Pseudo Almost Periodic Solutions for HCNNs with Time-Varying Leakage Delays

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1 DOI 1.763/s Moroccan J. Pure and Appl. Anal.(MJPAA) Volume 1(1), 215, Pages ISSN: RESEARCH ARTICLE Pseudo Almost Perodc Solutons for HCNNs wth Tme-Varyng Leakage Delays Ceml Tunç Abstract. In ths paper, we consder a class of hgh-order cellular neural networks (HCNNs) model wth tme-varyng delays n the leakage terms. We gve some suffcent condtons whch guarantee the exponental stablty of pseudo almost perodc solutons for the model. The obtaned results complement wth some recent ones n the lteature.the technque of proof nvolves the exponental dchotomy theory and the fxed pont theorem. An llustratve example s gven wth an applcaton. 2 Mathematcs Subject Classfcaton. 34C25, 34D4. Key words and phrases. Hgh-order cellular neural network; pseudo almost perodc soluton; exponental stablty; tme-varyng delay; leakage term. 1. Introducton It s well known that retarded functonal dfferental equatons descrbe those systems or processes whose rate of change of state s determned by ther past and present states. These equatons are frequently encountered as mathematcal models of most dynamcal process n mechancs, control theory, physcs, chemstry, bology, medcne, economcs, atomc energy, nformaton theory, etc. For example, t follows from lterature that the hgh-order recurrent neural networks (HCNNs), whch nclude both the Cohen- Grossberg neural networks and the Hopfeld neural networks as specal cases, allow hgh-order nteractons between neurons, and therefore have stronger approxmaton property, faster convergence rate, greater storage capacty, and hgher fault tolerance than the tradtonal frst-order neural networks (see Dembo et al. [1]). Hence, n the past years, hgh-order neural networks have been successfully appled n many areas, Receved Aprl 8, Accepted June 28, 215. c The Author(s) 215.Ths artcle s publshed wth open access by Sd Mohamed Ben Abdallah Unversty Ths work was supported by.yuzuncu Yl Unversty, BAP-214-FEN-B Correspondng author. Tel.:(432) ; fax:(432) Department of Mathematcs, Faculty of Scences, Yüzüncü Yıl Unversty, 658, VanTurkey. 51.

2 52 CEMIL TUNÇ such as bologcal scence, pattern recognton and optmzaton (see Psalts et al.[2], Karayanns and Venetsanopoulos [3]). In partcular, some attenton has been pad to the convergence behavor for HRNNs wth delays n the leakage terms (see [4 6] and the references theren). Recently, Xu [7] and Zhang [8] consdered the exstence and exponental stablty of the ant-perodc solutons for the followng HCNNs wth tme-varyng delays n the leakage terms: x (t) = c (t)x (t η (t)) a j (t)f j (x j (t τ j (t))) b jl (t)g j (x j (t α jl (t)))g l (x l (t β jl (t))) d jl (t) σ jl (u)h j (x j (t u))du ν jl (u)h l (x l (t u))du I (t), = 1, 2,, n, (1.1) n whch n corresponds to the number of unts n a neural network, x (t) corresponds to the state vector of the th unt at the tme t, c (t) represents the rate wth whch the th unt wll reset ts potental to the restng state n solaton when dsconnected from the network and external nputs, a j (t), b jl (t) and d jl (t) are the frst and second order connecton weghts of the neural network, respectvely, η (t) corresponds to the tme-varyng leakage delays, α jl (t), β jl (t) and τ j (t) correspond to the transmsson delays, σ jl (u) and ν jl (u) correspond to the transmsson delay kernels, I (t) denotes the external nputs at tme t, f j, g j and h j are the actvaton functons of sgnal transmsson. On the other hand, the dynamcs of delayed neural networks s manly affected by the varaton of the envronment. As mentoned n [9, p87-9] and [1, p77-94], perodcally and almost perodcally varyng envronments are the fundamental bass of the theory of natural selecton. In contrast wth perodcal effects, almost perodc effects can be encountered more often, and pseudo almost perodc effects regulate many phenomena excellently. Hence, complex repettve phenomena can be consdered as almost perodc process and an ergodc component. Therefore, the study of the exstence and stablty of almost perodc solutons and pseudo almost perodc solutons for the frst order cellular neural networks (CNNs) models wth leakage delays takes great attenton (see [11-14] and the references theren). It should be noted that to the best of our knowledge from the lterature, there s no result on the exstence of pseudo almost perodc solutons of the HCNNs wth tmevaryng delays n the leakage terms. The am of ths work s to prove the exstence and global exponental stablty of the pseudo almost perodc solutons for HCNNs (1.1). Our approach s based on the exponental dchotomy theory and contracton mappng fxed pont theorem developed n [15]. The ntal condtons assocated wth system (1.1) are of the form x (s) = ϕ (s), x (s) = ϕ (s), s (, ], = 1, 2,, n, (1.2)

