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

Abstract

Abstract 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) $(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})$. 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)) $$ \Biggl(\phi _{p} \Biggl(x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}) \Biggr)' \Biggr)'=\tilde{f} \bigl(t,x(t),x'(t)\bigr) $$ 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.

Keywords

• Neutral operator
• p-Laplacian
• Periodic solution
• Extension of Mawhin’s continuation theorem
• Singularity