Antiperiodic Solutions for a Generalized High-Order ( p , q ) -Laplacian Neutral Differential System with Delays in the Critical Case

Abstract

By applying the method of coincidence degree, some criteria are established for the existence of antiperiodic solutions for a generalized high-order $(p,q)$-Laplacian neutral differential system with delays ${({\phi }_p({(x(t)-cx(t-\tau ))}^{(k)}))}^{(m-k)}=F(t,x_{{\theta }_{\mathrm{0}}(t)},x_{{\theta }_{\mathrm{1}}(t)}^{\prime },\dots ,x_{{\theta }_k(t)}^{(k)},y_{{\vartheta }_{\mathrm{0}}(t)},y_{{\vartheta }_{\mathrm{1}}(t)}^{\prime },\dots ,y_{{\vartheta }_l(t)}^{(l)})$, $({\phi }_q((y(t)-$$dy(t-\sigma ){)}^{(l)}){)}^{(n-l)}=G(t,y_{{\mu }_{\mathrm{0}}(t)},y_{{\mu }_{\mathrm{1}}(t)}^{\prime },\dots ,y_{{\mu }_l(t)}^{(l)},x_{{\nu }_{\mathrm{0}}(t)},x_{{\nu }_{\mathrm{1}}(t)}^{\prime },\dots ,x_{{\nu }_k(t)}^{(k)})$ in the critical case $|c|=|d|=1$. The results of this paper are completely new. Finally, an example is employed to illustrate our results.