3 PSEUDO ALMOST PERIODIC SOLUTIONS 53 where ϕ ( ) and ϕ ( ) are real-valued bounded and contnuous functons defned on (, ]. For convenence, we denote by R n (R = R 1 ) the set of all ndmensonal real vectors (real numbers). Let J = {1, 2,, n} and {x } = (x 1, x 2,, x n ). For any {x } R n, we let x denote the absolute-value vector gven by x = { x }, and defne x = max J x. A matrx or vector A means that all entres of A are greater than or equal to zero. A > can be defned smlarly. For matrces or vectors A 1 and A 2, A 1 A 2 (resp. A 1 > A 2 ) means that A 1 A 2 (resp. A 1 A 2 > ). BC(R, R n ) denotes the set of bounded and contnues functons from R to R n. Note that (BC(R, R n ), ) s a Banach space, where denotes the sup norm f := sup t R f BC(R, R), we set f(t). For f = nf t R f(t), f = sup f(t). t R Defnton 1.1 (see [9, 1]). Let u(t) BC(R, R n ). u(t) s sad to be almost perodc on R f, for any ε >, the set T (u, ε) = {δ : u(tδ)u(t) < ε for all t R} s relatvely dense,.e., for any ε >, t s possble to fnd a real number l = l(ε) > wth the property that, for any nterval wth length l(ε), there exsts a number δ = δ(ε) n ths nterval such that u(t δ) u(t) < ε, for all t R. We denote by AP (R, R n ) the set of the almost perodc functons from R to R n. Precsely, defne the class of functons P AP (R, R n ) as follows: { f BC(R, R n 1 r ) lm f(t) dt = }. r 2r r A functon f BC(R, R n ) s called pseudo almost perodc f t can be expressed as f = h ϕ, where h AP (R, R n ) and ϕ P AP (R, R n ). The collecton of such functons wll be denoted by P AP (R, R n ). The functons h and ϕ n above defnton are respectvely called the almost perodc component and the ergodc perturbaton of the pseudo almost perodc functon f. Defnton 1.2. Let x (t) = (x 1(t), x 2(t),, x n(t)) T be the pseudo almost perodc soluton of system (1.1). If there exst constants α > and M > 1 such that for every soluton x(t) = (x 1 (t), x 2 (t),, x n (t)) T of system (1.1) wth any ntal value ϕ(t) = (ϕ 1 (t), ϕ 2 (t),, ϕ n (t)) T satsfyng (1.2), x(t)x (t) 1 = max {max{ x (t)x (t), x (t)x (t) }} M ϕx e αt, t >, =1, 2,, n where ϕ x = max{sup max ϕ (t) x (t), sup max t 1 n t 1 n ϕ (t) x (t) }, then x (t) s sad to be globally exponentally stable.

