Abstract
In this paper, we consider the existence of a positive periodic solution for the following kind of high-order p-Laplacian neutral singular Rayleigh equation with variable coefficient: (φp(x(t)−c(t)x(t−σ))(n))(m)+f(t,x′(t))+g(t,x(t))=e(t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(\\varphi_{p}\\bigl(x(t)-c(t)x(t-\\sigma)\\bigr)^{(n)} \\bigr)^{(m)}+f\\bigl(t,x'(t)\\bigr)+g\\bigl(t,x(t)\\bigr)=e(t). $$\\end{document} Our proof is based on Mawhin’s continuation theory.
Highlights
4 Conclusions In summary, a periodic solution of ( . ) with singularity is illustrated by Theorems . and
In Theorem . , we consider the existence of a periodic solution for ( . ) in the case
In Theorem . , we give a condition on f (t, u) that is weaker than |f (, u)| ≤ K in Theorem . , that is, we obtain the existence of periodic solution for ( . ) in the case where |f (t, u)| ≤ α|u|p– + β
Summary
Wang and Lu [ ] in investigated the existence of periodic solution for the following high-order neutral functional differential equation with distributed delay: x(t) – cx(t – σ ) (n) + f x(t) x (t) + g x(t + s) dα(s) = p(t). Applying the coincidence degree theory and some analysis skills, Xin et al [ ] discussed the existence of a positive periodic solution for the following neutral Liénard equation with singularity: φp x(t) – cx(t – τ ) (n) (m) + f x(t) x (t) + g t, x(t – σ ) = e(t).
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call