模糊赋范空间中非线性算子的不动点指数及应用研究

In this project, the problems on the fixed point index of nonlinear operator in fuzzy normed spaces are studied by using topological degree method, and thus the problems on the existence of the solution of nonlinear operators equations and the solution and multiple solutions of differential equations and integral equations in fuzzy normed spaces are considered．Firstly, the new concept of the fixed point index of compact continuous operators in fuzzy normed spaces is introduced, and some of their properties are studied, including normalization, homotopy invariance and solution property, etc. Secondly, as application, the problems on the existence of fixed point of compact continuous operators and the solution of fuzzy differential equations and fuzzy integral equations are studied, moreover, through calculating the fixed point index to determine the number of solutions of equations. Finally, the concept of the fixed point indexes of more generalized nonlinear operator in fuzzy normed spaces are introduced, including the fixed point index of strict-set-contraction operator, k-set-contraction operator, and the application of the other areas in theory of space are studied. The research of this project will provide new theory tools for the study on nonlinear operator and its equations in fuzzy normed spaces, meanwhile, it will provide new research methods for the solutions of fuzzy differential equation and fuzzy integral equationas. Therefore, the research of this project contributes important academic value and scientific significance to the enrichment and development of theory of space and nonlinear function analysis in both its theoretical and application.

本项目运用拓扑度方法对模糊赋范空间中非线性算子的不动点指数问题进行研究，并由此讨论该空间中非线性算子方程解的存在性以及微分方程、积分方程解的存在性与多解问题。首先，提出模糊赋范空间中紧连续算子的不动点指数的新概念，并研究该不动点指数的性质，包括正规性、同伦不变性以及可解性等。其次，作为应用，研究模糊赋范空间中紧连续算子不动点的存在性以及模糊微分方程与模糊积分方程解的存在性问题，并通过计算不动点指数来确定方程解的个数。最后，提出模糊赋范空间中更为广义的非线性算子的不动点指数的概念，包括严格集压缩算子、k-集压缩算子的不动点指数，研究它们在空间理论其它领域中的应用。 本项目的研究将为模糊赋范空间中非线性算子及其方程的研究提供新的理论工具，同时，也将为模糊微分方程与积分方程的求解提供新的研究方法。因此，本项目的研究对丰富和发展空间理论和非线性泛函分析理论及其应用都具有十分重要的理论价值与科学意义

四年来，本项目主要研究了非线性算子的不动点定理、拟线性和广义拟线性偏微分方程解的存在性和模糊多属性决策等若干问题。.首先，我们在度量空间中引入G-距离函数和G’-距离函数的概念，运用不动点指数方法讨论了度量空间中模糊映射的不动点定理、耦合重合点定理和耦合公共不动点定理；其次，在锥度量空间、偏度量空间、类度量空间、锥b度量空间、偏序G-度量空间、Menger PM-空间和广义Menger PM-空间等广义度量空间中引入新的压缩条件，运用迭代方法讨论了这些广义度量空间中非线性算子不动点定理、公共不动点定理和耦合不动点定理，全面推广了一系列非线性分析中的著名定理；再次，运用变分方法、临界点理论和对称山路定理讨论了广义拟线性椭圆方程、拟线性Schrödinger方程、广义拟线性Schrödinger方程、分数阶Schrödinger-Poisson系统、Schrödinger-Kirchhoff-Poisson系统、广义的拟线性Schrödinger- Maxwell系统基态变号解、径向解与非径向解、无穷多解等的存在性问题，推广和改进了已有的结果；最后，为了解决决策分析及信息融合中的若干问题，提出了犹豫三角直觉模糊集和对偶犹豫模糊概率的新概念，并在此基础上提出了犹豫三角直觉模糊集的信息集结算子和距离测度，提出了新的模糊多属性决策方法，并通过实例验证了其有效性和可行性。.本项目的研究已达到了预定目标，完满完成了各项研究任务，四年来，项目组共发表相关学术论文40篇，其中被SCI收录论文35篇。