Multiple solutions of nonlinear impulsive differential equations with Dirichlet boundary conditions via variational method
type: Item
instance of: scholarly article
title: Multiple solutions of nonlinear impulsive differential equations with Dirichlet boundary conditions via variational method
zbMATH Open document ID: 1296.34072
author: Dan Zhang
published in: Results in Mathematics
publication date: 2013-02-08T00:00:00Z
review text: Using a variational formulation of the second order nonlinear impulsive (Dirichlet) boundary value problem \[ -u''(t) + \lambda u(t) = f(t, u(t)), \;a.e. \; t\in [0,T], \] \[ \Delta u'(t_j) = I_j(u(t_j)), j = 1,2,\cdots,p, u(0) = u(T) = 0, \] where \(T > 0, f: [0,T]\times \mathbb{R} \rightarrow \mathbb{R}\) is continuous, \(0= t_0 < t_1 < \cdots < t_p < t_{p+1} = T\), \(\Delta u'(t_i) = u'(t_j^+) - u'(t_j^-), I_j: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, some existence and multiplicity theorems for the solutions are obtained. The impulsive functions \(I_j\) are assumed to be superlinear whereas \(f\) may be superlinear or sublinear. The results presented here extend and improve some recent results obtained in this direction.
Mathematics Subject Classification ID: 34B37
Mathematics Subject Classification ID: 58E50
zbMATH DE Number: 6135270
zbMATH Keywords: impulsive differential equations
zbMATH Keywords: critical point theory
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: variational structure
reviewed by: Gnana Bhaskar Tenali
MaRDI profile type: Publication
full work available at URL: s00025-011-0221-y
OpenAlex ID: W2013602472
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DOI: 10.1007/S00025-011-0221-Y
description: scientific article; zbMATH DE number 6135270