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含有最大值的二元积分不等式及其应用

发布时间:2019-06-13 08:45
【摘要】:随着微分方程理论的发展,积分不等式有了多种形式的推广.其中,GronwallBellman,Gamidov及Volterra型积分不等式在研究微分(积分)方程解的有界性,唯一性以及其它定性性质中被广泛应用.近几十年来,随着计算数学和数学模型在自动化理论应用中的发展,含有最大值的微分方程日益受到学者的关注.由此,含有最大值的积分不等式成为一个研究热点,其中,含有最大值的Gamidov型,Volterra-Fredholm型积分不等式的研究也取得了一些成果.本文在参考文献[6,16,26,27,33,35,42,45]的基础上,继续研究含有未知函数最大值的二元Gronwall-Bellman-Gamidov型积分不等式,Bihari型不等式,以及它们的弱奇异形式的推广,并且研究了一些含有最大值的二元非线性时滞Volterra-Fredholm型迭代积分不等式.利用分析技巧:比如变量替换,不等式放大,积分微分,反函数等,给出不等式中未知函数的估计.根据内容本文分为以下四章:第一章绪论,介绍本文研究的主要问题及其背景.第二章基于参考文献[26,27,42],研究含有未知函数最大值的二元GronwallBellman-Gamidov型积分不等式:(?)及其弱奇异形式:(?)并应用结论研究含有最大值的弱奇异积分方程解的有界性和唯一性.第三章基于文献[6,27,45],给出含有最大值的二元Gamidov-Bihari型积分不等式:(?)以及它的弱奇异形式:(?)并举例应用所得结果研究含有最大值的弱奇异积分方程解的有界性和唯一性.第四章参考文献[16,33,35],研究如下形式的时滞Volterra-Fredholm型迭代积分不等式:(?)并应用这些结论研究含有最大值的二元时滞Volterra-Fredholm型积分方程解的有界性.
[Abstract]:With the development of differential equation theory, integral inequality has been extended in many forms. Among them, GronwallBellman,Gamidov and Voltra type integral inequalities are widely used in the study of bounded, unique and other qualitative properties of solutions of differential (integral) equations. In recent decades, with the development of computational mathematics and mathematical models in the application of automation theory, differential equations with maximum values have been paid more and more attention by scholars. Therefore, the integral inequality with maximum value has become a hot research topic, in which the study of Gamidov type with maximum value and Volterra- Fred type integral inequality has also achieved some results. In this paper, on the basis of references [6, 16, 26, 27, 33, 35, 42, 45], we continue to study the binary Gronwall-Bellman- Gamidov type integral inequalities with the maximum unknown function, Bihari type inequalities and their weakly singular forms, and study some binary nonlinear delay Volterra- Fred type iterative integral inequalities with maximum values. By using analytical techniques, such as variable substitution, inequality amplification, integral differential, inverse function and so on, the estimation of unknown function in inequality is given. According to the content, this paper is divided into the following four chapters: the first chapter is the introduction, which introduces the main problems and background of this paper. In chapter 2, based on references [26, 27, 42], the binary GronwallBellman- Gamidov type integral inequality with the maximum value of unknown function is studied: (?) And its weakly singular form: (?) The boundary and uniqueness of solutions for weakly singular integral equations with maximum values are studied by using the results. In chapter 3, based on reference [6, 27, 45], the binary Gamidov- Bihari integral inequality with maximum value is given: (?) And its weakly singular form: (?) An example is given to study the boundary and uniqueness of the solution of the weakly singular integral equation with the maximum value. In chapter 4, with reference to [16, 33, 35], the following forms of Volterra- Fred integral inequalities with time delay are studied: (?) By using these results, the boundedness of solutions for binary delay Volterra- Fred integral equations with maximum values is studied.
【学位授予单位】:曲阜师范大学
【学位级别】:硕士
【学位授予年份】:2017
【分类号】:O178

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