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Therefore, we have that sup t ≥ 0 || ˜ Π w $^{1}$( t ) − ˜ Π w $^{2}$( t ) || ≤ α$_{2}$ sup t ≥ 0 || w $^{1}$( t ) − w $^{2}$( t ) || . Since α$_{2}$ < 1 , one can conclude by using a contraction mapping argument that there exists a unique fixed point ˜ w ( t ) = { ˜ w$_{ij}$ ( t ) } of the operator ˜ Π : Ψ$_{δ}$ → Ψ$_{δ}$ , which is a solution of (4.10). 

To complete the proof, we need to show that there does not exist a solution of (4.10) with σ = 0 different from ˜ w ( t ) . Suppose that θ$_{p}$ ≤ 0 < θ$_{p}$$_{+1}$ for some p ∈ Z . Assume that there exists a solution w ( t ) = { w$_{ij}$ ( t ) } of (4.10) different from ˜ w ( t ) . Denote by z ( t ) = { z$_{ij}$ ( t ) } the difference w ( t ) − ˜ w ( t ) , and let max t ∈ [0 ,θ$_{p}$$_{+1}$ ] || z ( t ) || = ¯ m. It can be verified for t ∈ [0 , θ$_{p}$$_{+1}$ ] that 

The last inequality yields ‖ z ( t ) ‖ ≤ α$_{3}$ ¯ m. Because α$_{3}$ < 1 we obtain a contradiction. Therefore, w ( t ) = ˜ w ( t ) for t ∈ [0 , θ$_{p}$$_{+1}$ ] . Utilizing induction one can easily prove the uniqueness for all t ≥ 0 . □ Remark 4.1 In the proof of Theorem 4.1, we make use of the contraction mapping principle to prove the 

exponential stability. In the literature, Lyapunov-Krasovskii functionals, LMI technique, free weighting matrix method and differential inequality technique were used to investigate the exponential stability in neural networks [12, 54, 57]. They may also be considered in the future to prove the exponential stability in networks of the form (2.3). 

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