Multi-point boundary value problem of first order impulsive functional differential equations
type: Item
instance of: scholarly article
title: Multi-point boundary value problem of first order impulsive functional differential equations
zbMATH Open document ID: 1209.34079
author: Fengqin Zhang
author: Yajing Zhang
published in: Journal of Applied Mathematics and Computing
publication date: 2009-11-09T00:00:00Z
review text: A monotone iterative technique is developed for first-order functional differential equations with impulses and nonlinear multi-point boundary conditions \[ x'(t)=f(t,x(t),x(\theta(t))),\qquad t \neq t_k, \qquad t \in J= [0,T], \] \[ \Delta x(t_k)= I_k(x(t_k)), \qquad k=1,2,\dots,p, \] \[ g(x(m_1),\dots, x(m_r))=0, \] where \(f \in C(J \times \mathbb{R}^2,\mathbb{R})\), \(g \in C(\mathbb{R}^n,\mathbb{R})\), \(\theta \in C(J,J)\), \(0 \leq \theta \leq T\), \(I_k \in C(\mathbb{R},\mathbb{R})\), \(0<t_1<t_2<\dots<t_p<T\), \(0=m_1<m_2<\dots<m_r=T\), \(m_i \neq t_j\) for \(i=1,\dots,r\), \(j=1,\dots,p\), \(\Delta x(t_k)=x(t_k^+)-x(t_k^-)\), and \(\Delta x'(t_k)=x'(t_k^+)-x'(t_k^-)\) for \(k=1,\dots,p\). For this purpose, the authors provide the concepts of upper and lower solutions, prove comparison results and discuss the existence of a unique solution to related linear boundary value problems for first-order functional differential equations with impulses by means of the method of upper and lower solutions. They establish an integral characterization of solutions and apply Schauder's fixed point theorem to derive the existence of extremal solutions to the problem of interest between the lower and upper solutions. The monotone method is developed for the case \[ g(\alpha(m_1),\dots, \alpha(m_r))\leq 0 \qquad \text{ and } \qquad g(\beta(m_1),\dots, \beta(m_r))\geq 0, \] but also for \[ g(\alpha(m_1),\dots, \alpha(m_r))> 0 \qquad \text{ and } \qquad g(\beta(m_1),\dots, \beta(m_r))< 0, \] where \(\alpha \leq \beta\) are, respectively, lower and upper solutions to the multi-point boundary value problem. An example is provided as an illustration of the main results of the paper.
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K45
Mathematics Subject Classification ID: 34K07
zbMATH DE Number: 5627270
zbMATH Keywords: impulsive functional differential equations
zbMATH Keywords: nonlinear multi-point boundary conditions
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: monotone iterative technique
MaRDI profile type: Publication
full work available at URL: s12190-008-0209-2
OpenAlex ID: W1993819518
cites work: Q3995625
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DOI: 10.1007/S12190-008-0209-2
description: scientific article; zbMATH DE number 5627270