Periodic and homoclinic solutions of some semilinear sixth-order differential equations

Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations

K e y w o r d s E x t e n d e d Fisher-Kolmogorov equation, Sixth-order equations, Eigenvalue problems~ Clark's theorem, Variational methods. 1. I N T R O D U C T I O N Bistable systems play an important role in the study of spatial patterns. Recently, interest has turned to higher-order model equations involving bistable dynamics, such as the extended Fisher-Kolmogorov (EFK) equation Ou 0% 02u...

Periodic Solutions for Some Fully Nonlinear Fourth Order Differential Equations

In this paper we present sufficient conditions for the existence of solutions to the periodic fourth order boundary value problem u(x) = f(x, u(x), u′(x), u′′(x), u′′′(x)) u(a) = u(b), i = 0, 1, 2, 3, for x ∈ [a, b], and f : [a, b] × R4 → R a continuous function. To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations w...

Existence of Periodic Solutions and Homoclinic Orbits for Third-order Nonlinear Differential Equations

The existence of periodic solutions for the third-order differential equation ̇̈ x+ ω2ẋ = μF(x,ẋ, ẍ) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bif...

Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations

Weighted Pseudo Almost Periodic Solutions for Semilinear Boundary Differential Equations

This paper is concerned with the study of weighted pseudo almost periodic mild solutions for semilinear boundary differential equations. Namely, some sufficient conditions for the existence and uniqueness of weighted pseudo almost periodic mild solutions of semilinear boundary differential equations are obtain.