本文選題：振動性 + 黎曼-斯蒂爾切斯　； 參考：《曲阜師范大學(xué)》2017年博士論文

【摘要】：眾所周知,對于大多數(shù)整數(shù)階和分數(shù)階的微分方程、差分方程來說,尋求其通解是非常困難的,有時甚至是不可能的.因而數(shù)學(xué)工作者們只能從方程本身去分析它的解可能具有的某些性質(zhì),比如:存在性、有界性、振動性、漸近性、穩(wěn)定性等,這些問題的研究促進了方程定性理論的發(fā)展.本文主要研究了幾類整數(shù)階微分方程解的振動性,脈沖微分系統(tǒng)正周期解的存在性,分數(shù)階微分方程和分數(shù)階差分方程解的有界性等問題,推廣并改進了文獻中的相關(guān)結(jié)果.主要內(nèi)容如下:第一章簡要概述了整數(shù)階微分方程解的振動性,具有脈沖效應(yīng)的捕食系統(tǒng)正周期解的存在性,分數(shù)階微分方程和分數(shù)階差分方程解的有界性等問題的研究背景與發(fā)展概況,同時介紹了本文的主要工作.在第二章中,利用廣義Riccati技巧、積分平均技巧以及微分不等式理論,我們討論了含有Riemann-Stieltjes積分項和變指數(shù)增長條件的二階強迫微分方程解的振動性.在2.1節(jié),研究了一類含有Riemann-Stieltjes積分項和變指數(shù)增長條件的二階強迫微分方程解的振動性,建立了方程振動的El-Sayed型準則和Kamenev型準則,改進了文獻中的已有結(jié)論.在2.2節(jié),研究了一類含有p-Laplacian以及Riemann-Stieltjes積分項和變指數(shù)增長條件的二階強迫微分方程解的振動性,建立了方程振動的El-Sayed型準則和Kamenev型準則,所得結(jié)果推廣和改進了相應(yīng)文獻中的己有結(jié)論,并通過一些實例,說明了相應(yīng)準則可應(yīng)用于以前所不能處理的若干情形.在第三章中,我們利用重合度理論中的延拓定理(Mawhin連續(xù)性定理)研究了一個具有脈沖效應(yīng)以及Holling-IV型功能反應(yīng)函數(shù)和干擾系數(shù)的捕食系統(tǒng)正周期解的存在性問題,獲得了一些新的結(jié)果.該結(jié)果表明在適當?shù)木�性周期脈沖擾動下,脈沖微分系統(tǒng)保持了原來相應(yīng)的非脈沖微分系統(tǒng)的周期性.在第四章中,我們通過建立兩類含弱奇異核的Gronwall-Bellman型積分不等式,推廣和改進了相應(yīng)文獻中的己有結(jié)果,并用這兩類不等式研究了兩類分數(shù)階微分方程解的有界性問題.在第五章中,我們對離散分數(shù)階方程解的有界性問題展開研究.在5.2節(jié),我們應(yīng)用Riemann-Liouville型分數(shù)階和分建立了一類非線性Gronwall-Bellman型分數(shù)階和分不等式.在5.3節(jié),我們應(yīng)用Riemann-Liouville型分數(shù)階和分建立一類非線性Volterra-Fredholm型分數(shù)階和分不等式.在5.4節(jié),利用上述不等式研究了一類分數(shù)階差分方程解的有界性和唯一性問題以及一類Volterra-Fredholm型分數(shù)階和分方程解的有界性問題.在第六章中,我們對今后的研究工作進行了展望.
[Abstract]:As we all know, for most differential equations of integer order and fractional order, it is very difficult, sometimes even impossible, to find the general solution of the difference equation.Therefore, the mathematical workers can only analyze some properties of its solution from the equation itself, such as existence, boundedness, oscillation, asymptotic property, stability and so on. The study of these problems has promoted the development of qualitative theory of equation.In this paper, we study the oscillation of solutions of several kinds of integer-order differential equations, the existence of positive periodic solutions of impulsive differential systems, the boundedness of solutions of fractional differential equations and fractional difference equations, and generalize and improve the related results in the literature.The main contents are as follows: in the first chapter, the oscillations of integer order differential equations and the existence of positive periodic solutions for predator-prey systems with impulsive effects are briefly summarized.The background and development of the research on the boundedness of the solutions of fractional differential equations and fractional difference equations are reviewed. The main work of this paper is also introduced.In chapter 2, by using generalized Riccati technique, integral averaging technique and differential inequality theory, we discuss the oscillation of solutions of second order forced differential equations with Riemann-Stieltjes integral term and variable exponential growth condition.In section 2.1, the oscillations of solutions of a class of second order forced differential equations with Riemann-Stieltjes integral term and variable exponential growth condition are studied. The El-Sayed type criterion and Kamenev type criterion for the oscillation of the equation are established, and the existing results in the literature are improved.In section 2.2, the oscillations of solutions of a class of second order forced differential equations with p-Laplacian and Riemann-Stieltjes integral terms and variable exponential growth conditions are studied. The El-Sayed type criteria and Kamenev type criteria for the oscillation of the equations are established.The results generalize and improve the existing conclusions in the relevant literatures and show that the corresponding criteria can be applied to some cases which can not be dealt with before by some examples.In chapter 3, we study the existence of positive periodic solutions of a predator-prey system with impulsive effect, Holling-IV type functional response function and interference coefficient by using continuation theorem Mawhin continuity theorem in coincidence degree theory.Some new results have been obtained.The results show that the impulsive differential system keeps the periodicity of the corresponding nonimpulsive differential system under the proper linear periodic impulsive perturbation.In chapter 4, by establishing two classes of Gronwall-Bellman type integral inequalities with weakly singular kernels, we generalize and improve the existing results in the corresponding literature, and study the boundedness of solutions of two kinds of fractional differential equations with these inequalities.In chapter 5, we study the boundedness of solutions of discrete fractional equations.In Section 5.2, we establish a class of nonlinear Gronwall-Bellman fractional order inequality by using the fractional sum of Riemann-Liouville type.In Section 5.3, we establish a class of nonlinear fractional order inequality of Volterra-Fredholm type by using the fractional order sum of Riemann-Liouville type.In Section 5.4, the boundedness and uniqueness of solutions for a class of fractional difference equations and the boundedness of solutions for a class of fractional and partial equations of Volterra-Fredholm type are studied by using the above inequalities.In the sixth chapter, we look forward to the future research work.
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

【參考文獻】

相關(guān)期刊論文 前1條

1 馬慶華,楊恩浩;弱奇性Volterra積分不等式解的估計[J];應(yīng)用數(shù)學(xué)學(xué)報;2002年03期

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本文編號：1741154