4 54 CEMIL TUNÇ 2. Prelmnary Lemmas In ths secton, we shall frst recall some basc defntons, lemmas whch are used n what follows. Throughout ths paper, t wll be assumed that c : R (, ) s an almost perodc functon, η, τ j, α jl, β jl : R [, ) and I, a j, b jl, d jl : R R are pseudo almost perodc on R, where, j, l J. We also make the followng assumptons whch wll be used later. (H 1 ) there exst nonnegatve constants L f j, Lg j, Lh j, M g j and M j h such that f j (u) f j (v) L f j u v, g j(u) g j (v) L g j u v, h j(u) h j (v) L h j u v, and g j (u) M g j, h j(u) M h j, where u, v R, j J. (H 2 ) For, j, l J, the delay kernels σ jl, ν jl : [, ) R are contnuous, σ jl (t) e κt and ν jl (t) e κt are ntegrable on [, ) for a certan postve constant κ. (H 3 ) For each J, there exst constants α > and ξ >, such that and c c η ξ 1 α, d jl a j Lf j ξ j ξ 1 σ jl (u) du b jl (M g j Lg l ξ l M g l Lg j ξ j) ν jl (u) du(m h j L h l ξ l M h l L h j ξ j ) c α c (1 α c ) < 1. Lemma 2.1 (see [11, Lemma 2.3 ]). Let B = {f f, f P AP (R, R n )} equpped wth the nduced norm defned by f B = max{ f, f } = max{sup t R f(t), sup f (t) }, t R then, B s a Banach space. Lemma 2.2 (see [11, Lemma 3.1 ]. Assume that assumptons (H 1 ) and (H 2 ) hold. Then, for ϕ j, ϕ l P AP (R, R), σ jl (u)h j (ξ j ϕ j (t u))du, Defnton 2.1 (see [9,1]). defned on R. The lnear system ν jl (u)h l (ξ l ϕ l (t u))du P AP (R, R),, j, l J. (2.1) Let x R n and Q(t) be an n n contnuous matrx x (t) = Q(t)x(t) (2.2)

5 PSEUDO ALMOST PERIODIC SOLUTIONS 55 s sad to admt an exponental dchotomy on R f there exst postve constants k, α, projecton P and the fundamental soluton matrx X(t) of (2.2) satsfyng X(t)P X 1 (s) ke α(ts) for t s, X(t)(I P )X 1 (s) ke α(st) for t s. Lemma 2.3 (see [9]). Assume that Q(t) s an almost perodc matrx functon and g(t) P AP (R, R n ). If the lnear system (2.2) admts an exponental dchotomy, then pseudo almost perodc system has a unque pseudo almost perodc soluton x(t), and x(t) = Lemma 2.4 (see [9,1]). M[c ] = Then the lnear system x (t) = Q(t)x(t) g(t) (2.3) X(t)P X 1 (s)g(s)ds lm T t X(t)(I P )X 1 (s)g(s)ds. (2.4) Let c (t) be an almost perodc functon on R and 1 T T t c (s)ds >, = 1, 2,, n. x (t) = dag (c 1 (t), c 2 (t),, c n (t))x(t) admts an exponental dchotomy on R (It s worthwhle to menton that the exponental dchotomy n that case s wth P = I). 3. Exstence and unqueness of pseudo almost perodc solutons In ths secton, we establsh suffcent condtons on the exstence of pseudo almost perodc solutons of (1.1). Theorem 3.1. Let (H 1 ), (H 2 ) and (H 3 ) hold. Then, there exsts a unque contnuously dfferentable pseudo almost perodc soluton of system (1.1). Proof. Set x (t) = ξ 1 x (t), (3.1) then we can transform (1.1) nto the followng system x (t) = c (t) x (t η (t)) ξ 1 I (t) a j (t)f j (ξ j x j (t τ j (t))) b jl (t)g j (ξ j x j (t α jl (t)))g l (ξ l x l (t β jl (t))) d jl (t) σ jl (u)h j (ξ j x j (t u))du ν jl (u)h l (ξ l x l (t u))du

