× ∣ f $^{(}$˜ v$_{kls}$ + w 1 $_{kls}$  ˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + w 1 klγ ( s $_{)}$) − f $^{(}$˜ v$_{kls}$ + w 2 $_{kls}$  ˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + w 2 klγ ( s $_{)}$) ∣ ds + ∫ t 0 e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl ij ∣ f $^{(}$˜ v$_{kls}$ + w 1 $_{kls}$  ˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + w 1 klγ ( s $_{)}$) ∣ ∣ w 1 $_{ij}$( s ) − w 2 $_{ij}$( s ) ∣ ds ≤ t 0 e − a$_{ij}$ ( t − s ) C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl $_{ij}$L ( H + K ( δ ) e − γ$_{0}$s/ 2 )( ∥ w 1 kls − w 2 kls ∥ 0 + ∥ w 1 klγ ( s ) − w 2 klγ ( s ) ∥ 0 ) ds + ∫ t 0 e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl $_{ij}$M ∣ w 1 $_{ij}$( s ) − w 2 $_{ij}$( s ) ∣ ds ≤ ( M + 2 LH ) sup t ≥ 0 ∥ w $^{1}$( t ) − w $^{2}$( t ) ∥ ∑ C$_{kl}$ ∈ N$_{r}$ ( i j $_{)}$C kl ij a$_{ij}$ ( 1 − e − a$_{ij}$t ) +4 L K ( δ ) sup t ≥ 0 ∥ w $^{1}$( t ) − w $^{2}$( t ) ∥ ∑ C$_{kl}$ ∈ N$_{r}$ ( i j $_{)}$C kl ij 2 a$_{ij}$ − γ$_{0}$ ( e − γ$_{0}$t/ 2 − e − a$_{ij}$t . Therefore  we have that sup t ≥ 0 || ˜ Π w $^{1}$( t ) − ˜ Π w $^{2}$( t ) || ≤ α$_{2}$ sup t ≥ 0 || w $^{1}$( t ) − w $^{2}$( t ) || . Since α$_{2}$ < 1   one can conclude by using a contraction mapping argument that there exists a unique fixed point ˜ w ( t ) = { ˜ w$_{ij}$ ( t ) } of the operator ˜ Π : Ψ$_{δ}$ → Ψ$_{δ}$   which is a solution of (4.10). To complete the proof  we need to show that there does not exist a solution of (4.10) with σ = 0 different from ˜ w ( t ) . Suppose that θ$_{p}$ ≤ 0 < θ$_{p}$$_{+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}$$_{+1}$ ] . Utilizing induction one can easily prove the uniqueness for all t ≥ 0 . □ Remark 4.1 In the proof of Theorem 4.1  we make use of the contraction mapping principle to prove the exponential stability. In the literature  Lyapunov-Krasovskii functionals  LMI technique  free weighting matrix method and differential inequality technique were used to investigate the exponential stability in neural networks [12  54  57]. They may also be considered in the future to prove the exponential stability in networks of the form (2.3). | z$_{ij}$ ( t ) | ≤ ∫ t 0 e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl ij $^{|}$˜ v$_{ij}$ ( s ) + ˜ w$_{ij}$ ( s ) | × ∣ f $^{(}$˜ v$_{kls}$ + w$_{kls}$$^{ }$˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + w$_{klγ}$$_{(}$$_{s}$$_{)}$ ) − f $^{(}$˜ v$_{kls}$ + w$_{kls}$$^{ }$˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + ˜ w$_{klγ}$$_{(}$$_{s}$$_{)}$ ) ∣ ds + ∫ t 0 e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl ij ∣ f $^{(}$˜ v$_{kls}$ + w$_{kls}$$^{ }$˜ v$_{klγ}$$_{(}$$_{s}$$_{)}$ + w$_{klγ}$$_{(}$$_{s}$$_{)}$ ) ∣ | z$_{ij}$ ( s ) | ds ≤ t 0 e − a$_{ij}$ ( t − s ) C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl $_{ij}$L ( H + K ( δ )) ( ‖ z$_{kls}$ ‖ 0 + ∥ z$_{klγ}$$_{(}$$_{s}$$_{)}$ ∥ 0 ) ds + ∫ t 0 e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl $_{ij}$M | z$_{ij}$ ( s ) | ds ≤ ¯ θ ¯ m [ M + 2 L ( H + K ( δ ))] ∑ C$_{kl}$ ∈ N$_{r}$ ( i j ) C kl $_{ij}$. 12