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Abstract We consider fuzzy shunting inhibitory cellular neural networks (FSICNNs) with time-varying coefficients and constant delays. By virtue of continuation theorem of coincidence degree theory and Cauchy–Schwartz inequality, we prove the existence of periodic solutions for FSICNNs. Furthermore, by employing a suitable Lyapunov functional we establish sufficient criteria which ensure global exponential stability of the periodic solutions. Numerical simulations that support the theoretical discussions are depicted.