李文勝，周　千,楊　青

(西安航空學(xué)院 理學(xué)院，陜西 西安 710077)

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隨機(jī)脈沖抽象積分方程溫和解的存在性

李文勝，周千,楊青

(西安航空學(xué)院 理學(xué)院，陜西 西安 710077)

利用凝聚映射不動(dòng)點(diǎn)定理結(jié)合積分預(yù)解算子理論，研究了一類隨機(jī)脈沖一階抽象積分方程，建立并證明了此類問(wèn)題溫和解的存在性。

積分方程;積分預(yù)解算子;隨機(jī)脈沖

0　引言

近年來(lái)，積分微分方程溫和解的存在性越來(lái)越受關(guān)注[1-4],有關(guān)隨機(jī)脈沖的相關(guān)知識(shí)可參見(jiàn)文獻(xiàn)[5-8].

本文主要考慮一類隨機(jī)脈沖一階抽象積分方程:

t≠ξk,τ≤t≤T

(1)

(2)

1　預(yù)備知識(shí)

即：

2　主要結(jié)果

為了證明系統(tǒng)(1)-(2)溫和解的存在性,假設(shè)下面條件成立：

并且,對(duì)于t∈Rτ,存在一個(gè)常數(shù)L0>0,使得

H3.存在常數(shù)Q>0，使得

定理2.1 假設(shè)條件H1-H3成立,如果

(3)

(4)

則系統(tǒng)(1)-(2)的溫和解是存在的.

為了應(yīng)用引理1.2,證明分為以下三步：

所以

由此,對(duì)t∈Rτ,可得

第二步,Γ是壓縮的。

由(4)知,Γ有一個(gè)不動(dòng)點(diǎn)x∈Λ.

第三步，類似于文獻(xiàn)[8],Γ是全連續(xù)映射。由引理1.2知,隨機(jī)脈沖積分方程問(wèn)題(1)-(2)至少有一個(gè)溫和解.

3　結(jié)語(yǔ)

本文研究了一類隨機(jī)脈沖一階抽象積分方程，首先將模型轉(zhuǎn)化成定義1.2中的積分形式，然后在給定的條件下，利用積分預(yù)解算子理論結(jié)合凝聚映射不動(dòng)點(diǎn)定理，證明了此類積分方程溫和解的存在性。

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[責(zé)任編輯、校對(duì)：周千]

Existence Results of Mild Solution to A Random Impulsive Abstract Integral Equations

LiWen-sheng,ZhouQian,YangQing

(Faculty of Science,Xi'an Aeronautical University,Xi'an 710077)

This paper is concerned with the existence of mild solution to a random impulsive abstract integral equation.Using the condensing mapping fixed point theorem and the integral resolvent operator theory,the existence of mild solutions is established and proven.

Integral Equations;Integral Resolvent Operator;Random Impulsive

2016-07-06

陜西省教育廳科研項(xiàng)目(15JK1379);西安航空學(xué)院科研基金(2014KY1210)

李文勝(1984-)，男，陜西禮泉人，講師，從事泛函微分方程理論研究。

O175.22

A

1008-9233(2016)05-0069-03