On nonlinear distributional and impulsive Cauchy problems
type: Item
instance of: scholarly article
title: On nonlinear distributional and impulsive Cauchy problems
zbMATH Open document ID: 1238.34017
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-12-08T00:00:00Z
review text: There are established qualitative results for the unique, smallest, greatest, minimal, and maximal solutions of the nonlinear distributional Cauchy problem \[ y'=f(y),\qquad y(a)=c.\tag{P} \] In the first part of the paper, a conversion of problem (P) into an integral equation is given; some related fixed point results are also presented. The next section is devoted to the study of conditions under which problem (P) is well-posed, that is, it has a unique solution that depends continuously on the initial value \(c\). There are also proved existence and comparison results for the smallest and greatest solutions of the Cauchy problem (P). The case when solutions also monotone is also treated in this paper. Existence results are established for minimal and maximal solutions of the Cauchy problem (P). The main tools in the proofs are fixed point results, which are established by means of ordinary and generalized iteration methods. Concrete examples are solved to illustrate the obtained results. Iteration methods, induction and/or Maple programming are used to determine solutions.
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34A45
zbMATH DE Number: 5987812
zbMATH Keywords: distribution
zbMATH Keywords: primitive
zbMATH Keywords: integral
zbMATH Keywords: regulated
zbMATH Keywords: continuous
zbMATH Keywords: Cauchy problem
zbMATH Keywords: impulsive
zbMATH Keywords: solution
zbMATH Keywords: fixed point
zbMATH Keywords: unique
zbMATH Keywords: smallest
zbMATH Keywords: greatest
zbMATH Keywords: minimal
zbMATH Keywords: maximal
author: S. Heikkilä
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.na.2011.09.018
OpenAlex ID: W2077566670
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DOI: 10.1016/J.NA.2011.09.018
description: scientific article; zbMATH DE number 5987812