Measures of noncompactness in the study of solutions of nonlinear differential and integral equations
type: Item
instance of: scholarly article
title: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations
zbMATH Open document ID: 1277.47067
published in: Central European Journal of Mathematics
publication date: 2013-02-18T00:00:00Z
review text: The paper presents a survey on some existence results for nonlinear differential and integral equations obtained by the author and his co-workers during last years. No proofs are given. First of all, the axiomatic measures of noncompactness in a Banach space \(E\) are defined and a Darbo-type fixed point theorem is recalled. Then, after the definition of measure of noncompactness in the space \(BC(\mathbb R_+)\), it is illustrated how the technique of measures of noncompactness can be applied to provide existence theorems for nonlinear functional integral equations in the space \(BC(\mathbb R_+)\). Finally, the author considers an existence result for an infinite system of ordinary differential equations in the classical sequence space \(\ell^\infty\). The system is handled as a special case of a Cauchy problem of the form \(x'=f(t,x)\), \(x(0)=x_0\), and it is treated by using the sublinear measure of noncompactness in the space \(\ell^\infty\).
reviewed by: Paola Rubbioni
Mathematics Subject Classification ID: 47H08
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 6137125
zbMATH Keywords: Banach space
zbMATH Keywords: compact set
zbMATH Keywords: fixed point theorem
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: integral equation
zbMATH Keywords: integral equation of fractional order
zbMATH Keywords: differential equation
zbMATH Keywords: infinite system of ordinary differential equations
author: Józef Banaś
MaRDI profile type: Publication
full work available at URL: s11533-012-0120-9
OpenAlex ID: W2083453457
cites work: Measures of noncompactness and condensing operators. Transl. from the Russian by A. Iacob
cites work: Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations
cites work: Measures of noncompactness in metric fixed point theory
cites work: Q3883053
cites work: Solvability of infinite systems of differential equations in Banach sequence spaces
cites work: Q4329071
cites work: Q4464983
cites work: Monotonic solutions of a quadratic integral equation of fractional order
cites work: Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity
cites work: Punti uniti in trasformazioni a codominio non compatto
cites work: Ordinary differential equations in Banach spaces
cites work: Q3254057
cites work: Q4421936
cites work: Q5699297
cites work: On the Cauchy problem for ordinary differential equations in Banach spaces
cites work: Applications of measures of noncompactness to the infinite system of differential equations in \(\ell_p\) spaces
cites work: Q5507864
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
DOI: 10.2478/S11533-012-0120-9
description: scientific article; zbMATH DE number 6137125