半序Banach空间中两类非线性算子方程的多重解

In recent years, much attention has been attached to non-local problems of ordinary differential equations and more general dynamic equations on time scales. However, these nonlinear problems usually have not variational structures. Therefore, critical point theory cannot be applied to these problems. Nonlinear operators discussed in our project need not have variational structures. By using the topological degree and fixed point index theory, the existence of multiple solutions(including positive solutions and sign-changing solutions) for superlinear and asymptotically linear operator equations is studied. Firstly, under non-cone mappings, the multiple nonzero fixed points of superlinear operators satisfing upper and lower solutions in reversed order and paralleled lower and upper solutions are considered, respectively. Furthermore, some existence theorems of multiple fixed points for the sum of concave and convex operators are established. These results provide some ideas for constructing paralleled lower and upper solutions. Secondly, under the assumptions that asymptotically linear operators have two pairs of lower and upper solutions and a pair of lower and upper solution, respectively, the existence of multiple fixed points and sign-changing fixed points is discussed. Lastly, the abstract results obtained in our works can be applied to superlinear Hammerstein integral equations and non-local problems of nonlinear ordinary differential equations, the existence results of multiple solutions and sign-changing solutions for the above equations and problems are obtained. We do not only unify the results of some concrete integral and differential equations, but also extend some conditions of the above results and obtain some new multiplicity results.

近年来，非线性常微分方程及更一般的时标上动力学方程的非局部问题受到了广泛的关注，这些非线性问题通常不具有变分结构，因此，临界点理论难以奏效。本项目拟针对不具有变分结构的非线性算子，借助于拓扑度和不动点指数理论，研究一类超线性和一类渐近线性算子方程的多重解(包括正解和变号解)的存在性。首先，在非锥映射下，考察一类分别满足反向上下解和平行上下解的超线性算子的多重非零不动点，进而建立一类凸凹算子之和的多重不动点的存在性定理，从而为构造平行上下解提供思路。然后，在分别假定渐近线性算子存在两对上下解和一对上下解的前提下，讨论其多重不动点特别是变号不动点的存在性。最后，将获得的抽象结果应用于超线性Hammerstein型积分方程和非线性常微分方程的非局部问题，得到其多重解与变号解的存在性结果。我们的工作不仅将统一一些具体方程的结论，扩展其中的部分条件，而且将获得其存在多重解的一些新结果。

一年来，本项目针对不具变分结构的非线性算子，开展了以下两方面的工作：其一、分别建立了超线性算子与凹凸算子和的多重不动点的存在性定理。其二、在渐近线性算子分别满足两对及一对正向上下解的前提下，获得了其正、负不动点与变号不动点的存在性，进而分别构造了适当的上下解条件。所采用的主要思路与工具及取得的成果为：其一、在非锥映射下，利用一致正算子和不动点指数的性质，给出了超线性算子方程存在三个及六个非平凡解的充分条件与所在区域，此时的算子分别满足反向与平行上下解。进一步，将凸算子与平行上下解相结合，借助于序形式的锥拉伸与锥压缩不动点定理，获得了一类较广泛凹凸型算子方程之和的两解定理。其二、利用可微映射零点指数定理，分别构造了一对正向与反向上下解，从而获得了存在两个与一个变号解的多重解结论。本项目所获得的抽象结果，可直接应用于核函数下方有界的超线性Hammerstein型积分方程与非线性常微分方程非局部问题，不仅改进了其中的某些条件，而且获得了其存在其变号解与多重解的一些新结果。