【摘要】：對(duì)于積分一微分方程解的漸近性的研究是方程領(lǐng)域的重要研究問題,由于在某些特定的條件下,利用積分不等式,可以得到非線性積分一微分方程解的漸近狀態(tài)與某個(gè)齊次方程解的漸近狀態(tài)一致.因此在推廣的過程中也產(chǎn)生了系統(tǒng)的研究類似問題的統(tǒng)一方法.Gronwall-Bellman和Bihari積分不等式及其推廣在積分一微分方程解的漸近性方面起著重要的作用.許多學(xué)者和研究者為了達(dá)到不同的目標(biāo),己經(jīng)在過去幾年內(nèi)建立了一些重要的Gronwall-Bellman和Bihari積分不等式,并用此研究了幾類積分-微分方程解的漸近性.在2004年,孟凡偉[6]研究了下列的具有偏差變?cè)亩A積分一微分方程解的漸近性:在2013年,孟凡偉和姚建麗[7]研究了下列形式的具有偏差變?cè)母唠A非線性積分-微分方程解的漸近性:本文在此基礎(chǔ)上,利用推廣的Gronwall-Bellman和Bihari積分不等式,對(duì)上述積分一微分方程進(jìn)行推廣,并研究了其解的漸近狀態(tài),得到一些新的結(jié)果.最后,通過一種推廣的離散Bihari型不等式,我們可以得到一類三階非線性差分方程的解的有界性與漸近性.根據(jù)內(nèi)容本論文由以下五章構(gòu)成:第一章 緒論,介紹本論文研究的主要問題和背景.第二章 利用新的Gronwall-Bellman和Bihari積分不等式,對(duì)積分-微分方程進(jìn)行推廣,得到具有偏差變?cè)娜A積分-微分方程,并研究其解的漸近性:其中a=a(t)是在R+=[0, ∞)上的正的連續(xù)可微函數(shù),使得a(0) = 1;b (t),c(t),d(t)是在R+上的連續(xù)函數(shù);f∈C[R+×R7,R]和g∈C[R+2×R6,R];α(t),β(t)是連續(xù)可微的并且滿足α(t)≤t,β(t)≤t;α'(t)0,β'(t)0并且α(t),β(t)最終是正的.第三章利用新的Gronwall-Bellman和Bihari積分不等式,對(duì)積分-微分方程進(jìn)行推廣,得到具有偏差變?cè)母唠A積分-微分方程,并研究其解的漸近性:其中p(t)是定義在R+=[0,∞)上的一個(gè)可微函數(shù),并且p(t)0,p(0)=1;ci=ci(t)(i=1,2,...,n)是R+上的連續(xù)函數(shù);φ∈C[R+,R],α(t)≤t,α'(t0,β(t)≤t,β'(t) 0,并且α(t),β(t)最終是正的,f∈C[R+×R2n+1, R],g∈C[R+2×Rn,R].第四章 利用新的Gronwall-Bellman和Bihari積分不等式,對(duì)積分-微分方程進(jìn)行推廣,得到具有偏差變?cè)母唠A非線性積分-微分方程,并研究其解的漸近性:其中p=p(t)是一個(gè)定義在R+=[0,∞)上的正的連續(xù)可微函數(shù),使得p(0) = 1;ci(t)(i=1,2,…,n)是R+上的連續(xù)函數(shù);f∈C[R+×R2n+1,R]并且g∈C[R+2×R2n,R];α(t),β(t)是連續(xù)可微的,并且滿足α(t)≤t,β(t)≤t;α'(t)0, β'(t)0同時(shí)α(t),β(t)最終是正的.第五章通過一種推廣的離散Bihari型不等式,研究一類三階非線性差分方程解的有界性和漸近性:△(r2(n)△(r1(n)△(xp(n))))+f(n,x(n))=0其中n ∈N+(n0) = {n0,n0 + 1,...},n0∈N+, △為向目前差分算子,r(n)是實(shí)序列,f是定義在N(n0) × R × R上的實(shí)值函數(shù).
[Abstract]:The study of asymptotic behavior of solutions of integro-differential equations is an important problem in the field of equations. The asymptotic state of the solution of the nonlinear integro-differential equation is consistent with that of the solution of a homogeneous equation. Therefore, in the process of generalization, the unified method of studying similar problems. Gronwall-Bellman and Bihari integral inequalities and their generalization also play an important role in the asymptotic behavior of the solutions of integro-differential equations. In order to achieve different goals, many scholars and researchers have established some important Gronwall-Bellman and Bihari integral inequalities in the past few years, and used this to study the asymptotic behavior of solutions of several kinds of integro-differential equations. In 2004, Meng Fanwei [6] studied the asymptotic behavior of solutions to second order integro-differential equations with deviating arguments: in 2013, Meng Fanwei and Yao Jianli [7] have studied the asymptotic behavior of solutions of higher order nonlinear integro-differential equations with deviating arguments: in this paper, we use generalized Gronwall-Bellman and Bihari integral inequalities. In this paper, we generalize the above integro-differential equation and study the asymptotic state of its solution, and obtain some new results. Finally, by means of a generalized discrete Bihari type inequality, we can obtain the boundedness and asymptotic behavior of solutions for a class of third-order nonlinear difference equations. According to the content, this paper is composed of the following five chapters: the first chapter introduces the main problems and background of this paper. In chapter 2, by using the new Gronwall-Bellman and Bihari integral inequalities, we generalize the integro-differential equations and obtain the third-order integro-differential equations with deviating arguments. The asymptotic behavior of the solution is studied: where a (t) is a positive continuous differentiable function on R = [0, 鈭瀅. Such that a (0) = 1b (t) c (t) d (t) is a continuous function on R f 鈭，

本文編號(hào)：2260904