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Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition
Author(s) -
Lu Shiping,
Zhong Tao
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4230
Subject(s) - mathematics , homoclinic orbit , order (exchange) , mountain pass theorem , differential equation , class (philosophy) , mathematical analysis , pure mathematics , nonlinear system , bifurcation , physics , computer science , finance , quantum mechanics , artificial intelligence , economics
In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equationsu ( 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.