Extremal solutions for nonlinear functional \(\varphi\)-Laplacian impulsive equations
type: Item
instance of: scholarly article
title: Extremal solutions for nonlinear functional \(\varphi\)-Laplacian impulsive equations
zbMATH Open document ID: 1124.34012
author: Alberto Cabada
author: Jan Tomeček
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-05-30T00:00:00Z
review text: This interesting article is devoted to second-order nonlinear \(\varphi\)-Laplacian boundary value problems with impulses of the form \[ (\varphi(u'(t)))' =f(t,u(t),u'(t)); \] \[ g_1(u(0),u(T))=0, \] \[ g_2(u(0), u(T),u'(0),u'(T),u)=0; \] \[ I_k(u(t_k),u(t_k^+))=0, \] \[ M_k(u(t_k), u(t_k^+),u'(t_k),u'(t_k^+),u)=0, \] where \(k=1,\dots,p\), the points \(t_1<\cdots<t_p\) in \((0,T)\) are fixed. The boundary and impulsive conditions are nonlinear and some of them can contain functional dependence on the solution, some monotonicity assumptions are imposed on the corresponding maps. The existence of solutions and of minimal and maximal solutions is studied. Reviewer's remark: Problems with nonlinear functional boundary conditions satisfying monotonicity assumptions were studied in a different situation in [\textit{S. A. Brykalov}, Differ. Equations 29, No. 5, 633--639 (1993); translation from Differ. Uravn. 29, No. 5, 744--750 (1993; Zbl 0835.34087)].
reviewed by: Sergei A. Brykalov
Mathematics Subject Classification ID: 34B37
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5159591
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: nonlinear equations
zbMATH Keywords: boundary value problems
zbMATH Keywords: functional equations
MaRDI profile type: Publication
full work available at URL: j.na.2006.06.043
OpenAlex ID: W2033576842
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DOI: 10.1016/J.NA.2006.06.043
description: scientific article; zbMATH DE number 5159591