New approach for the existence and uniqueness of periodic solutions to p-Laplacian prescribed mean curvature equations

Bo Du,Weigao Ge

doi:10.1186/s13661-016-0689-1

Abstract

Using a generalized Mawhin continuation theorem, we obtain some sufficient conditions which guarantee the existence and uniqueness of periodic solutions for two types of prescribed mean curvature p-Laplacian equations.

Highlights

Periodic phenomena are omnipresent in the real world
In the face of such a fact, we cannot but ask ’how can we find the different ways for studying the p-Laplacian equations?’ In order to investigate BVPs with a p-Laplacian, Ge and Ren [ ] extended Mawhin’s continuation theorem
To the best of our knowledge, there are few results for studying the existence and uniqueness of periodic solutions to p-Laplacian prescribed mean curvature equations by using the generalized Mawhin continuation theorem, the main purpose of this paper is to introduce a new method for studying the above equations

Summary

Introduction

Periodic phenomena are omnipresent in the real world. Since Hale [ ] first put forward the concept of functional differential equation (FDE), a large amount of results for periodic solutions of FDEs have been obtained; see e.g. [ – ]. This paper is devoted to an investigation of the existence and uniqueness of periodic solutions for a prescribed mean curvature p-Laplacian Rayleigh equation and a p-Laplacian Duffing equation as follows: φp + x (t) + f x (t) + g x t – τ (t) = e (t) and x (t) φp + x (t) + g x t – τ (t) = e (t), where φp(s). Mawhin’s continuation theorem can be used to study the existence and uniqueness of periodic solutions to equation ). To the best of our knowledge, there are few results for studying the existence and uniqueness of periodic solutions to p-Laplacian prescribed mean curvature equations by using the generalized Mawhin continuation theorem, the main purpose of this paper is to introduce a new method for studying the above equations. [A ] there is a constant D > such that g(x) < –|e| – f ( ) for x > D and g(x) > |e| + f ( ) for x < –D; [A ] there is a constant D > such that g(x) < –|e| for x > D and g(x) > |e| for x < –D ; [A ] there is a constant σ ≥ such that g(x) ≥ σ |x|m, ∀x ∈ R, m ∈ N or g(x) ≤ –σ |x|m, ∀x ∈ R, m ∈ N

Existence of periodic solutions

Conclusion

Conclusions