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

Li, Yongxiang; Liu, Zhe

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 1, 83-92 (2007).

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

Keywords:

impulsive integro-differential equation; initial value problem; cone; lower and upper solution; measure of noncompactness

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