若干非线性算子与非线性方程的讨论

【作者】 王文霞；

【导师】 梁展东；

【作者基本信息】 郑州大学 ， 基础数学， 2003， 博士

【摘要】 在本文中，我们主要讨论了Banach空间中两类非线性方程，其一为具有凹凸性的非线性算子方程，其二为非线性积分-微分方程．所使用的主要方法为半序方法。 全文共分为三章，在第一章（引言）中，我们对具有凹凸性的非线性算子的研究现状及我们在本文中将要做的工作进行了阐述；同时给出了我们处理本文所讨论的非线性积分-微分方程的总体思路。 在第二章（凹凸算子的不动点及应用）中，我们主要讨论具有凹凸性的非线性单调算子存在唯一正不动点的条件。在§2.1中，我们给出了诸凹算子的关系，诸-凸算子（-u_o凸，-φ凸，-α凸及序凸算子等）的关系。在此基础上，我们证明了u_o-凹增算子存在唯一正不动点的充分必要条件；-u_o凸减算子存在唯一正不动点的充分条件。在本节中，我们也讨论了具有序凹（凸）性的单调算子及具有u_o凹凸性的混合单调算子唯一正不动点的存在性问题。 在§2.2中，我们给出了增的u_o凸算子，增的α（＞1）齐次算子算于存在唯一正不动点的充分条件。 在§2.3中，作为前两节抽象结果的应用，我们讨论了具有凹凸性的Hammerstein积分方程正解的存在唯一性问题。 所得结论推广并改善了现有相关结果。 在第三章（Banach空间中非线性积分-微分方程）中，我们主要研究了Banach空间中带有一阶微分项的二阶非线性积分-微分方程（脉冲积分-微分方程）。在§3.1中，通过建立比较定理，我们讨论了如下带有非线性算子的一阶非线性积分-微分方程的最小最大解的存在性问题，其中，B=h+∫₀^tx（s）ds，h∈E. 在§3.2，利用§3.1的结果。我们讨论了如下带有一阶微分项的的二阶非线性积分-微分方程的最小最大解的存在性问题：{·犷“‘，一刃，，，，T\x，“双，，、L工LU)=工0，x LU)=工L上)t〔J 在够.3及笋.4中，’我们利用笋.1及笋.2的思路，讨论了相应的一阶、二阶脉冲积分一微分方程.所得结果发展并推广了相关的结论.更多还原

【Abstract】 The purpose of this paper is to discuss two classes of nonlinear equations, one of which is nonlinear operator equations with concavity or convexity and the other is nonlinear integro-difFerential equations in Banach space.By means of partial ordering method and iterative techniques, the existence of solutions for these nonlinear equations is studied.This paper includes three chapter.Chapter 1. Introduction. In the first chapter we provide a research summary of the operators with concavity or convexity since the beginning when these definitions of concave (convex) operator were introduced . and the general thoughts in which we can deal with the nonlinear integro-differential equations in Banach space.Chapter 2 Fixed Point Theorems of Concave(Convex) Operators and Applications. The chapter is devoted to the existence and uniqueness of fixed point of monotone operators with concavity or convexity. In 2.1. we present some relations: among the various concave operators, among the various -convex operators. In particular, we give a series of sufficient and necessary conditions for the existence and uniqueness of fixed point of increasing operators with u0-concavity and decreasing operators with -u0-convexity. Moreover, we discuss increasing operators with order concavity, decreasing operators with order convexity and mixed monotone operators with u0-concavity and convexity.In 2.2, we give a series of sufficient conditions for the existence and uniqueness of fixed point of increasing operator with u0-convexity or Q(> 1)- homogeneity.In 2.3, we apply the abstract results in 2.1 and 2.2 to nonlinear Hammerstein integral equations.Chapter 3 Integro-differential Equations in Banach Space . In this final chapter, our purpose is to investigate the existence of maximal and minimal solutions of second-order integro-difTerential equations in Banach Space by results of first-order integro-differential equations in Banach Space. In 3.1, by establishing comparison results, we study the existence of maximal and minimal solutions of initial value problems and boundary value problem for first-order integro-differential equations with nonlinear operator in Banach Space as follow:where,In 3.2. we discuss the initial value value problems and boundary value problem for second-order integro-differential equation of mixed type in Banach space as follow:And give some results of maximal and minimal solutions of them.In 3.3 and 3.4, following the same idea as 3.3 and 3.4, we discuss the existence of maximal and minimal solutions of first-order and second-order impulsive integro-differential equations corresponding to the equations in 3.1 and3.2.更多还原