In this paper, the authors study the existence of periodic solutions to a p-Laplacian Rayleigh differential equation with a delay as follows:where p > 1 is a constant, ϕ p : R → R, ϕ p (u) = |u| p−2 u, f, g, e, τ ∈ C(R, R), τ (t + T ) ≡ τ (t) with τ (t) 0, ∀t ∈ [0, T ], and e(t + T ) ≡ e(t), T > 0 is a constant. By using Mawhin's continuation theorem, some new results are obtained.
By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness, and global exponential stability of periodic solution for shunting inhibitory cellular neural networks with impulses, dx(ij)dt=-a(ij)x(ij)- summation operator(C(kl)inN(r)(i,j))C(ij) (kl)f(ij)[x(kl)(t)]x(ij)+L(ij)(t), t>0,t not equal t(k); Deltax(ij)(t(k))=x(ij)(t(k) (+))-x(ij)(t(k) (-))=I(k)[x(ij)(t(k))], k=1,2,...] . Furthermore, the numerical simulation shows that our system can occur in many forms of complexities, including periodic oscillation and chaotic strange attractor. To the best of our knowledge, these results have been obtained for the first time. Some researchers have introduced impulses into their models, but analogous results have never been found.
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