6 56 CEMIL TUNÇ = c (t) x (t) c (t) x (s)ds ξ 1 tη (t) a j (t)f j (ξ j x j (t τ j (t))) b jl (t)g j (ξ j x j (t α jl (t)))g l (ξ l x l (t β jl (t))) d jl (t) σ jl (u)h j (ξ j x j (t u))du ν jl (u)h l (ξ l x l (t u))du I (t), J. (3.2) Let ϕ B. Obvously, the boundedness of ϕ mples that ϕ s a unformly contnuous functon on R for J. Set f(t, z) = ϕ (t z), ( J). By Theorem 5.3 n [9, p. 58] and Defnton 5.7 n [9, p. 59], we can obtan that f P AP (R Ω) and f s contnuous n z K and unformly n t R for all compact subset K of Ω R. Ths, together wth η P AP (R, R) and Theorem 5.11 n [9, p. 6], mples that Smlarly, we have ϕ (t η (t)) P AP (R, R), J. ϕ j (t τ j (t)), ϕ j (t α jl (t)), ϕ j (t β jl (t)) P AP (R, R),, j, l J. (3.3) From (3.3), (H 1 ) and Corollary 5.4 n [9, p. 58], we have f j (ξ j ϕ j (t τ j (t))), g j (ξ j ϕ j (t α jl (t))), g l (ξ l ϕ l (t β jl (t))) P AP (R, R),, j, l J, (3.4) whch, together wth Lemma 2.2 and the fact that ϕ (t η (t)) P AP (R, R), mples and c (t) ξ 1 tη (t) ϕ (s)ds = c (t)ϕ (t) c (t)ϕ (t η (t)) P AP (R, R), J, (3.5) a j (s)f j (ξ j ϕ j (t τ j (t))) b jl (t)g j (ξ j ϕ j (t α jl (t)))g l (ξ l ϕ l (t β jl (t))) d jl (t) σ jl (u)h j (ξ j ϕ j (t u))du ν jl (u)h l (ξ l ϕ l (t u))du I (t) P AP (R, R), J. (3.6) For ϕ B, we consder the pseudo almost perodc soluton x ϕ (t) of the followng nonlnear pseudo almost perodc dfferental equatons: x (t) = c (t) x (t) c (t) ϕ (s)ds ξ 1 a j (t)f j (ξ j ϕ j (t τ j (t))) tη (t)

7 PSEUDO ALMOST PERIODIC SOLUTIONS 57 b jl (t)g j (ξ j ϕ j (t α jl (t)))g l (ξ l ϕ l (t β jl (t))) d jl (t) σ jl (u)h j (ξ j ϕ j (t u))du ν jl (u)h l (ξ l ϕ l (t u))du I (t), = 1, 2,, n. (3.7) Then, notce that M[c ] >, = 1, 2,, n, t follows from Lemma 2.4 that the lnear system x (t) = c (t) x (t), J, (3.8) admts an exponental dchotomy on R. Thus, by (3.5), (3.6) and Lemma 2.3, we obtan that the system (3.7) has exactly one pseudo almost perodc soluton: x ϕ (t) = {x ϕ (t)} Let = { F (t) = c (t) e s c (u)du [c (s) s sη (s) ϕ (u)du ξ 1 a j (s)f j (ξ j ϕ j (s τ j (s))) b jl (s)g j (ξ j ϕ j (s α jl (s)))g l (ξ l ϕ l (s β jl (s))) d jl (s) I (s)]ds}. tη (t) ϕ (u)du ξ 1 σ jl (u)h j (ξ j ϕ j (s u))du a j (t)f j (ξ j ϕ j (t τ j (t))) b jl (t)g j (ξ j ϕ j (t α jl (t)))g l (ξ l ϕ l (t β jl (t))) d jl (t) I (t), J. σ jl (u)h j (ξ j ϕ j (t u))du Then, {F } P AP (R, R n ), and (3.9) mples that { From (3.5), (3.6) and (3.1), we get ν jl (u)h l (ξ l ϕ l (s u))du ν jl (u)h l (ξ l ϕ l (t u))du e s c (u)du F (s)ds} P AP (R, R n ). (3.1) (3.9) (x ϕ (t)) = {(x ϕ (t)) }

8 58 CEMIL TUNÇ = {F (t) c (t) e s c (u)du F (s)ds} P AP (R, R n ). Thus, x ϕ B. Now, we defne a mappng T : B B by settng (T ϕ)(t) = x ϕ (t), ϕ B. We next prove that the mappng T s a contracton mappng of the B. In fact, n vew of (3.9), (H 1 ) and (H 3 ), for ϕ, ψ B, we have = { ((T ϕ)(t) (T ψ)(t)) } { e s c(u)du {c (s) s sη (s) (ϕ (u) ψ (u))du a j (s)(f j (ξ j ϕ j (s τ j (s))) f j (ξ j ψ j (s τ j (s)))) b jl (s)[(g j (ξ j ϕ j (s α jl (s)))g l (ξ l ϕ l (s β jl (s))) g j (ξ j ϕ j (s α jl (s)))g l (ξ l ψ l (s β jl (s)))) (g j (ξ j ϕ j (s α jl (s)))g l (ξ l ψ l (s β jl (s))) g j (ξ j ψ j (s α jl (s)))g l (ξ l ψ l (s β jl (s))))] d jl (s)[( σ jl (u)h j (ξ j ϕ j (s u))du ( { σ jl (u)h j (ξ j ϕ j (s u))du σ jl (u)h j (ξ j ϕ j (s u))du σ jl (u)h j (ξ j ψ j (s u))du e s c (u)du [c (s) s sη (s) ν jl (u)h l (ξ l ψ l (s u))du) ν jl (u)h l (ξ l ϕ l (s u))du ν jl (u)h l (ξ l ψ l (s u))du } ν jl (u)h l (ξ l ψ l (s u))du)]}ds ϕ (u) ψ (u) du a j (s) L f j ξ j ϕ j (s τ j (s)) ψ j (s τ j (s)) b jl (s) [M g j Lg l ξ l ϕ l (s β jl (s)) ψ l (s β jl (s)) M g l Lg j ξ j ϕ j (s α jl (s)) ψ j (s α jl (s)) ]

