Existence of weak solutions for a boundary value problem of a second order ordinary differential equation
type: Item
instance of: scholarly article
title: Existence of weak solutions for a boundary value problem of a second order ordinary differential equation
zbMATH Open document ID: 1439.34030
author: Shiyou Weng
author: Fuli Wang
published in: Boundary Value Problems
publication date: 2018-12-20T00:00:00Z
review text: The authors investigate the existence of weak solutions for a boundary value problem of second-order differential equations of the form \[ \left(\frac{x(t)-g(t,x(t))}{h(t,x(t))}\right)''+f(t,x(t)=0,\;\;\;0<t<1, \tag{1} \] \[ x(0)=g(0,x(0)),\;\;\;x(1)=f(1,x(1)), \tag{2} \] where $f, g:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$, and $g:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}\setminus \{0\}$ satisfy the Carathéodory conditions. They transform (1) and (2) to the following perturbed quadratic integral equation: \[ x(t)=g(t,x(t))+h(t,x(t))\int_0^1 G(t,s)f(s,x(s))ds,\;\;0\leq t \leq 1. \] Upper and lower bounds of the Green function $G$ are given in order to apply the well-known Krasnosel'skii-type fixed point theorem in conjunction with the technique of measures of weak noncompactness in Banach spaces. Two examples are presented to illustrate the validity of the main result.
reviewed by: Ramzi Alsaedi
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47H08
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6996317
zbMATH Keywords: boundary value problem
zbMATH Keywords: measure of weak noncompactness
zbMATH Keywords: quadratic integral equation
Wikidata QID: Q115241514
MaRDI profile type: Publication
full work available at URL: s13661-018-0929-7
OpenAlex ID: W2782737389
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DOI: 10.1186/S13661-018-0929-7
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description: scientific article; zbMATH DE number 6996317