Exponential Convergence for Cellular Neural Networks with Time-Varying Delays in the Leakage Terms

and Applied Analysis 3 H2 for all t > 0 and i, j ∈ {1, 2, . . . , n}, there exist constants η > 0, λ > 0 and ξi > 0 such that ∫∞ 0 ∣ ∣Kij u ∣ ∣edu < ∞, −η > − [ ci t − λe−ληi t − ηi t ci t ( λ c i e λη i )] ei t ξi n ∑ j 1 L̃j (∣ ∣aij t ∣ ∣eij t a ijηi t ci t e ληi t e ij ) ξj n ∑ j 1 Lj ∫∞ 0 ∣ ∣Kij u ∣ ∣edu (∣ ∣bij t ∣ ∣ b ijηi t ci t e ληi t ) ξj ; 1.5 H3 Ii t O e−λt t → ±∞ , i 1, 2, . . . , n. The initial conditions associated with system 1.1 are of the form xi s φi s , s ∈ −∞, 0 , i 1, 2, . . . , n, 1.6 where φi · denotes real-valued-bounded continuous function defined on −∞, 0 . 2. Main Results Theorem 2.1. Let H1 , H2 , and H3 hold. Then, for every solution Z t x1 t , x2 t , . . . , xn t T of system 1.1 with any initial value φ φ1 t , φ2 t , . . . , φn t T , there exists a positive constant K such that |xi t | ≤ Kξie−λt ∀t > 0, i 1, 2, . . . , n. 2.1 Proof. Let Z t x1 t , x2 t , . . . , xn t T be a solution of system 1.1 with any initial value φ φ1 t , φ2 t , . . . , φn t T , and let Xi t exi t , i 1, 2, . . . , n. 2.2 4 Abstract and Applied Analysis In view of 1.1 , we have X′ i t λXi t e λt ⎡ ⎣−ci t xi ( t − ηi t ) n ∑ j 1 aij t fj ( xj ( t − τij t )) n ∑ j 1 bij t ∫∞ 0 Kij u gj ( xj t − u ) du Ii t ⎤ ⎦ λXi t − ci t ei t Xi ( t − ηi t ) e ⎡ ⎣ n ∑ j 1 aij t fj ( e−λ t−τij t Xj ( t − τij t )) n ∑ j 1 bij t ∫∞ 0 Kij u gj ( e−λ t−u Xj t − u ) du Ii t ⎤ ⎦, i 1, 2, . . . , n. 2.3 Let M max i 1,2,..., n sup s≤0 { e ∣φi s ∣ } . 2.4 From 1.3 , H2 , and H3 , we can choose a positive constant K such that Kξi > M, η > [ ηi t ci t ei t 1 ] supt∈R ∣Ii t e ∣ K , ∀t > 0, i 1, 2, . . . , n. 2.5 Then, it is easy to see that |Xi t | ≤ M < Kξi ∀t ≤ 0, i 1, 2, . . . , n. 2.6 We now claim that |Xi t | < Kξi ∀t > 0, i 1, 2, . . . , n. 2.7 If this is not valid, then, one of the following two cases must occur: 1 there exist i ∈ {1, 2, . . . , n} and t∗ > 0 such that Xi t∗ Kξi, ∣Xj t ∣∣ < Kξj ∀t < t∗, j 1, 2, . . . , n, 2.8 2 there exist i ∈ {1, 2, . . . , n} and t∗∗ > 0 such that Xi t∗∗ −Kξi, ∣Xj t ∣∣ < Kξj ∀t < t∗∗, j 1, 2, . . . , n. 2.9 Now, we consider two cases. Abstract and Applied Analysis 5 Case i. If 2.8 holds. Then, from 2.3 , 2.5 , and H1 − H3 , we have 0 ≤ X′ i t∗ λXi t∗ − ci t∗ ei t∗ Xi ( t∗ − ηi t∗ ) e ∗ ⎡ ⎣ n ∑ j 1 aij t∗ fj ( e−λ t ∗−τij t∗ Xj ( t∗ − τij t∗ ))and Applied Analysis 5 Case i. If 2.8 holds. Then, from 2.3 , 2.5 , and H1 − H3 , we have 0 ≤ X′ i t∗ λXi t∗ − ci t∗ ei t∗ Xi ( t∗ − ηi t∗ ) e ∗ ⎡ ⎣ n ∑ j 1 aij t∗ fj ( e−λ t ∗−τij t∗ Xj ( t∗ − τij t∗ )) n ∑ j 1 bij t∗ ∫∞ 0 Kij u gj ( e−λ t ∗−u Xj t∗ − u ) du Ii t∗ ⎤ ⎦ λXi t∗ − ci t∗ ei t∗ Xi t∗ ci t∗ ei t∗ [ Xi t∗ −Xi ( t∗ − ηi t∗ )] e ∗ ⎡ ⎣ n ∑ j 1 aij t∗ fj ( e−λ t ∗−τij t∗ Xj ( t∗ − τij t∗ )) n ∑ j 1 bij t∗ ∫∞ 0 Kij u gj ( e−λ t ∗−u Xj t∗ − u ) du Ii t∗ ⎤ ⎦ − [ ci t∗ ei t ∗ − λ ] Xi t∗ ci t∗ ei t ∗ ∫ t∗ t∗−ηi t∗ X′ i s ds e ∗ ⎡ ⎣ n ∑ j 1 aij t∗ fj ( e−λ t ∗−τij t∗ Xj ( t∗ − τij t∗ )) n ∑ j 1 bij t∗ ∫∞ 0 Kij u gj ( e−λ t ∗−u Xj t∗ − u ) du Ii t∗ ⎤ ⎦ − [ ci t∗ ei t ∗ − λ ] Xi t∗ ci t∗ ei t ∗ ∫ t∗ t∗−ηi t∗ ⎡ ⎣λXi s − ci s ei s Xi ( s − ηi s ) e ⎛ ⎝ n ∑ j 1 aij s fj ( e−λ s−τij s Xj ( s − τij s )) n ∑ j 1 bij s ∫∞ 0 Kij u gj ( e−λ s−u Xj s − u ) du Ii s ⎞ ⎠ ⎤ ⎦ds e ∗ ⎡ ⎣ n ∑ j 1 aij t∗ fj ( e−λ t ∗−τij t∗ Xj ( t∗ − τij t∗ )) n ∑ j 1 bij t∗ ∫∞ 0 Kij u gj ( e−λ t ∗−u Xj t∗ − u ) du Ii t∗ ⎤ ⎦ 6 Abstract and Applied Analysis ≤ − [ ci t∗ ei t ∗ − λ − ηi t∗ ci t∗ ei t∗ ( λ c i e λη i )] ξiK n ∑ j 1 L̃j (∣ ∣aij t∗ ∣ ∣eij t ∗ a ijηi t ∗ ci t∗ ei t ∗ e ij ) ξjK n ∑ j 1 Lj ∫∞ 0 ∣ ∣Kij u ∣ ∣edu (∣ ∣bij t∗ ∣ ∣ b ijηi t ∗ ci t∗ ei t ∗ ) ξjK [ ηi t∗ ci t∗ ei t ∗ 1 ] supt∈R ∣ ∣ ∣Ii t e ∣ ∣ ∣ { − [ ci t∗ − λe−ληi t∗ − ηi t∗ ci t∗ ( λ c i e λη i )] ei t ∗ ξi n ∑ j 1 L̃j (∣ ∣aij t∗ ∣ ∣eij t ∗ a ijηi t ∗ ci t∗ ei t ∗ e ij ) ξj n ∑ j 1 Lj ∫∞ 0 ∣Kij u ∣eλudu (∣bij t∗ ∣ b ijηi t ∗ ci t∗ ei t ∗ ) ξj [ ηi t∗ ci t∗ ei t ∗ 1 ] supt∈R ∣Ii t e ∣ K } K < { −η [ ηi t∗ ci t∗ ei t ∗ 1 ] supt∈R ∣Ii t e ∣ K } K