9 PSEUDO ALMOST PERIODIC SOLUTIONS 59 d jl (s) [ σ jl (u) dum h j σ jl (u) L h j ξ j ϕ j (s u) ψ j (s u) du { e s c(u)du [c η ξ 1 a j Lf j ξ j b jl (M g j Lg l ξ l M g l Lg j ξ j) { { e d jl σ jl (u) du } s c(u)du (c (s) α )ds ϕ ψ B e s c(u)du d( {1 α { 1 α c s c (u)du) α ν jl (u) L h l ξ l ϕ l (s u)) ψ l (s u) du } ν jl (u) duml h ]}ds } ν jl (u) du(mj h L h l ξ l Ml h L h j ξ j )]ds ϕ ψ B ) } e s c(u)du ds ϕ ψ B } e s c du ds ϕ ψ B } ϕ ψ B, (3.12) and = { ((T ϕ) (t) (T ψ) (t)) } { [c (t) (ϕ (u) ψ (u))du tη 1 (t) a j (t)(f j (ξ j ϕ j (t τ j (t))) f j (ξ j ψ j (t τ j (t)))) b jl (t)(g j (ξ j ϕ j (t α jl (t)))g l (ξ l ϕ l (t β jl (t))) g j (ξ j ψ j (t α jl (t)))g l (ξ l ψ l (t β jl (t)))) d jl (t)( σ jl (u)h j (ξ j ϕ j (t u))du c (t) σ jl (u)h j (ξ j ψ j (t u))du ν jl (u)h l (ξ l ψ l (t u))du)] e s t s c(u)du [c (s) (ϕ (u) ψ (u))du sη (s) ν jl (u)h l (ξ l ϕ l (t u))du

10 6 CEMIL TUNÇ a j (s)(f j (ξ j ϕ j (s τ j (s))) f j (ξ j ψ j (s τ j (s)))) b jl (s)(g j (ξ j ϕ j (s α jl (s)))g l (ξ l ϕ l (s β jl (s))) g j (ξ j ψ j (s α jl (s)))g l (ξ l ψ l (s β jl (s)))) d jl (s)( σ jl (u)h j (ξ j ϕ j (s u))du σ jl (u)h j (ξ j ψ j (s u))du { [c η ξ 1 a j Lf j ξ j b jl (M g j Lg l ξ l M g l Lg j ξ j) d jl c { c e s c (u)du [c η σ jl (u) du ξ 1 ν jl (u)h l (ξ l ϕ l (s u))du } ν jl (u)h l (ξ l ψ l (s u))du)]ds ν jl (u) du(m h j L h l ξ l M h l L h j ξ j )]ds ϕ ψ B a j Lf j ξ j b jl (M g j Lg l ξ l M g l Lg j ξ j) From (H 3 ), we have and d jl σ jl (u) du } ν jl (u) du(mj h L h l ξ l Ml h L h j ξ j )]ds ϕ ψ B α c (1 α } c ) ϕ ψ B. (3.13) K = max{ max {1 α 1 n c < 1 α c }, max 1 n {c whch, together wth (3.12) and (3.13), yeld < 1, T ϕ T ψ B K ϕ ψ B, α c (1 α c )} < 1, whch mples that the mappng T : B B s a contracton mappng. Therefore, the mappng T possesses a unque fxed pont x = (x 1 (t), x 2 (t),, x n (t)) T B, T x = x.

