## Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces.(English)Zbl 1109.34005

Summary: We use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro-differential equation of Volterra type in a Banach space $$E$$ $\begin{cases} u'(t)=f\bigl(t,u(t),Tu(t)\bigr),\quad t\in J, \quad t\neq t_k,\\ \Delta u|_{t=t_k}=I_k\bigl(u(t_k)\bigr),\quad k=1,2,\dots,m,\\ u(0)=x_0, \end{cases}$ with $$f\in C(J\times E\times E, E)$$, $$J=[0,a]$$, $$0<t_1<t_2<\cdots< t_m<a$$, and $$I_k\in C(E,E)$$, $$k= 1,2,\dots,m$$. Under wide monotone conditions and the noncompactness measure condition of the nonlinearity $$f$$, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. Our result improves and extends some relevant results on abstract differential equations.

### MSC:

 34A45 Theoretical approximation of solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses 34G20 Nonlinear differential equations in abstract spaces 45J05 Integro-ordinary differential equations 34K30 Functional-differential equations in abstract spaces
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### References:

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