Implicit nonlinear discontinuous functional boundary value \(\phi\)-Laplacian problems: Extremality results.
type: Item
instance of: scholarly article
title: Implicit nonlinear discontinuous functional boundary value \(\phi\)-Laplacian problems: Extremality results.
zbMATH Open document ID: 1041.34010
DOI: 10.1016/S0096-3003(01)00061-3
author: Alberto Cabada
author: S. Heikkilä
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: The authors consider nonlinear functional equations of the form \[ L_0 u(t)= f(t,u,L_0 u)\tag{\(*\)} \] on the interval \([a,b]\) with nonlinear boundary conditions of the form \[ L_1(u)= B_1(u,L_1(u)),\quad L_2(u(a), u(b))= 0, \] where \(L_0\) and \(L_1\) are defined by \[ L_0 u(t)=- {d\over dt} \varphi(t, u(t), u'(t))- g(t, u,u(t), u'(t)) \] and \[ L_1(u)= L_1(u(a), u(b),u'(a), u'(b),u), \] while \(f\), \(B_1\), \(L_2\), \(\varphi\) and \(g\) are given functions. \(u\) appearing in expressions of \(f\), \(B_1\), \(g\) and \(L_1\) refers to functionals on \(u(t)\). The goal is to show that under suitable conditions on \(f\), \(B_1\), \(L_2\), \(\varphi\) and \(g\), there are extremal solutions to \((*)\) in a sector formed by a pair of lower and upper solutions \(\alpha(t)\) and \(\beta(t)\), namely: \(\alpha(t)\leq u(t)\leq \beta(t)\). To this end, they consider first the case when (\(f\equiv 0\), \(B_1\equiv 0\)) and prove three existence theorems. Then they generalize the results to a more general case.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 34A36
zbMATH DE Number: 1861107
reviewed by: Mithat Idemen
MaRDI profile type: Publication
cites work: Pairs of positive solutions for the one-dimensional p-Laplacian
cites work: A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for \((| u'| ^{p-2}u')'+f(t,u)=0\), \(u(0)=u(T)=0\), \(p>1\)
cites work: Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian
cites work: Existence results for the problem (φ(u′))′=f(t,u,u′) with nonlinear boundary conditions
cites work: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions
cites work: Optimal existence conditions for \(\varphi\)-Laplacian equations with upper and lower solutions in the reversed order
cites work: Monotone iterative methods for boundary value problems
cites work: Periodic solution for nonlinear systems with \(p\)-Laplacian-like operators
cites work: Some General Existence Principles and Results for $(\phi (y')) = qf(t,y,y'),0 < t < 1$
cites work: Q4344231
cites work: Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions
cites work: Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem
cites work: Existence theory for functional \(p\)-Laplacian equations with variable exponents
cites work: Q4492741
cites work: Q5758481
cites work: Q4303531
description: scientific article; zbMATH DE number 1861107