11 PSEUDO ALMOST PERIODIC SOLUTIONS 61 By (3.7) and (3.9), x satsfes (3.7). So (1.1) has a unque contnuously dfferentable pseudo almost perodc soluton x = (ξ 1 x 1 (t), ξ 2 x 2 (t),, ξ n x n (t)) T. The proof of Theorem 3.1 s now completed. 4. Exponental stablty of the pseudo almost perodc soluton In ths secton, we wll dscuss the global exponental stablty of the pseudo almost perodc soluton of system (1.1). Theorem 4.1. Suppose that all condtons n Theorem 3.1 are satsfed. Moreover, assume that (1 c c )(c η < 1, J. d jl ξ 1 a j Lf j ξ j ξ 1 σ jl (u) du b jl (M g j Lg l ξ l M g l Lg j ξ j) ν jl (u) du(m g j Lg l ξ l M g l Lg j ξ j) (4.1) Then system (1.1) has at least one pseudo almost perodc soluton x (t), and x (t) s globally exponentally stable. Proof. By Theorem 3.1, (1.1) has a unque contnuously dfferentable almost perodc soluton x (t) = (x 1(t), x 2(t),, x n(t)) T. Suppose that x(t) = (x 1 (t), x 2 (t),, x n (t)) T s an arbtrary soluton of (1.1) assocated wth ntal value ϕ(t) = (ϕ 1 (t), ϕ 2 (t),, ϕ n (t)) T satsfyng (1.2). Let Then y(t) = (y 1 (t), y 2 (t),, y n (t)) T = (ξ 1 1 (x 1 (t) x 1(t)), ξ 1 2 (x 2 (t) x 2(t)),, ξ 1 n (x n (t) x n(t))) T. y (t) = c (t)y (t η (t)) ξ 1 a j (t)(f j (x j (t τ j (t))) f j (x j(t τ j (t)))) b jl (t)(g j (x j (t α jl (t)))g l (x l (t β jl (t))) g j (x j(t α jl (t)))g l (x l (t β jl (t)))) d jl (t)( σ jl (u)h j (x j (t u))du where = 1, 2,, n. σ jl (u)h j (x j(t u))du ν jl (u)h l (x l (t u))du ν jl (u)h l (x l (t u))du), (4.2)

12 62 CEMIL TUNÇ and We can choose a constant λ (, mn{κ, mn J c }) such that Γ (λ) = c λ c η eλη ξ 1 a j Lf j ξ je λτ j b jl [M g j Lg l ξ le λβ jl M g l Lg j ξ je λα jl ] d jl [ σ jl (u) dumj h σ jl (u) L h j ξ j e λu du β ν jl (u) dum h l ν jl (u) L h l ξ l e λu du = (c λ)( c λ 1) <, (4.3) Π (λ) = (1 c c λ )[c η eλη ξ 1 a j Lf j ξ je λτ j b jl [M g j Lg l ξ le λβ jl M g l Lg j ξ je λα jl ] d jl [ σ jl (u) dumj h σ jl (u) L h j ξ j e λu du ν jl (u) dum h l ] ν jl (u) L h l ξ l e λu du where β = c η eλη ξ 1 = (1 c c λ )β < 1, (4.4) a j Lf j ξ je λτ j b jl [M g j Lg l ξ le λβ jl M g l Lg j ξ je λα jl ] d jl [ σ jl (u) dumj h σ jl (u) L h j ξ j e λu du ν jl (u) L h l ξ l e λu du ν jl (u) dum h l, = 1, 2,, n.

