{"title":"Infinitely many fast homoclinic orbits for a class of superquadratic damped vibration systems","authors":"M. Timoumi","doi":"10.23952/jnfa.2020.8","DOIUrl":null,"url":null,"abstract":". In this paper, we consider the following damped vibration system ¨ u ( t )+ q ( t ) ˙ u ( t ) − L ( t ) u ( t )+ ∇ W ( t , u ( t )) = 0 , ∀ t ∈ R , where q ∈ C ( R , R ) , L ∈ C ( R , R N 2 ) is a symmetric matrix valued function and W ( t , x ) ∈ C 1 ( R × R N , R ) . We prove the existence of infinitely many fast homoclinic solutions for the system when Q ( t ) = (cid:82) t 0 q ( s ) ds → + ∞ as | t | → ∞ , L is neither coercive nor uniformly positive definite and W ( t , x ) is superquadratic at infinity in the second variable but does not satisfy the well-known superquadratic growth conditions like the Ambrosetti-Rabinowitz or the Fei’s conditions.","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23952/jnfa.2020.8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
. In this paper, we consider the following damped vibration system ¨ u ( t )+ q ( t ) ˙ u ( t ) − L ( t ) u ( t )+ ∇ W ( t , u ( t )) = 0 , ∀ t ∈ R , where q ∈ C ( R , R ) , L ∈ C ( R , R N 2 ) is a symmetric matrix valued function and W ( t , x ) ∈ C 1 ( R × R N , R ) . We prove the existence of infinitely many fast homoclinic solutions for the system when Q ( t ) = (cid:82) t 0 q ( s ) ds → + ∞ as | t | → ∞ , L is neither coercive nor uniformly positive definite and W ( t , x ) is superquadratic at infinity in the second variable but does not satisfy the well-known superquadratic growth conditions like the Ambrosetti-Rabinowitz or the Fei’s conditions.
期刊介绍:
Journal of Nonlinear Functional Analysis focuses on important developments in nonlinear functional analysis and its applications with a particular emphasis on topics include, but are not limited to: Approximation theory; Asymptotic behavior; Banach space geometric constant and its applications; Complementarity problems; Control theory; Dynamic systems; Fixed point theory and methods of computing fixed points; Fluid dynamics; Functional differential equations; Iteration theory, iterative and composite equations; Mathematical biology and ecology; Miscellaneous applications of nonlinear analysis; Multilinear algebra and tensor computation; Nonlinear eigenvalue problems and nonlinear spectral theory; Nonsmooth analysis, variational analysis, convex analysis and their applications; Numerical analysis; Optimal control; Optimization theory; Ordinary differential equations; Partial differential equations; Positive operator inequality and its applications in operator equation spectrum theory and so forth; Semidefinite programming polynomial optimization; Variational and other types of inequalities involving nonlinear mappings; Variational inequalities.