Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions
zbMATH Open document ID: 0941.34008
DOI: 10.1016/S0377-0427(99)00243-5
author: Nikolaos Halidias
author: Nikolaos S. Papageorgiou
published in: Journal of Computational and Applied Mathematics
publication date: 2000-07-28T00:00:00Z
review text: Continuing their previous work [J. Differ. Equations 147, No. 1, 123-154 (1998; Zbl 0912.34020)], the authors prove the existence of a solution \(x(.)\in C^1(T,\mathbb{R}^N), \;\|x'(.)\|^{p-2}x'(.)\in W^{1,q}(T,\mathbb{R}^N)\) to a boundary value problem of the form: \[ (\|x'(t)\|^{p-2}x'(t))'\in F(t,x(t),x'(t)) \quad \text{a.e. }(T=[0,b]), \] \[ (\|x'(0)\|^{p-2}x'(0),- \|x'(b)\|^{p-2}x'(b))\in \xi (x(0),x(b)), \] defined the convex-valued multifunction \(F(.,.,.)\), by the maximal monotone operator \(\xi (.,.)\) and by \(p\in [2,\infty)\). The rather technical proofs of the auxiliary results and of the main one rely on the construction of some approximate selectors of Carathéodory type and on the use of the Leray-Schauder fixed point theorem for compact single-valued operators. The authors show that their general formulation contains as particular cases the Dirichlet, Neumann, periodic solutions and some Sturm-Liouville-type problems.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 1398395
zbMATH Keywords: boundary value problem
zbMATH Keywords: second-order differential inclusion
zbMATH Keywords: compact operator
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: approximate selector
zbMATH Keywords: Leray-Schauder fixed point theorem
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1398395