 n - 1 ） th order derivatives are continuous. Secondly, a projector operator Q is defined in the Lebesgue integrable functions＇ space Y such that the composition of the inverse operator of L, I-Q and the nonlinear term f is compact, where I is the identity mapping in Y. Finally, imposing growth conditions on f, the existence of at least one solution for the 2 n-order m-point boundary value problem at resonance is obtained by using coincidence degree theory of Mawhin. An example is given to demonstrate the result. The interest is that the nonlinear term f may be noncontinuous.

关 键 词：共振 FREDHOLM算子 多点边值问题 重合度理论 

分 类 号：O175.8[理学—数学]