Boundary value problems for systems of nonlinear second order differential difference equations
type: Item
instance of: scholarly article
title: Boundary value problems for systems of nonlinear second order differential difference equations
zbMATH Open document ID: 0828.34055
author: Shumei Miao
published in: Chinese Annals of Mathematics. Series B
publication date: 1995-07-03T00:00:00Z
review text: The boundary value problem for the system of nonlinear second-order differential difference equations (1) \(x''= f(t, x(t- s), x, x')\), \(0\leq t\leq 1\), \(s= \text{const}> 0\), \(x(t)= \phi(t)\), \(t\in [- s, 0]\), \(x(1)= A\), is considered, where \(x\), \(f\in \mathbb{R}^n\), \(\phi\in C([- s, 0],\mathbb{R}^n)\), \(A\in \mathbb{R}^n\). The author adopts to the above problem a new-type Nagumo condition in which the control function is a vector-valued function. This function enables the finding of a priori estimates of each component of the derivative of the solutions. In the paper the function \(f\) satisfies Nagumo or Lipschitz conditions. Under suitable assumptions by Schauder's fixed point theorem the existence of a solution to (1) is demonstrated.
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 770011
zbMATH Keywords: boundary value problem
zbMATH Keywords: system of nonlinear second-order differential difference equations
zbMATH Keywords: Nagumo condition
zbMATH Keywords: a priori estimates
zbMATH Keywords: Schauder's fixed point theorem
reviewed by: Tadeusz Dłotko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 770011