Exponential stability criteria for fuzzy bidirectional associative memory Cohen Grossberg neural networks with mixed delays and impulses He and Chu Advances in Difference Equations (2017) 2017 61 DOI[.]
He and Chu Advances in Difference Equations (2017) 2017:61 DOI 10.1186/s13662-017-1082-9 RESEARCH Open Access Exponential stability criteria for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses Weina He and Longxian Chu* * Correspondence: chulongxian_pdsu@126.com Software College, Pingdingshan University, Pingdingshan, 467000, PR China Abstract This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses By constructing an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model The results in this paper extend and complement the previous publications An example is given to illustrate the effectiveness of our obtained results MSC: 34C20; 34K13; 92B20 Keywords: fuzzy BAM Cohen-Grossberg neural networks; exponential stability; mixed delays; periodic solution; impulse Introduction In recent years, considerable attention has been paid to bidirectional associative memory (BAM) Cohen-Grossberg neural networks [] due to their potential applications in various fields such as neural biology, pattern recognition, classification of patterns, parallel computation and so on [–] In real life, numerous application examples appear, for example, emerging parallel/distributed architectures were explored for the digital VLSI implementation of adaptive bidirectional associative memory (BAM) [], Teddy and Ng [] applied a novel local learning model of the pseudo self-evolving cerebellar model articulation controller (PSECMAC) associative memory network to produce accurate forecasts of ATM cash demands Chang et al [] proposed a maximum-likelihood-criterion based on BAM networks to evaluate the similarity between a template and a matching region Sudo et al [] proposed a novel associative memory that operated in noisy environments and performed well in online incremental learning applying self-organizing incremental neural networks On the one hand, the existence and stability of the equilibrium point of BAM Cohen-Grossberg neural networks plays an important role in practical application On the other hand, time delay is inevitable due to the finite switching speed of amplifiers in the electronic implementation of analog neural networks, moreover, time delays may have important effect on the stability of neural networks and lead to periodic oscillation, © The Author(s) 2017 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 bifurcation, chaos and so on [, , ] Thus many interesting stability results on BAM Cohen-Grossberg neural networks with delays have been available [–] As is well known, numerous dynamical systems of electronic networks, biological neural networks, and engineering fields often undergo abrupt change at certain moments due to instantaneous perturbations which leads to impulsive effects [, , , –] Many scholars [, ] think that uncertainty or vagueness often appear in mathematical modeling of real world problems, thus it is necessary to take vagueness into consideration Fuzzy neural networks (FNNs) pay an important role in image processing and pattern recognition [] and some results have