Periodic solutions for p-Laplacian neutral differential equation with multiple delay and variable coefficients

In this paper, we first discuss some properties of the neutral operator with multiple delays and variable coefficients (Ax)(t):=x(t)−∑i=1nci(t)x(t−δi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})$\end{document}. Afterwards, by using an extension of Mawhin’s continuation theorem, a second order p-Laplacian neutral differential equation (ϕp(x(t)−∑i=1nci(t)x(t−δi))′)′=f˜(t,x(t),x′(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Biggl(\phi _{p} \Biggl(x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}) \Biggr)' \Biggr)'=\tilde{f} \bigl(t,x(t),x'(t)\bigr) $$\end{document} is studied. Some new results on the existence of a periodic solution are obtained. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from those known in the literature.