Periodic solutions of some fourth-order nonlinear differential equations

Abstract In this paper, using Mawhin’s continuation theorem, we obtain existence results concerning T -periodic solutions of fourth-order nonlinear differential equations of the type u ⁗ − p u ″ − g ( t , u ) = e ( t ) , where p ∈ R , g : R 2 → R is continuous, T -periodic in t and e : R → R is continuous, T -periodic with mean value zero. Applications are given for the extended Fisher–Kolmogorov and Swift–Hohenberg equations with forcing term.

[1]  A. Cabada,et al.  Maximum principles for fourth and sixth order periodic boundary value problems , 1997 .

[2]  Julia Chaparova.  Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations , 2002 .

[3]  J. Mawhin,et al.  A continuation approach to fourth order superlinear periodic boundary value problems , 1993 .

[4]  Jin Shan,et al.  Periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument☆ , 2008 .

[5]  William C. Troy,et al.  Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions , 1997 .

[6]  P. Amster,et al.  Oscillating solutions of a nonlinear fourth order ordinary differential equation , 2007 .

[7]  Fuyi Li,et al.  Existence and multiplicity of solutions to 2mth-order ordinary differential equations , 2007 .

[8]  J. Mawhin.  Topological degree methods in nonlinear boundary value problems , 1979 .

[9]  Olímpio H. Miyagaki,et al.  Periodic solutions for extended Fisher–Kolmogorov and Swift–Hohenberg equations by truncature techniques , 2007 .

[10]  James R. Ward,et al.  Asymptotic conditions for periodic solutions of ordinary differential equations , 1981 .

[11]  F. Zanolin,et al.  On forced nonlinear oscillations in nth order differential systems with geometric conditions , 1984 .

[12]  Stepan Tersian,et al.  Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations☆ , 2001 .

[13]  Qingliu Yao,et al.  Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem , 2005 .

[14]  S. Terracini,et al.  Infinitely many solutions to fourth order superlinear periodic problems , 2003 .