been reported on stability and periodicity of FNNs [, – ] Here we would like to point out that most neural networks involve negative feedback terms and not possess amplification functions or behaved functions The model (.) of this paper has amplifications function and behaved functions which differ from most neural networks with negative feedback term Up to now, there are rare papers that consider exponential stability of this kind of fuzzy bidirectional associative memory CohenGrossberg neural networks with mixed delays and impulses Inspired by the discussion above, in this paper, we are to consider the following fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses, ⎧ x˙ i (t) = ιi (xi (t))[–ai (t, xi (t)) + m ⎪ j= cji (t)fj (yj (t – τ (t))) ⎪ ⎪ t m m ⎪ ⎪ + j= αji (t) –∞ Kji (t – s)fj (yj (s)) ds + m ⎪ j= Tji uj + j= Hji uj ⎪ ⎪ t m ⎪ ⎪ + β (t) K (t – s)f (y (s)) ds + I (t)], t = t , i ⎪ j j i k ∈ , j= ji –∞ ji ⎪ ⎪ ⎨ x (t ) = x (t ) – x (t – ) = –γ x (t – ) + m e (t – )E (y (t – – τ )), k ∈ Z , i k i k i k ik i k j j k + j= ij k n ⎪ y˙ j (t) = ϑj (yj (t))[–bj (t, yj (t)) + i= dij (t)gi (xi (t – τ (t))) ⎪ ⎪ t ⎪ ⎪ ⎪ + ni= pij (t) –∞ Nij (t – s)gi (xi (s)) ds + ni= Sij ui + ni= Lij ui ⎪ ⎪ t ⎪ ⎪ ⎪ + ni= qij (t) –∞ Nij (t – s)gi (xi (s)) ds + Jj (t)], t = tk , j ∈ , ⎪ ⎪ ⎩ yj (tk ) = yj (tk ) – yj (tk– ) = –δjk yi (tk– ) + ni= hji (tk– )Hi (xi (tk– – τ )), k ∈ Z+ , (.) with initial conditions xi (s) = φi (s), s ∈ (–∞, ], i ∈ , yj (s) = φi (s), s ∈ (–∞, ], j ∈ , (.) where n and m correspond to the number of neurons in X-layer and Y -layer, respectively xi (t) and yj (t) are the activations of the ith neuron and the jth neurons, respectively ιi (·) and ϑj (·) are the abstract amplification functions, (t, ·) and bj (t, ·) stand for the rate functions with which the ith neuron and jth neuron will reset its potential to the resting state in isolation when disconnected from the network and external inputs; αji (t), βji (t), Tji and Hji are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in X-layer, respectively; pij (t), qij (t), Sij and Lij are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in Y -layer, respectively; and denote the fuzzy AND and fuzzy OR operation, respectively; uj , ui denote external input of the ith neurons in X-layer and external input of the jth neurons in Y -layer, respectively; Ii (t) and Jj (t) are external bias of X-layer and Y layer, respectively, fj (·) and gi (·) are signal transmission functions, Kji (t) and Nij (t) are delay He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 kernels, = {, , , n}, = {, , , m}, Z+ denotes the set of positive integral numbers, the impulse times tk satisfy = t < t < t < · · · < tk < · · · , limk→∞ tk = ∞, φi (·), φj (·) ∈ C, where C denotes real-valued continuous functions defined on (–∞, ], τ (t) is the transmission delay such that ≤ τ (t) ≤ τ , τ is a positive constant, eij (tk– ) represents impulsive perturbations of the ith unit at time tk , hji (tk– ) represents impulsive perturbations of the jth unit at time tk , Ej (yj (tk– )) represents impulsive perturbations of the jth unit at time tk and yj (tk– ) denotes impulsive perturbations of the jth unit at time tk caused by the transmission delays, Hi (xi (tk– )) represents impulsive perturbations of the ith unit at time tk and xi (tk– ) denotes impulsive perturbations of the ith unit at time tk which caused by the transmission delays For details, see [–] The main purpose of this paper is to investigate the existence and global exponential stability of a periodic solution of fuzzy BAM Cohen-Grossberg neural networks with mixed delays and impulses By constructing a suitable Lyapunov function and a new differential inequality, we establish some sufficient conditions to ensure the existence and global exponential stability of a periodic solution of the model (.) The results obtained in this paper extend and complement the previous studies in [, ] Two examples are given to illustrate the effectiveness of our theoretical findings To the best of our knowledge, there are very few papers that deal with this aspect Therefore we think that the study of the fuzzy BAM Cohen-Grossberg neural networks with mixed delays and impulses has important theoretical and practical value Here we shall mention that since the existence of amplifications function and behaved functions in model (.), thus there are some difficulties in dealing with the exponential stability We will apply some inequality techniques, meanwhile, the construction of Lyapunov function is a key issue The remaining part of this paper is organized as follows In Section , the necessary definitions and lemmas are introduced In Section , we present some new sufficient conditions to ensure the existence and global exponential stability of a periodic solution of model (.) In Section , an illustrative example is given to show the effectiveness of the proposed method A brief conclusion is drawn in Section Preliminaries Let R denote the set of real number, Rn the n-dimensional real space equipped with the Euclidean norm | · |, R+ the set of positive numbers Denote PC(R, R+ ) = {φ : R → Rn : φ(t) is continuous for t = tk , φ(tk+ ), φ(tk– ) ∈ Rn and φ(tk– ) = φ(tk )} Throughout this paper, we make the following assumptions: (H) For i ∈ , j ∈ , cji (t), αji (t), βji (t), eij (t), dij (t), pij (t), qij (t), hji (t), τ (t), Ii (t) and Jj (t) are all continuously periodic functions defined on t ∈ [, ∞) with common period ω > g f (H) For i ∈ , j ∈ , there exist positive constants Lj , LEj , Li and LH j such that fj (u) – fj (v) ≤ Lf |u – v|, j gi (u) – gi (v) ≤ Lg |u – v|, i Ej (u) – Ej (v) ≤ LE |u – v|, j Hi (u) – Hi (v) ≤ LH |u – v| j for all u, v ∈ R (H) For i ∈ , j ∈ , ιi (·) and ϑj (·) are continuous and satisfy ≤ ιi ≤ ιi (·) ≤ ιi , ≤ ϑ j ≤ ϑj (·) ≤ ϑ j , where ιi , ιi , ϑ j , ϑ j are some positive constants He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 (H) For i ∈ , j ∈ , there exist continuous positive ω-periodic functions i (t) and σj (t) such that bj (t, u) – bj (t, v) ≥ σj (t) u–v (t, u) – (t, v) ≥ i (t), u–v for all u, v ∈ R (H) For i ∈ , j ∈ , the delay kernels Kij (·), Nji (·) ∈ C(R+ , R+ ) are piecewise continuous ˜ and Nji (s) ≤ K(s) ˜ for all s ∈ R+ , where K(s) ˜ ∈ C(R+ , R+ ) and satisfy Kij (s) ≤ K(s) ∞ μs ˜ and integrable, satisfying K(s)e ds < ∞, in which the constant μ denotes some positive number (H) For i ∈ , j ∈ ω > , there exists q ∈ Z+ such that tk + ω = tk+q and γik = γi(k+q) , δjk = δj(k+q) , k ∈ Z+ (H) For i ∈ , j ∈ , c∗ji = maxt∈[,ω] |cji (t)|, αji∗ = maxt∈[,ω] |αji (t)|, βji∗ = maxt∈[,ω] |βji (t)|, e∗ij = maxt∈[,ω] |eij (t)|, dij∗ = maxt∈[,ω] |dij (t)|, p∗ij = maxt∈[,ω] |pij (t)|, qij∗ = maxt∈[,ω] |qij (t)|, h∗ji = maxt∈[,ω] |hji (t)|, i = mint∈[,ω] | i (t)|, σi = mint∈[,ω] |σj (t)| In this paper, we use the following norm of Rn+m : u = n i= |xi | + m |yj |, φ = sup s∈(–∞,] j= n m φi (s) + φj (s) i= j= for u = (x , x , , xn , y , y , , ym )T ∈ Rn+m , φ = (φ , φ , , φn , φ , φ , , φm )T ∈ Cn+m Lemma . ([]) Let x and y be two states of system (.) Then n n n αij (t) gj (x) – gj (y) α (t)g (x) – α (t)g (y) ≤ ij j ij j j= j= j= and n n n βij (t) gj (x) – gj (y) βij (t)gj (y) ≤ βij (t)gj (x) – j= j= j= Lemma . ([]) Let p, q, r and τ denote nonnegative constants and f ∈ PC(R, R+ ) satisfies the scalar impulsive differential inequality σ D+ f (t) ≤ –pf (t) + q supt–τ ≤s≤t f (s) + r k(s)f (t – s) ds, f (tk ) ≤ ak f (tk– ) + bk f (tk– – τ ), k ∈ Z+ , t = tk , t ≥ t , (.) σ where < σ ≤ +∞, ak , bk ∈ R, k(·) ∈ PC([, σ ], R+ ) satisfies k(s)eη s ds < ∞ for some positive constant η > in this case when σ = +∞ Moreover, when σ = +∞, the interσ val [t – σ , t] is understood to be replaced by (–∞, t] Assume that (i) p > q + r k(s) ds (ii) There exist constant M > , η > such that n k= max , ak + bk eλτ ≤ Meη(tn –t ) , n ∈ Z+ , He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 where λ ∈ (, η ) satisfies σ λ < p – qeλτ – r k(s)eλs ds Then f (t) ≤ Mf (t )e–(λ–η)(t–t ) , t ≥ t , where f (t ) = supt –max{σ ,τ } f (s) Global exponential stability of the periodic solution In this section, we will discuss the global exponential stability of the periodic solution for (.) Theorem . Assume that (H)-(H) hold, then there exists a unique ω-periodic solution of system (.) which is globally exponentially stable if the following conditions are fulfilled (H) min{ i , σj } mini∈,j∈ {ιi , ϑ j } maxi∈,j∈ {ιi , ϑ j } n m ∗ f ∗ g > max max cji Lj , max dij Li j∈ i= i∈ j= n n m m ∗ f ∗ f ∗ g ∗ g max αji Lj , max βji Lj , max pij Li , max qij Li + max i= ∞ × j∈ j∈ i= j= i∈ j= i∈ ˜ ds K(s) (H) There exist constants M ≥ , λ ∈ (, λ ) and η ∈ (, λ) such that n ηtn for all n ∈ Z+ holds and l= max{, χl } ≤ Me λ< min{ i , σj } mini∈,j∈ {ιi , ϑ j } n m ∗ f ∗ g – max max cji Lj , max dij Li eλτ i∈ j∈ maxi∈,j∈ {ιi , ϑ j } i= j= n n m m ∗ f ∗ f ∗ g ∗ g – max max αji Lj , max βji Lj , max pij Li , max qij Li i= ∞ × j∈ i= j∈ j= i∈ j= i∈ ˜ ds, K(s) where n m maxi∈,j∈ {ιi , ϑ j } ∗ E ∗ H χl = max |–γil |, |–δjl |+max max eij Lj , max hji Li eλτ i∈ j∈ mini∈,j∈ {ιi , ϑ j } i∈,j∈ i= j= Proof Assume that u(t) = (x (t, φ ), x (t, φ ), , xn (t, φ ), y (t, φ ), y (t, φ ), , ym (t, φ ))T is an arbitrary solution of system (.) through (t, φ , φ ), where φ = (φ , φ , , φn )T , He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 φ = (φ , φ , , φm )T Define t ≤ , i ∈ , xi (t + ω, φ ) = ϕi , t ≤ , j ∈ , yj (t + ω, φ ) = ϕj , then ϕ = (ϕ , ϕ , , ϕn )T ∈ Cn , ϕ = (ϕ , ϕ , , ϕm )T ∈ Cm Now we construct the following Lyapunov function: V (t) = n i= + xi (t+ω) xi (t) m ds sgn xi (t + ω) – xi (t) ιi (s) yj (t+ω) yj (t) j= ds sgn yj (t + ω) – yj (t) ϑj (s) (.) It is easy to see that i∈,j∈ n m xi (t + ω) – xi (t) + yj (t + ω) – yj (t) , ιi ϑ j i= j= n m xi (t + ω) – xi (t) + yj (t + ω) – yj (t) , ≤ V (t) ≤ max i∈,j∈ ιi ϑ j i= j= (.) When t = tk , calculating the derivative of D+ V (t) along the solution of (.), we have D+ V (t) ≤ n + D xi (t + ω) ιi (xi (t + ω)) i= + m + D yj (t + ω) ϑj (yj (t + ω)) j= = – n D+ xi (t) sgn xi (t + ω) – xi (t) ιi (xi (t)) – D+ yj (t) sgn yj (t + ω) – yj (t) ϑj (yj (t)) m cji (t)fj yj t + ω – τ (t + ω) –ai t, xi (t + ω) + t, xi (t) + i= – j= m cji (t)fj yj t – τ (t) + m j= – m m αji (t) Kji (t – s)fj yj (s) ds + t βji (t) m i= t+ω βji (t) Kji (t + ω – s)fj yj (s) ds –∞ j= n –bj t, yj (t + ω) + bj t, yj (t) + dij (t)gi xi t + ω – τ (t + ω) i= n n –∞ dij (t)gi xi t – τ (t) + i= – m Kji (t – s)fj yj (s) ds sgn xi (t + ω) – xi (t) j= – Kji (t + ω – s)fj yj (s) ds –∞ –∞ j= + t t+ω αji (t) j= j= – n i= t pij (t) –∞ t+ω pij (t) Nij (t + ω – s)gi xi (s) ds –∞ n Nij (t – s)gi xi (s) ds + qij (t) i= t+ω –∞ Nij (t + ω – s)gi xi (s) ds He and Chu Advances in Difference Equations (2017) 2017:61 – n Nij (t + ω – s)gi xi (s) ds sgn yj (t + ω) – yj (t) –∞ – i i∈ + n n m f xi (t + ω) – xi (t) + c∗ji Lj yj t + ω – τ (t) – yj t – τ (t) i= n m i= j= αji∗ Lj f n m f βji∗ Lj j∈ + ∞ ˜ yj (t + ω – s) – yj (t – s) ds K(s) m n m g yj (t + ω) – yj (t) + dij∗ Li xi t + ω – τ (t) – xi t – τ (t) j= m n j= i= p∗ij Li g m n qij∗ Li j= i= ∞ ˜ xi (t + ω – s) – xi (t – s) ds K(s) ∞ ˜ xi (t + ω – s) – xi (t – s) ds K(s) j= i= + ˜ yj (t + ω – s) – yj (t – s) ds K(s) i= j= – σj ∞ i= j= + t+ω qij (t) i= ≤ Page of 16 g n ≤ – i , σj xi (t + ω) – xi (t) + i= + n + n f j∈ + n f + m max βji∗ Lj f max dij∗ Li g i∈ + j= + m max qij∗ Li g i∈ ∞ j= ∞ ˜ K(s) n xi (t + ω – s) – xi (t – s) ds i= ∞ ˜ K(s) n xi (t + ω – s) – xi (t – s) ds i= xi (t + ω) – xi (t) + i= n i= m yj (t + ω – s) – yj (t – s) ds n xi t + ω – τ (t) – xi t – τ (t) ≤ – i , σj + max ˜ K(s) n ˜ yj (t + ω – s) – yj (t – s) ds K(s) i= g max p∗ij Li i∈ j= ∞ j∈ j= m j∈ i= yj (t + ω) – yj (t) j= j= max αji∗ Lj i= m yj t + ω – τ (t) – yj t – τ (t) max c∗ji Lj i= m f max c∗ji Lj , j∈ m j= m yj (t + ω) – yj (t) j= g max dij∗ Li i∈ n m xi t + ω – τ (t) – xi t – τ (t) + yj t + ω – τ (t) – yj t – τ (t) × i= j= He and Chu Advances in Difference Equations (2017) 2017:61 + max n i= Page of 16 f max αji∗ Lj , j∈ n i= f max βji∗ Lj , j∈ m j= g max p∗ij Li , i∈ m j= g max qij∗ Li i∈ n ∞ m ˜ xi (t + ω – s) – xi (t – s) + yj (t + ω – s) – yj (t – s) ds × K(s) i= j= (.) In view of (.), it follows from (.) that D+ V (t) ≤ – i , σj {ιi , ϑ j }V (t) + max + max i∈,j∈ n i= n i= ∞ × f max c∗ji Lj , j∈ m f max αji∗ Lj , j∈ j= g max dij∗ Li i∈ n i= f max βji∗ Lj , j∈ max {ιi , ϑ j }V t – τ (t) i∈,j∈ m j= g max p∗ij Li , i∈ m j= g max qij∗ Li i∈ ˜ K(s)V (t – s) ds (.) When t = tk , in view of (H), (H), and (.), we get , V (tk ) ≤ max i∈,j∈ ιi ϑ j n m xi (tk + ω) – xi (tk ) + yj (tk + ω) – yj (tk ) i= j= n m xi (tk+q ) – xi (tk ) + yj (tk+q ) – yj (tk ) , = max i∈,j∈ ιi ϑ j i= j= , ≤ max i∈,j∈ ιi ϑ j + n m n – – xi tk– | – γik | xi tk+q i= – e∗ij LEj yj tk+q – τ – yj tk– – τ i= j= + m m n – – – – yj tk– + | – δjk | yj tk+q h∗ji LH i xi tk+q – τ – xi tk – τ j= j= i= n – xi t + ω – xi t – , max | – γik | ≤ max k k i∈ i∈,j∈ ιi ϑ j i= + n max e∗ij LEj j∈ i= j= + max | – δjk | j∈ + n i= ≤ max i∈,j∈ m – yj t + ω – τ – yj t – – τ k k m – yj t + ω – yj t – k k j= max h∗ji LH i i∈ , ιi ϑ j m – xi t + ω – τ – xi t – – τ k k j= max | – γik |, | – δjk | i∈,j∈ He and Chu Advances in Difference Equations (2017) 2017:61 Page of 16 n m – – xi t + ω – xi t – + yj t + ω – yj t – × k k k k i= j= n n ∗ E ∗ H + max max eij Lj , max hji Li i= j∈ i= i∈ n m – – xi t + ω – xi t – + yj t + ω – yj t – × k k k k i= ≤ j= maxi∈,j∈ {ιi , ϑ j } max | – γik |, | – δjk | V tk– mini∈,j∈ {ιi , ϑ j } i∈,j∈ n n maxi∈,j∈ {ιi , ϑ j } ∗ E ∗ H + max eij Lj , max hji Li V tk– – τ max i∈ j∈ mini∈,j∈ {ιi , ϑ j } i= i= (.) In view of (.)-(.) and (H)-(H), using Lemma ., we have t ≥ , V (t) ≤ MV ()e–(λ–η)t , (.) where V () = sup–∞≤s≤ V (s) It follows from (.) that n m xi (t + ω) – xi (t) + yj (t + ω) – yj (t) ≤ ν φ – ϕ e–(λ–η)t , i= t ≥ , (.) j= where ν=M maxi∈,j∈ {ιi , ϑ j } ≥ , mini∈,j∈ {ιi , ϑ j } λ satisfies the condition (H) Notice that xi (t + kω) = xi (t) + k xi (t + lω) – xi t + (l – )ω , i ∈ , yj (t + lω) – yj t + (l – )ω , j ∈ l= yj (t + kω) = yj (t) + k l= In view of (.), we have ∞ xi (t + lω) – xi t + (l – )ω l= = lim k→∞ k xi (t + lω) – xi t + (l – )ω l= ≤ ν φ – ϕ lim k→∞ k e–(λ–η)(t+(l–)ω) l= ≤ ν φ – ϕ e–(λ–η)t ∞ l= e–(λ–η)(l–)ω < ∞, as k → ∞, (.) He and Chu Advances in Difference Equations (2017) 2017:61 Page 10 of 16 for any given t ≥ By (.), we know that limk→∞ xi (t + kω) exists Similarly, we know that limk→∞ yj (t + kω) also exists Set (x∗ (t), y∗ (t))T = (x∗ (t), x∗ (t), , x∗n (t), y∗ (t), y∗ (t), , y∗m (t))T , where x∗i = limk→∞ xi (t + kω), y∗j = limk→∞ yj (t + kω), then (x∗ (t), y∗ (t))T is a periodic function with period ω for system (.) Assume that system (.) has another ω-periodic solution (x∗∗ (t), y∗∗ (t))T as follows: x∗∗ (t, ψ ), y∗∗ (t, ψ ) T ∗∗ ∗∗ = x∗∗ (t, ψ ), x (t, ψ ), , xn (t, ψ ), ∗∗ ∗∗ y∗∗ (t, ψ ), y (t, ψ ), , ym (t, ψ ) , T where ψ ∈ Cn , ψ ∈ Cm It follows from (.) that n m ∗ ∗ x (t) – x∗∗ (t) + y (t) – y∗∗ (t) i i j j i= = j= n m ∗ ∗ x (t + kω) – x∗∗ (t + kω) + y (t + kω) – y∗∗ (t + kω) i i j i= ≤ ν φ – ψ e(λ–η)(t+kω) , j j= t ≥ (.) ∗ ∗∗ Let k → ∞, then x∗i (t) = x∗∗ i (t), yj (t) = yj (t), t ≥ Thus we can conclude that system (.) has a unique ω-periodic solution which is globally exponentially stable The proof of Theorem . is complete Remark . Li [] investigated the existence and global exponential stability of a periodic solution for impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays, the model in [] is not concerned with fuzzy terms Bao [] discussed the existence and exponential stability of a periodic solution for BAM fuzzy CohenGrossberg neural networks with mixed delays, the model in [] is not concerned with impulsive effects Yang [] considered the periodic solution for fuzzy Cohen-Grossberg BAM neural networks with both time-varying and distributed delays and variable coefficients, the model in [] is not concerned with impulsive effect and distributed delays Balasubramaniam et al [] analyzed the global asymptotic stability of stochastic fuzzy cellular neural networks with multiple time-varying delays, the model in [] is not concerned with impulsive effect and distributed delays, Balasubramaniam and Vembarasan [] studied the robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses, the authors did not discuss the existence and global exponential stability of a periodic solution of neural networks and the model in [] is also not concerned with distributed delays In this paper, we study the exponential stability for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses All the obtained results in [, , , , ] cannot be applicable to model (.) to obtain the exponential stability of model (.) From this viewpoint, our results on the exponential stability for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses are essentially new and complement earlier works to some extent He and Chu Advances in Difference Equations (2017) 2017:61 Page 11 of 16 Examples In this section, we consider the following neural networks with mixed delays and impulses ⎧ ⎪ x˙ (t) = ι (x (t))[–a (t, x (t)) + j= cj (t)fj (yj (t – τ (t))) ⎪ ⎪ t ⎪ ⎪ ⎪ + j= αj (t) –∞ Kj (t – s)fj (yj (s)) ds + j= Tj uj + j= Hj uj ⎪ ⎪ ⎪ t ⎪ ⎪ + j= βj (t) –∞ Kj (t – s)fj (yj (s)) ds + I (t)], t = tk , ⎪ ⎪ ⎪ ⎪ ⎪ x˙ (t) = ι (x (t))[–a (t, x (t)) + j= cj (t)fj (yj (t – τ (t))) ⎪ ⎪ ⎪ t ⎪ ⎪ + j= αj (t) –∞ Kj (t – s)fj (yj (s)) ds + j= Tj uj + j= Hj uj ⎪ ⎪ t ⎪ ⎪ + j= βj (t) –∞ Kj (t – s)fj (yj (s)) ds + I (t)], t = tk , ⎪ ⎪ ⎪ ⎪ ⎪ x (tk ) = x (tk ) – x (tk– ) = –γk xi (tk– ) + j= ej (tk– )Ej (yj (tk– – τ )), k ∈ Z+ , ⎪ ⎪ ⎪ ⎨ x (tk ) = x (tk ) – x (tk– ) = –γk x (tk– ) + j= ej (tk– )Ej (yj (tk– – τ )), k ∈ Z+ , ⎪ y˙ (t) = ϑ (y (t))[–b (t, yj (t)) + i= di (t)gi (xi (t – τ (t))) ⎪ ⎪ ⎪ t ⎪ ⎪ + i= pi (t) –∞ Ni (t – s)gi (xi (s)) ds + i= Si ui + i= Li ui ⎪ ⎪ t ⎪ ⎪ ⎪ + i= qi (t) –∞ Ni (t – s)gi (xi (s)) ds + J (t)], t = tk , ⎪ ⎪ ⎪ ⎪ ⎪ y˙ (t) = ϑ (y (t))[–b (t, yj (t)) + i= di (t)gi (xi (t – τ (t))) ⎪ ⎪ t ⎪ ⎪ + i= pi (t) –∞ Ni (t – s)gi (xi (s)) ds + i= Si ui + i= Li ui ⎪ ⎪ ⎪ t ⎪ ⎪ + i= qi (t) –∞ Ni (t – s)gi (xi (s)) ds + J (t)], t = tk , ⎪ ⎪ ⎪ ⎪ ⎪ y (tk ) = y (tk ) – y (tk– ) = –δk y (tk– ) + i= hi (tk– )Hi (xi (tk– – τ )), k ∈ Z+ , ⎪ ⎪ ⎪ ⎩ y (t ) = y (t ) – y (t – ) = –δ y (t – ) + h (t – )H (x (t – – τ )), k ∈ Z , k k k k k i i k + i= i k where c (t) c (t) . + . sin t = . + . sin t c (t) c (t) . + . sin t d (t) d (t) = . + . sin t d (t) d (t) α (t) α (t) . + . cos t = . + . cos t α (t) α (t) . + . cos t p (t) p (t) = . + . cos t p (t) p (t) . + . sin t β (t) β (t) = . + . sin t β (t) β (t) q (t) q (t) . + . cos t = . + . cos t q (t) q (t) S T T = , T t S L H H = , H H L u u + sin t + cos t = , I J + sin t + cos t . + . cos t , . + . cos t . + . cos t , . + . cos t . + . sin t , . + . sin t . + . sin t , . + . sin t . + . cos t , . + . cos t . + . sin t , . + . sin t S = , S L = , L (.) He and Chu Advances in Difference Equations (2017) 2017:61 Page 12 of 16 e (t) e (t) tanh(.t) tanh(.t) = , tanh(.t) tanh(.t) e (t) e (t) h (t) h (t) tanh(.t) tanh(.t) γk (s) γk (s) – – = , = , tanh(.t) tanh(.t) – – h (t) h (t) δk (s) δk (s) ι (x (t)) . + . cos(x (t)) x (t) ϑ (y (t)) = , a (t, x (t)) b (t, y (t)) . + . sin(y (t)) y (t) . + . sin(x (t)) x (t) ι (x (t)) ϑ (y (t)) = , a (t, x (t)) b (t, y (t)) . + . cos(y (t)) y (t) K (s) K (s) N (s) N (s) e–s e–s e–s e–s = –s –s , = –s –s , e e e e K (s) K (s) N (s) N (s) E (y (tk– – τ )) E (y (tk– – τ )) tanh(.y (tk– – .)) tanh(.y (tk– – .)) = H (x (tk– – τ )) H (x (tk– – τ )) tanh(.x (tk– – .)) tanh(.x (tk– – .)) Let tk = .