Existence of solutions for time second order nonmonotone differential equations involving convex potentials with periodic and anti-periodic boundary conditions
type: Item
instance of: scholarly article
title: Existence of solutions for time second order nonmonotone differential equations involving convex potentials with periodic and anti-periodic boundary conditions
author: Hanen Mhemdi
author: Sahbi Boussandel
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2022-12-13T00:00:00Z
review text: The paper is devoted to the second order ordinary differential equation \[ \left\{\begin{array}{l} u\in W^{2,2}(0,T;\mathbb{R}^N),\\ -u''+u'+A(u)=f(t),\text{ for a.e. }t\in (0,T), \end{array}\right.\tag{1} \] subject to the periodic boundary conditions \[ u(0)=u(T),\quad u'(0)=u'(T),\tag{2} \] or the anti-periodic boundary conditions \[ u(0)=-u(T),\quad u'(0)=-u'(T).\tag{3} \] Here \(A(u)=\nabla_{g(t)}E(u)\) or \(A(u)=Q(t,u,u')^{-1}\nabla \widetilde E(u)\), where \(\nabla_{g(t)}E\) denotes the gradient of a potential \(E\) on \(\mathbb{R}^N\) which is taken with respect to time-dependent inner products \(\langle\cdot,\cdot\rangle_{g(t)}\) on \(\mathbb{R}^N\), and \(\nabla \widetilde E\) denotes the euclidian gradient operator on \(\mathbb{R}^N\), \(Q:[0,T]\times \mathbb{R}^N\times \mathbb{R}^N\to \mathcal{L}(\mathbb{R}^N)\) and \(f:[0,T]\times \mathbb{R}^N\to \mathbb{R}^N\) are given functions. The authors prove the existence of solutions for problems \((1),(2)\) and \((1),(3)\) by using the continuity method, a global inverse functions theorem and the Schaefer fixed point theorem. Based on the Galerkin method and the monotonicity method, they also study the existence of solutions for nonmonotone second order differential equations in infinite dimensional spaces with periodic and anti-periodic boundary conditions. Some applications for partial differential equations involving Dirichlet and Neumann \(p\)-Laplace operators are finally presented.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7632572
zbMATH Keywords: periodic solutions
zbMATH Keywords: global inverse functions theorem
zbMATH Keywords: Schäefer fixed point theorem
zbMATH Keywords: Galerkin method
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: s00574-022-00321-9
OpenAlex ID: W4312050528
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DOI: 10.1007/S00574-022-00321-9
description: scientific article; zbMATH DE number 7632572