• 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

1. 安徽建筑工业学院数理系, 合肥 230601
• 收稿日期:2005-09-21 修回日期:2006-10-16 出版日期:2008-04-25 发布日期:2008-04-25

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

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

XIE Shengli

1. Department of Mathematics and Physics, Anhui University of Architecture, Hefei 230601
• Received:2005-09-21 Revised:2006-10-16 Online:2008-04-25 Published:2008-04-25

K=\inf\Big\{d\geq1:\int_0^am(s){\rm d}s\leq d\min\limits_{0\leq k\leq m}
\int_{t_k}^{t_{k+1}}m(s){\rm d}s\Big\}.\end{eqnarray*}则$m(t)=0,~t\in J$.

Assume that $m(t)\in C[J_k,{\bf R^+}](k=1,2,\cdots,m)$ and $$m(t)\leq (L_1+L_2t)\int_0^tm(s){\rm d}s+L_3t\int_0^am(s){\rm d}s +\sum\limits_{0<t_k<t}M_km(t_k),$$where $L_i\geq0(i=1,2,3),~M_k\geq0$ satisfy either $$KaL_3\big({\rm e}^{\delta(L_1+aL_2)}-1\big)<L_1+aL_2,$$ or
$$a(2L_1+aL_2+aKL_3)<2$$ with $$\delta=\max\limits_{0\leq k\leq m}(t_{k+1}-t_k),\q K=\inf\Big\{d\geq1:\int_0^am(s){\rm d}s\leq d\min\limits_{0\leq k\leq m}\int_{t_k}^{t_{k+1}}m(s){\rm d}s\Big\}.$$
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.

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