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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 \[ dx_i/dt=-b_i(t)x_i(t)+ \sum^n_{j=1}a_{ij}(t)f_j(x_j(t))+ \sum^n_{j=1}b_{ij}(t)f_j(x_j (\zeta_{ij}(t,x_j(t)))) +I_i(t), \] and cellular neural networks (CNNs) with state-dependent delays \[ dx_i/dt=-b_i(t)x_i(t)+ \sum^n_{j=1}a_{ij}(t)f_j(x_j(t))+ \sum^n_{j=1}b_{ij}(t)f_j\left(\int^\infty_0 k_{ij}(u)x_j(t-u)\,du\right)+I_i(t), \] \(i,j=1,2\dots,n\).

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
34K13 Periodic solutions to functional-differential equations
68T05 Learning and adaptive systems in artificial intelligence
34K20 Stability theory of functional-differential equations
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References:

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