Abstract
In this paper, we are concerned with a class of quaternion-valued cellular neural networks with time-varying transmission delays and leakage delays. By applying a continuation theorem of coincidence degree theory and the Wirtinger inequality as well as constructing a suitable Lyapunov functional, sufficient conditions are derived to ensure the existence and global exponential stability of anti-periodic solutions via direct approaches. Our results are completely new. Finally, numerical examples are also provided to show the effectiveness of our results.
Highlights
A quaternion, which was invented by Hamilton in 1843 [1], consists of a real and three imaginary parts
Up to date, very few papers have been published on the existence of anti-periodic solutions of the quaternion differential equations [19,20,21]
4 Global exponential stability we study the global exponential stability of anti-periodic solutions of (1) by constructing a suitable Lyapunov functional
Summary
A quaternion, which was invented by Hamilton in 1843 [1], consists of a real and three imaginary parts. The author of [15, 16] studied the existence of periodic solution of the quaternion Riccati equation with two-sided coefficients. Up to date, very few papers have been published on the existence of anti-periodic solutions of the quaternion differential equations [19,20,21]. Our main purpose of this paper is by applying a continuation theorem of coincidence degree theory and the Wirtinger inequality as well as constructing a suitable Lyapunov functional to study the existence and global exponential stability of anti-periodic solutions via a direct method. Example 5.2 Consider the following quaternion-valued cellular neural network with time-varying transmission delays: xp(t) = –ap(t)xp(t) + bpq(t)fq xq(t) + cpq(t)gq xq t – τpq(t) q=1 q=1.
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