Existence result of second-order differential equations with integral boundary conditions at resonance
type: Item
instance of: scholarly article
title: Existence result of second-order differential equations with integral boundary conditions at resonance
zbMATH Open document ID: 1180.34016
author: Xuemei Zhang
author: Weigao Ge
author: Meiqiang Feng
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-29T00:00:00Z
review text: The paper concerns the boundary value problem \[ x''(t)=f(t,x(t),x'(t)), \quad 0<t<1, \] \[ x'(0)=\int^1_0h(t)x'(t)\,dt, \qquad x'(1)=\int^1_0g(t)x'(t)\,dt. \] It is assumed that the functions \(h\) and \(g\) are continuous and nonnegative and such that the conditions \(\int^1_0h(t)\,dt=1\) and \(\int^1_0g(t)\,dt=1\) hold. The linear operator \(L,\) \(Lx:=x''\), defined on the subspace of functions that belong to the Sobolev space \(W^{2,1}(0,1)\) and satisfy the boundary conditions is a Fredholm operator with index zero when the functions \(h\) and \(g\) satisfy a condition formulated in the preliminary part of the paper. Moreover, its kernel is two-dimensional. Next, a fixed point theorem due to \textit{J. Mawhin} [Topological degree methods in nonlinear boundary value problems. Regional Conference Series in Mathematics. No. 40. R.I.: The American Mathematical Society (1979; Zbl 0414.34025)] is recalled. The existence of at least one solution of the boundary value problem is proved when certain conditions are satisfied. They are formulated in the main result which proof is based on Mawhin's fixed point theorem. An example is presented.
reviewed by: Tihomir Gyulov
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5549430
zbMATH Keywords: second order boundary value problem
zbMATH Keywords: integral boundary conditions
zbMATH Keywords: Fredholm operator
zbMATH Keywords: resonance
zbMATH Keywords: coincidence degree theory
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.11.082
OpenAlex ID: W2040875612
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DOI: 10.1016/J.JMAA.2008.11.082
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description: scientific article; zbMATH DE number 5549430