πk, τ (t) = .| sin t|, f (u) = |u + |, g (u) = |u – |, then we get ι = ., ι = ., ϑ = ., ϑ = ., ι = ., ι = ., ϑ = ., ϑ = ., c∗ = ., c∗ = ., c∗ = ., ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ c∗ = ., d = ., d = ., d = ., d = ., α = ., α = ., α = ., α = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ., p = ., p = ., p = ., p = ., β = ., β = ., β = ., β = ., q = ∗ ∗ ∗ ., q = ., q = ., q = ., h∗ = , h∗ = , h∗ = , h∗ = , τ = ., K(s) = e–s , g f H λ = ., ω = π , L = L = , LE = ., LE = ., LH = ., L = . It is easy to check that mini∈,j∈ { i , σj } mini∈,j∈ {ιi , ϑ j } maxi∈,j∈ {ιi , ϑ j } = ., n m g f max max c∗ji Lj , max dij∗ Li i= j∈ j= i∈ n n m m ∗ f ∗ f ∗ g ∗ g + max max αji Lj , max βji Lj , max pij Li , max qij Li i= j∈ i= j∈ j= i∈ j= i∈ ≈ . Choose λ = . < λ such that ∞ λ < . – e.λ – . e(λ–)s ds Then we obtain χl = maxi∈,j∈ {ιi , ϑ j } mini∈,j∈ {ιi , ϑ j } n m ∗ E ∗ H max eij Lj , max hji Li eλτ × max | – γil |, | – δjl | + max i∈,j∈ ≈ . i= j∈ j= i∈ ∞ ˜ ds K(s) He and Chu Advances in Difference Equations (2017) 2017:61 Page 13 of 16 Figure Numerical solutions of system (3.1): times series of x1 Figure Numerical solutions of system (3.1): times series of x2 Thus we can choose η = . < λ such that nl= max{, χl } = .n < .n ≈ eηtn for all n ∈ Z+ Then all the conditions of Theorem . hold Thus (.) has exactly one π periodic solution which is globally exponentially stable These results are illustrated in Figures , , , Conclusions In this article, we have analyzed the global exponential stability of fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses By constructing a suitable Lyapunov function and a new differential inequality, some sufficient criteria which ensure the existence and global exponential stability of a periodic solution of the model have been established The obtained conditions are easy to check in practice The results in this paper extend and complement some previous studies Finally, He and Chu Advances in Difference Equations (2017) 2017:61 Page 14 of 16 Figure Numerical solutions of system (3.1): times series of y1 Figure Numerical solutions of system (3.1): times series of y2 an example with their numerical simulations is carried out to illustrate the correctness To the best of our knowledge, there are only rare results on the exponential stability for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with proportional delays, which will be our future research direction Competing interests The authors declare that there is no conflict of interest regarding the publication of this paper Authors’ contributions The authors have equally made contributions All authors read and approved the final manuscript Acknowledgements The 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consider the following fuzzy bidirectional associative memory Cohen- Grossberg neural networks with mixed. .. neural networks with negative feedback term Up to now, there are rare papers that consider exponential stability of this kind of fuzzy bidirectional associative memory CohenGrossberg neural networks. .. of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in Y -layer, respectively; and denote the fuzzy AND and fuzzy