13 Let ϕ x ξ = max{sup t PSEUDO ALMOST PERIODIC SOLUTIONS 63 max 1 n ξ1 ϕ (t) x (t), sup t max 1 n ξ1 ϕ (t) x (t) } (4.5) and M be a constant such that M > c λ > 1, β for all = 1, 2,..., n, (4.6) whch, together wth (4.3), yelds 1 M β c λ <, β c λ Consequently, for any ε >, t s obvous that < 1, for all J, (4.7) y(t) 1 < ( ϕ x ξ ε)e λt < M( ϕ x ξ ε)e λt for all t (, ]. In the followng, we wll show y(t) 1 < M( ϕ x ξ ε)e λt for all t >. (4.8) Otherwse, there must exst {1, 2,, n} and θ > such that { y(θ) 1 = max{ y (θ), y (θ) } = M( ϕ x ξ ε)e λθ, y(t) 1 < M( ϕ x ξ ε)e λt for all t (, θ). Note that y (s) c (s)y (s) = c (s) s sη (s) y (u)du a j (s)(f j (x j (s τ j (s))) f j (x j(s τ j (s)))) b jl (s)(g j (x j (s α jl (s)))g l (x l (s β jl (s))) g j (x j(s α jl (s)))g l (x l (s β jl (s)))) d jl (s)( σ jl (u)h j (x j (s u))du σ jl (u)h j (x j(s u))du ν jl (u)h l (x l (s u))du (4.9) ν jl (u)h l (x l (s u))du), s [, t], t [, θ].(4.1) Multplyng both sdes of (4.1) by e s c(u)du, and ntegratng t on [, t], we get y (t) = y ()e c (u)du e s c (u)du [c (s) s sη (s) y (u)du a j (s)(f j (x j (s τ j (s))) f j (x j(s τ j (s))))

14 64 CEMIL TUNÇ b jl (s)(g j (x j (s α jl (s)))g l (x l (s β jl (s))) g j (x j(s α jl (s)))g l (x l (s β jl (s)))) d jl (s)( σ jl (u)h j (x j (s u))du σ jl (u)h j (x j(s u))du ν jl (u)h l (x l (s u))du ν jl (u)h l (x l (s u))du)]ds, t [, θ]. Thus, wth the help of (4.7), we have y (θ) = y ()e θ θ c(u)du e s θ s c(u)du [c (s) y (u)du sη (s) a j (s)(f j (x j (s τ j (s))) f j (x j(s τ j (s)))) b jl (s)(g j (x j (s α jl (s)))g l (x l (s β jl (s))) g j (x j(s α jl (s)))g l (x l (s β jl (s)))) d jl (s)( σ jl (u)h j (x j (s u))du σ jl (u)h j (x j(s u))du ( ϕ x ξ ε)e c θ θ a j Lf j ξ jm( ϕ x ξ ε)e λ(sτ j(s)) ν jl (u)h l (x l (s u))du)]ds ν jl (u)h l (x l (s u))du e θ s c (u)du [c (s)η (s)m( ϕ x ξ ε)e λ(sη (s)) b jl [M g j Lg l ξ lm( ϕ x ξ ε)e λ(sβ jl(s)) M g l Lg j ξ jm( ϕ x ξ ε)e λ(sαjl(s)) ] d jl [ σ jl (u) dumj h ν jl (u) L h l ξ l M( ϕ x ξ ε)e λ(su) du σ jl (u) L h j ξ j M( ϕ x ξ ε)e λ(su) du ( ϕ x ξ ε)e c θ θ e θ s c (u)du e λs [c η eλη ν jl (u) dum h l ]ds

15 PSEUDO ALMOST PERIODIC SOLUTIONS 65 a j Lf j ξ je λτ j b jl [M g j Lg l ξ le λβ jl M g l Lg j ξ je λα jl ] d jl [ σ jl (u) dumj h σ jl (u) L h j ξ j e λu du ν jl (u) L h l ξ l e λu du ν jl (u) dum h l ]dsm( ϕ x ξ ε) θ ( ϕ x ξ ε)e c θ e c θ e (c λ)s dsβ M( ϕ x ξ ε) = M( ϕ x ξ ε)e λθ [( 1 M β c λ )e(λc )θ β c λ ] (4.11) < M( ϕ x ξ ε)e λθ, (4.12) whch, together wth (4.9), mples that y(θ) 1 = max{ y (θ), y (θ) } = y (θ) = M( ϕ x ξ ε)e λθ. (4.13) From (4.4) and (4.7), (4.1) and (4.11) yeld y (θ) c (θ)y (θ) c (θ) θ θη (θ) y (u) du a j (θ) f j (x j (θ τ j (θ))) f j (x j(θ τ j (θ))) b jl (θ) [ g j (x j (θ α jl (θ)))g l (x l (θ β jl (θ))) g j (x j (θ α jl (θ)))g l (x l (θ β jl (θ))) g j (x j (θ α jl (θ)))g l (x l (θ β jl (θ))) g j (x j(θ α jl (θ)))g l (x l (θ β jl (θ))) ] d jl (θ) [ σ jl (u)h j (x j (θ u))du σ jl (u)h j (x j (θ u))du σ jl (u)h j (x j (θ u))du σ jl (u)h j (x j(θ u))du ν jl (u)h l (x l (θ u))du ν jl (u)h l (x l (θ u))du ν jl (u)h l (x l (θ u))du ] ν jl (u)h l (x l (θ u))du

