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Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients
Zhibo Cheng,Zhonghua Bi
Keywords:neutral operator with multiple variable coefficients; $p$-Laplacian; periodic solution; extension of Mawhin's continuation theorem; singularity.
Abstract:
      In this paper, we first discuss some properties of the neutral operator with multiple variable coefficients $(Ax)(t):=x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)$. Afterwards, by using an extension of Mawhin's continuous theorem, a kind of second order $p$-Laplacian neutral differential equation with multiple variable coefficients as follows $$\left(\phi_p\left(x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)\right)'\right)'=\tilde{f}(t,x(t),x'(t))$$ is studied. Finally, we consider two kinds of second-order $p$-Laplacian neutral Rayleigh equations with singularity and without singularity. Some new results on the existence of periodic solution are obtained. It is worth noting that $c_i(t)$ is no longer a constant which is different from the corresponding ones of past work.