【摘要】：隨著微分方程理論的發(fā)展,積分不等式有了多種形式的推廣.其中,GronwallBellman,Gamidov及Volterra型積分不等式在研究微分(積分)方程解的有界性,唯一性以及其它定性性質(zhì)中被廣泛應(yīng)用.近幾十年來,隨著計算數(shù)學(xué)和數(shù)學(xué)模型在自動化理論應(yīng)用中的發(fā)展,含有最大值的微分方程日益受到學(xué)者的關(guān)注.由此,含有最大值的積分不等式成為一個研究熱點,其中,含有最大值的Gamidov型,Volterra-Fredholm型積分不等式的研究也取得了一些成果.本文在參考文獻[6,16,26,27,33,35,42,45]的基礎(chǔ)上,繼續(xù)研究含有未知函數(shù)最大值的二元Gronwall-Bellman-Gamidov型積分不等式,Bihari型不等式,以及它們的弱奇異形式的推廣,并且研究了一些含有最大值的二元非線性時滯Volterra-Fredholm型迭代積分不等式.利用分析技巧:比如變量替換,不等式放大,積分微分,反函數(shù)等,給出不等式中未知函數(shù)的估計.根據(jù)內(nèi)容本文分為以下四章:第一章緒論,介紹本文研究的主要問題及其背景.第二章基于參考文獻[26,27,42],研究含有未知函數(shù)最大值的二元GronwallBellman-Gamidov型積分不等式:(?)及其弱奇異形式:(?)并應(yīng)用結(jié)論研究含有最大值的弱奇異積分方程解的有界性和唯一性.第三章基于文獻[6,27,45],給出含有最大值的二元Gamidov-Bihari型積分不等式:(?)以及它的弱奇異形式:(?)并舉例應(yīng)用所得結(jié)果研究含有最大值的弱奇異積分方程解的有界性和唯一性.第四章參考文獻[16,33,35],研究如下形式的時滯Volterra-Fredholm型迭代積分不等式:(?)并應(yīng)用這些結(jié)論研究含有最大值的二元時滯Volterra-Fredholm型積分方程解的有界性.
[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.
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O178

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