16 66 CEMIL TUNÇ M( ϕ x ξ ε)e λθ [c ( 1 M β c λ )e(λc )θ β ( c λ 1)] < M( ϕ x ξ ε)e λθ, whch contradcts (4.13). Hence, (4.8) holds. Lettng ε, we have from (4.8) that y(t) 1 M ϕ x ξ e λt for all t >, whch mples x(t) x (t) 1 M ϕ x e λt for all t >. Ths completes the proof. 5. Example and Remark In ths secton, some examples and remarks are provded to demonstrate the effectveness of our results. Example 5.1. Consder the followng for HCNNs wth tme-varyng leakage delays: x 1(t) = [ cos t cos 2t ]x1 (t 1 sn t ) (cos t cos 2t e t4 sn 2 t )f 1 (x 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) 25 1 (cos t cos 3t e t4 sn 2 t )f 2 (x 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 1 (cos t cos 2t sn 2 t et4 )g1(x 2 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) 2 7 (cos t cos 5t e t4 sn 2 t )g2(x 2 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 1 (cos t cos 7t e t4 sn 2 t )g 1 (x 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) g 2 (x 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 3 (cos t cos 2t et4 sn 2 t ) 5 1 sn t et4 cos t, x 2(t) = [ where c e u h 1 (x 1 (t u))du e u h 2 (x 2 (t u))du cos t cos 3t ]x1 (t 1 cos t ) (cos t cos πt sn 2 t et4 )f 1 (x 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 9 (cos t cos 2t sn 2 t et4 )f 2 (x 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 1 (cos t cos πt sn t et4 )g1(x 2 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) 5 11 (cos t cos 2t e t4 sn t )g2(x 2 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 1 13 (cos t cos 3t e t4 sn t )g 1 (x 1 (t cos 2 2t cos 2 t e t4 sn 2 t )) g 2 (x 2 (t cos 2 2t cos 2 t e t4 sn 2 t )) 7 3 (cos t cos 2t et4 sn 2 t ) 1 5 et4 sn t cos t, f j (x) = 1 4 ( x cos x), g j(x) = h j (x) = cos x, e u h 1 (x 1 (t u))du e u h 2 (x 2 (t u))du (5.1)

17 PSEUDO ALMOST PERIODIC SOLUTIONS 67 and b jl (u) =, (jl 112, jl 212), σ jl (u) = v jl (u) = e u,, j, l = 1, 2. Comparng (5.1) wth (1.1) and usng some basc nformaton, t follows that c 1 = 1 4, c 1 = 1 2, c 2 = 1 4, c 2 = 1 2, η j = 1 1, Lf j = 1 2, L g j = Lh j = L h l = 1, M g j = M j h = Ml h = 1, ξ = ξ 1 = ξ k = ξ l = 1, c c η ξ 1 a j Lf j ξ ξ 1 b jl (M g j Lg l ξ l M g l Lg j ξ j), 2 2 d jl σ jl (u) du v jl (t) du(m h j L h l ξ l M h l L h j ξ j ), = ( ) ( ), (1 c c )(c η ( c ξ 1 2 d jl e u du e u du) = = α, α c (1 α c ) = < 1, 2 a j Lf j ξ ξ 1 σ jl (u) du 1 = 3[ 2 ( )] 8 1 ( 2 2 b jl (M g j Lg l ξ l M g l Lg j ξ j), v jl (t) du(m h j L h l ξ l M h l L h j ξ j ) e u du e u du)] = < 1, whch mply that (5.1) satsfes all the condtons n Theorem 3.1 and Theorem 4.1. Hence, we can conclude that system (5.1) has a unque contnuously dfferentable pseudo almost perodc soluton x (t), whch s globally exponentally stable wth the exponental convergent rate λ.1.

18 68 CEMIL TUNÇ Open Access: Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense (CC-BY 4.) whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and the source are credted.

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