Extremal solutions of nonlinear first-order impulsive integro-differential equations in Banach spaces
type: Item
instance of: scholarly article
title: Extremal solutions of nonlinear first-order impulsive integro-differential equations in Banach spaces
zbMATH Open document ID: 0943.45004
author: Huiqin Lu
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2000-09-05T00:00:00Z
review text: The object of the paper is presented in its title. A nonlinear first-order impulsive integro-differential initial value problem in a Banach space \((E,\|\cdot\|)\) is formulated as: \[ x'= f(t,x,Tx), \quad t\in J,\;t\neq t_k, \quad \Delta x|_{t=t_k}= I_k [x(t_k)], \quad (k=1,2,\dots, p) \text{ and }x(t_0)= x_0, \] where \(f\in C[J\times E\times E,E]\), \(J= [t_0, t_0+a]\), \(a>0\), \(I_k\in C[E,E]\), \(k=1,2,\dots, p\); \(t_0< t_1< \dots< t_k<\dots< t_p< t_0+a\), \(x_0\in E\), \(T(x)(t)= \int_{t_0}^t k(t,s) x(s) ds\), \(k\in C[ D,\mathbb{R}^+]\), \(D= \{(t,s)\in J\times J: t\geq s\}\), \(\mathbb{R}^+= [0,\infty)\), \(\Delta x|_{t=t_k}= x(t_k^+)- x(t_k^-)\), where \(x(t_k^+)\) and \(x(t_k^-)\) are the right and left limits of \(x(t)\) at \(t=t_k\), respectively. For the above formulated problem, the author proves a theorem giving the conditions in which it has minimal and maximal solutions, defined in the paper, very difficult to reproduce in such a review. His results improve and generalize some similar results obtained previously by other authors, quoted in the bibliography of the paper. It ends with an example illustrating his results.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45L05
zbMATH DE Number: 1375297
zbMATH Keywords: extremal solutions
zbMATH Keywords: monotone iterative technique
zbMATH Keywords: contraction mapping principle
zbMATH Keywords: cone ordering
zbMATH Keywords: partial ordering
zbMATH Keywords: nonlinear first-order impulsive integro-differential initial value problem
zbMATH Keywords: Banach space
zbMATH Keywords: minimal and maximal solutions
reviewed by: Radu Iordănescu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1375297