关于Banach空间一阶非线性脉冲积分-微分方程初值问题解存在性的注记

谢胜利

doi:10.12341/jssms10099

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系统科学与数学 ›› 2008, Vol. 28 ›› Issue (4) : 482-489. DOI: 10.12341/jssms10099

论文

关于Banach空间一阶非线性脉冲积分-微分方程初值问题解存在性的注记

谢胜利

作者信息 +

A Remrk on the Existence of Solutions of Initial Value Problem for First order Nonlinear Impulsive Integro-Differential Equation in Banach Space

XIE Shengli

Author information +

文章历史 +

摘要

设

m (t) \in C [J_{k}, R^{+}] (k = 1, 2, \dots, m)

,且满足不等式

m (t) \leq (L_{1} + L_{2} t) \int_{0}^{t} m (s) d s + L_{3} t \int_{0}^{a} m (s) d s + \sum_{0 < t_{k} < t} M_{k} m (t_{k}),

其中

L_{i} \geq 0 (i = 1, 2, 3), M_{k} \geq 0

满足

K a L_{3} (e^{δ (L_{1} + a L_{2})} - 1) < L_{1} + a L_{2}

或者

a (2 L_{1} + a L_{2} + a K L_{3}) < 2,

这里

\begin{array}{rcl} δ = max_{0 \leq k \leq m} (t_{k + 1} - t_{k}), \q K = inf {d \geq 1 : \int_{0}^{a} m (s) d s \leq d min_{0 \leq k \leq m} \int_{t_{k}}^{t_{k + 1}} m (s) d s} . \end{array}

则

m (t) = 0, t \in J

.
我们首先指出上述的下确界

K

不存在,然后在比较宽松的条件下,获得了Banach空间中一阶非线性脉冲积分--微分方程初值问题解的存在性定理,本质上改进和更正了现有的结果.

Abstract

Assume that

m (t) \in C [J_{k}, R^{+}] (k = 1, 2, \dots, m)

and

m (t) \leq (L_{1} + L_{2} t) \int_{0}^{t} m (s) d s + L_{3} t \int_{0}^{a} m (s) d s + \sum_{0 < t_{k} < t} M_{k} m (t_{k}),

where

L_{i} \geq 0 (i = 1, 2, 3), M_{k} \geq 0

satisfy either

K a L_{3} (e^{δ (L_{1} + a L_{2})} - 1) < L_{1} + a L_{2},

a (2 L_{1} + a L_{2} + a K L_{3}) < 2

with

δ = max_{0 \leq k \leq m} (t_{k + 1} - t_{k}), \q K = inf {d \geq 1 : \int_{0}^{a} m (s) d s \leq d min_{0 \leq k \leq m} \int_{t_{k}}^{t_{k + 1}} m (s) d s} .

Then

m (t) = 0, t \in J

. Firstly, it is shown that the above infimum

K

is not meaning, and then the existence theorem of solutions of initial value problems is obtained for first order nonlinear impulsive integro-differential equations in Banach spaces under some looser conditions, and hence the existing results are improved.

导出引用

谢胜利. 关于Banach空间一阶非线性脉冲积分-微分方程初值问题解存在性的注记. 系统科学与数学, 2008, 28(4): 482-489. https://doi.org/10.12341/jssms10099

XIE Shengli. A Remrk on the Existence of Solutions of Initial Value Problem for First order Nonlinear Impulsive Integro-Differential Equation in Banach Space. Journal of Systems Science and Mathematical Sciences, 2008, 28(4): 482-489 https://doi.org/10.12341/jssms10099

中图分类号： 34K45 45J05

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收稿日期	修回日期	出版日期
2005-09-21	2006-10-16	2008-04-25
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2008-04-25

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