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Existence and stability of periodic solutions of delayed cellular neural networks. (English) Zbl 1086.92002

Summary: We use the continuation theorem of coincidence degree theory [see R. E. Gaines and J. L. Mawhin, Coincidence degree, and nonlinear differential equations. (1977; Zbl 0339.47031); J. L. Mawhin, Topological degree methods in nonlinear boundary value problems. (1979; Zbl 0414.34025)] and Lyapunov functions to study the existence and stability of positive periodic solutions for cellular neural networks (CNNs) with distributed delays

dx i /dt=-b i (t)x i (t)+ j=1 n a ij (t)f j (x j (t))+ j=1 n b ij (t)f j (x j (ζ ij (t,x j (t))))+I i (t),

and cellular neural networks (CNNs) with state-dependent delays

dx i /dt=-b i (t)x i (t)+ j=1 n a ij (t)f j (x j (t))+ j=1 n b ij (t)f j 0 k ij (u)x j (t-u)du+I i (t),


92B20General theory of neural networks (mathematical biology)
34K13Periodic solutions of functional differential equations
68T05Learning and adaptive systems
34K20Stability theory of functional-differential equations