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

设m(t)∈CJk,R ](k=1,2,…,m),且满足不等式m(t)<(L1 L2t)∫tn(s)ds L3t∫a m(s)ds ∑o0满足KaLs(eδ(L1 aL2)-1)

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

Assume that $m(t)\in CJ_k,{\bf R^+}](k=1,2,\cdots,m)$ 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.