【摘要】：隨著微分方程理論的發(fā)展,微分方程在社會(huì)科學(xué)和自然科學(xué)中有著廣泛的應(yīng)用.Gronwall不等式,Gronwall-Bellman不等式及其推廣形式是研究微分方程與積分方程解的存在性、穩(wěn)定性、唯一性、有界性、漸近性和振動(dòng)性等定性性質(zhì)的一種重要工具,眾多學(xué)者對(duì)其進(jìn)行了一系列的推廣.眾所周知,脈沖微分方程在微分方程理論中占有重要地位,許多作者致力于利用脈沖微分不等式和脈沖積分不等式來考察具有脈沖擾動(dòng)問題的解的定性性質(zhì).Borysenko在[1]中研究了具有兩個(gè)獨(dú)立變量的函數(shù)的脈沖積分不等式:其中φ(t,x)除了點(diǎn)(ti,xi)外是非負(fù)連續(xù)的,u(t_i+ 0,x_i + 0)≠u(t_i -0,x_i-0),i = 1,2, ….本文在參考文獻(xiàn)[1,3,5,9,15,19,20,21]的基礎(chǔ)上,對(duì)具有兩個(gè)獨(dú)立變量的不連續(xù)函數(shù)的Wendroff型脈沖積分不等式進(jìn)行了推廣.根據(jù)內(nèi)容本文分為以下三章:第一章 緒論,介紹本文研究的主要問題及其背景.第二章 本章推廣出如下的Wendroff型積分不等式:這里m 0, t_0 ≥ 0, x_0 ≥ 0;γ_i= const ≥0,β_i = const ≥ 0, 0≤τ_k≤σ_k≤t_k-t_(k-1),0≤δ_k≤λ_k≤x_k-x_(k-1),對(duì)任意的(t,x) ∈Ω,a(t,x)0.且關(guān)于(t,x)是非減函數(shù),即(?)(p,g)∈ Ω, (P,Q)∈Ω,當(dāng)(?)p ≤ P,(?)q ≤ Q時(shí),有a(p,q)≤a(P,Q),并且滿足條件:b(t,x)≥0并且b(ζ,η) = 0,若(ζ,η)∈Ωij,i≠j,(?)i,j=1,2,….當(dāng)t_kt_(k+1),x_kx_(k+1),(?)k=0,1,2,…時(shí),有(t_k,x_k)(t_(k+1),x_(k+1)),且本章得到了更為廣泛的結(jié)論,為研究偏微分方程解的有界性提供了有力的工具.第三章將第二章中具有積分跳躍條件的Wemdroff型積分不等式推廣至帶有時(shí)滯的且具有積分跳躍條件的二元積分不等式:推廣了已有的研究結(jié)果,并用這些不等式研究方程解的性質(zhì).
[Abstract]:With the development of differential equation theory, differential equation has been widely used in social science and natural science. Gronwall inequality, Gronwall-Bellman inequality and its extended form are to study the existence and stability of solutions for differential equations and integral equations. Uniqueness, bounded, asymptotic and oscillatory are an important tool for qualitative properties, which have been extended by many scholars. As we all know, impulsive differential equations play an important role in differential equation theory. Many authors devote themselves to studying the qualitative properties of solutions with impulsive perturbation problems by using impulsive differential inequalities and impulsive integral inequalities. Borysenko studied the impulsive integral inequality of functions with two independent variables in [1]. Formula: where 蠁 (t, X) in addition to the point (ti,xi), there is a nonnegative continuous, u (t 鈮，

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