On nonlinear Volterra integral equations with state dependent delays in several variables
type: Item
instance of: scholarly article
title: On nonlinear Volterra integral equations with state dependent delays in several variables
zbMATH Open document ID: 1209.45004
DOI: 10.4171/ZAA/1399
author: Agata Gołaszewska
author: Jan Turo
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2010-02-10T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Consider \(G= [0,a]\subset\mathbb{R}^n\), \(a\in\mathbb{R}^n_+\), \(\Omega= [-r,a]\subset\mathbb{R}^n\), \(r\in\mathbb{R}^n_+\), \(B=\Omega\setminus G\subset \mathbb{R}^n\), \(E\) a Banach space, \(u:\Omega\to E\), a function defined by \(u_x(\tau)= u(x+\tau)\) for \(x\in G\) and \(\tau\in B\), \(F\in C(G\times E^m\times C(B,E),E)\) for \(m\in N\), \(f\in C(G^2\times C(B,E), E^m)\), \(\theta\in C(G\times C(B,E),G)\), \(\psi\in C(G\times C(B, E), G^m)\), \(\beta\in C(G,G)\), \(\alpha\in C(G,G^m)\), \(\varphi\in C(B,E)\) and \(H= (H_1,\dots, H_m): G\to \mathbb{R}^m\), where for \(1\leq i\leq m\) and \(x\in G\), \(H_i(x)\) is a subset of a \(p_i\)-dimensional hyperplane parallel to the coordinate axes, being Lebesgue measurable for \(1\leq p_i\leq n\). The authors consider the Cauchy problem \[ \begin{cases} u(x)= F(x, \int_{H(x)} f(x,s,u_{\psi(s,u_\alpha(s))})\,ds, u_{\theta(x, u_\beta(x))}),\quad & x\in G,\\ u(x)= \varphi(x),\quad x\in B,\end{cases}\tag{1} \] and prove that the Cauchy problem (1) has a unique solution in a class defined in Lemmas 2.2 and 2.3, under hypotheses given in these lemmas.
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 5668459
zbMATH Keywords: initial problems
zbMATH Keywords: comparison method
zbMATH Keywords: neutral equations with state dependent delays
zbMATH Keywords: nonlinear Volterra integral equations in several variables
zbMATH Keywords: Cauchy problem
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
cites work: Existence and uniqueness of the global solution of the initial value problem for neutral type differential-functional equations in Banach space
cites work: On some nonlinear Volterra integral-functional equation in several variables
cites work: Q3431371
cites work: Q3519682
cites work: Q5443891
cites work: Existence, uniqueness and continuous dependence for hereditary systems
cites work: On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays
cites work: Parameter identification in classes of neutral differential equations with state-dependent delays
cites work: Q3767626
cites work: Q4834986
cites work: On the existence and uniqueness of solutions of systems of differential equations with a deviated argument
cites work: Q4699469
cites work: Initial problems for neutral functional differential equations with unbounded delay
cites work: Q5462039
cites work: On the Cauchy Problem for Differential-Delay Equations in a Banach Space
cites work: On non-linear Volterra integral-functional equations in several variables
cites work: Carathéodory solutions to hyperbolic functional differential systems with state dependent delays
cites work: Q4087605
cites work: On the existence and uniqueness of solutions of a certain integral-functional equation
cites work: Q4046582
cites work: On the existence and uniqueness of solutions of Darboux problem for partial differential-functional equations
cites work: On the existence and uniqueness of solutions of the Darboux problem for partial differential-functional equations in a Banach space
cites work: Q3338742
cites work: On the existence and uniqueness of integrable solutions of functional equations in a Banach space
cites work: Existence, uniqueness und successive approximations for a class of integral-functional equations
cites work: Q4095375
cites work: On the existence and convergence of successive approximation for some functional equations in a Banach space
cites work: Q3818696
cites work: Q3834429
cites work: Q5528907
description: scientific article; zbMATH DE number 5668459