Solvability of 2n-Order m-Point Boundary Value Problem at Resonance
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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A·D2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．
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关键词:
- 共振 /
- Fredholm算子 /
- 多点边值问题 /
- 重合度理论
Abstract: 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. -
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