On the existence of periodic solutions to p-Laplacian Rayleigh differential equation with a delay

Abstract

In this paper, the authors study the existence of periodic solutions to a p-Laplacian Rayleigh differential equation with a delay as follows:where p > 1 is a constant, ϕ p : R → R, ϕ p (u) = |u| p−2 u, f, g, e, τ ∈ C(R, R), τ (t + T ) ≡ τ (t) with τ (t) 0, ∀t ∈ [0, T ], and e(t + T ) ≡ e(t), T > 0 is a constant. By using Mawhin's continuation theorem, some new results are obtained.

Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coefficient of nonlinear term

“…We mention the works [1][2][3][4][5][6][7][8][9][10][11][12] and references therein. In the aforementioned works, Lu and Gui [9] discussed the existence of periodic solutions for a Rayleigh type p-Laplacian equation which is of the following form ðu p ðx 0 ðtÞÞÞ 0 þ f ðx 0 ðtÞÞ þ gðxðt À sðtÞÞÞ ¼ eðtÞ:…”

Homoclinic solutions for Rayleigh type p-Laplacian equations with a deviating argument

“…During the past few years, there are many papers about the periodic solutions of the Rayleigh type p-Laplacian equations, see [1][2][3][4][5][6][7][8] and the references therein. For example, in [3], Xiong and Shao discussed the following Rayleigh type p-Laplacian equation d dt [φ p (u (t))] + f (t, u (t)) + g(t, u(t)) = e(t).…”

New existence criteria for periodic solution to a Duffing p-Laplacian-Like equation

“…(1), the difficulty lies in that φ is more complicated than p-Laplacian operator and φ has no concrete form. There are many results about p-Laplacian operator, see examples [1][2][3][4][5][6][7].…”