Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition. (14th November 2016)

• Record Type: Journal Article
• Authors: Lu, Shiping; Zhong, Tao
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 8(2017)
• Page Start: 3163
• Page End: 3172
• Publication Date: 2016-11-14
• Subjects: fourth‐order differential equation -- homoclinic solution -- variational method -- mountain pass theorem; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4230
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equations u ( 4 ) + w u ′ ′ + a ( x ) u = f ( x, u ), where w is a constant, a ∈ C ( R, R ) and f ∈ C ( R × R, R ) . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a ( x ) and f ( x, u ) are nonperiodic in x, a does not fulfil the coercive condition, and f does not satisfy the well‐known ( A R )‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.