一类非线性分数量子方程研究

In this project, a class of nonlinear equation in the sense of quantum-fractional calculus will be considered. Firstly, some comparison theorems within quantum-fractional calculus will be established. After that, by applying the upper and lower solutions method combined with the monotone iterative technique, we will not only establish the conditions for the existence and uniqueness of solution of the problem, but also develop monotone iterative sequence for approximating the solution together with an error estimate for the approximation. Furthermore, we will also study stability of solution for the problem. On the other hand, combing topological degree theory with vector lattice theory, we investigate existence, multiplicity of positive solutions and eigenvalue for quantum-fractional equation with sign changing nonlinearity.

本项目主要研究量子分数微积分框架下的一类非线性方程. 一方面，拟建立若干量子分数微积分框架下的比较原理，并结合上下解方法和单调迭代技巧，将不仅给出解的存在性，而且给出极值解、唯一解及一致收敛于解的迭代序列和相应误差估计；进一步地，将讨论相应方程解的稳定性.另一方面，本项目拟将格结构和拓扑度理论相结合，有效解决非线性项变号的分数量子方程正解的存在性及个数，特征值性质等问题.

本项目调查了量子分数差分方程、量子分数Schrödinger差分方程、分数微分方程、分数反应扩散方程，分数湍流模型等问题。一方面，建立了量子分数微积分框架下的比较原理，通过应用上下解方法、单调迭代技巧、不动点指数理论等现代分析工具，不仅给出了非线性量子分数方程（正）解、极值解、唯一解的存在性条件，而且给出了一致收敛于解的迭代逼近序列和相应近似解的误差估计式；另一方面，也确立了几类分数微分方程解的存在唯一性、稳定性等定性性质. 依托本项目，主持三项省级课题，发表SCI学术论文13篇...