具有共振的2n阶m点边值问题的可解性

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.

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

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 invertible in the intersection of the domain of L and the kernel of P,where X is the space of functions whose(2n-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 2n-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.