PERIODIC SOLUTIONS FOR IMPULSIVE NEUTRAL DYNAMIC EQUATIONS WITH INFINITE DELAY ON TIME SCALES

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1 Kragujevac Journal of Mahemaics Volume 42(1) (218), Pages PERIODIC SOLUTIONS FOR IMPULSIVE NEUTRAL DYNAMIC EQUATIONS WITH INFINITE DELAY ON TIME SCALES A. ARDJOUNI 1 AND A. DJOUDI 2 Absrac. Le T be a periodic ime scale. We use he Krasnoselskii s fixed poin heorem o show ha he impulsive neural dynamic equaions wih infinie delay x () = A()x σ () + g (, x( h())) + x( + j ) = x( j ) + I j(x( j )), j Z + D (, u) f(x(u)) u, j, T, have a periodic soluion. Under a slighly more sringen condiions we show ha he periodic soluion is unique using he conracion mapping principle. 1. Inroducion In 1988, Sephan Hilger 9 inroduced he heory of ime scales (measure chains) as a means of unifying discree and coninuum calculi. Since Hilger s iniial work here has been significan growh in he heory of dynamic equaions on ime scales, covering a variey of differen problems; see 7,8,17 and references herein. The sudy of impulsive iniial and boundary value problems is exensive. For he heory and classical resuls, we direc he reader o he monographs 6, 16, 18. Recenly Alhubii, Makhzoum and Raffoul 2 invesigaed he exisence and uniqueness of periodic soluions for he neural differenial equaion wih infinie delay x () = a()x() + d d g(, x( h())) + D (, u) f(x(u))du. By employing he Krasnoselskii s fixed poin heorem and he conracion mapping principle, he auhors obained exisence and uniqueness resuls for periodic soluions. Key words and phrases. Periodic soluions, neural dynamic equaions, impulses, Krasnoselskii fixed poin, infinie delay, ime scales. 21 Mahemaics Subjec Classificaion. Primary: 34N5, 34K13. Secondary: 34K4, 34K45. Received: Ocober 13, 216. Acceped: January 13,

2 7 A. ARDJOUNI AND A. DJOUDI The nonlinear impulsive dynamic equaion x () = a()x σ () + f(, x()), j, T, x( + j ) = x( j ) + I j( j, x( j )), j = 1, 2,..., n, has been invesigaed in 1. By using Schaeffer s heorem, he exisence of periodic soluions has been esablished. In his aricle, we are ineresed in he analysis of qualiaive heory of periodic soluions of impulsive neural dynamic equaions. Inspired and moivaed by he works menioned above and he papers 1 5, 1 15, 2 22 and he references herein, we are concerned wih he sysem (1.1) x () = A()x σ () + g (, x( h())) + D (, u) f(x(u)) u, j, T, x( + j ) = x( j ) + I j(x( j )), j Z +, where T is an ω-periodic ime scale, T and x σ = x σ. For each inerval U of R, we denoe by U T = U T, x( + j ) and x( j ) represen he righ and he lef limi of x( j) in he sense of ime scales, in addiion, if j is lef-scaered, hen x( j ) = x( j), A() = diag(a i ()) n n (a i R + ) and D (, u) = diag(d i (, u)) n n (D i C(T, R)) are diagonal marices wih coninuous real-valued funcions as is elemens, R + = {a C(T, R) : 1+ µ()a() > } where µ() = σ(), h C(T, T), g = (g 1, g 2,..., g n ) C(T R n, R n ),,..., I (n) j f = (f 1, f 2,..., f n ) C(R n, R n ), I j = (I (1) j, I (2) j ) C(R n, R n ) and A(), h(), g(, x( h())) are all ω-periodic funcions wih respec o, D ( + ω, u + ω) = D (, u), ω > is a consan. There exiss a posiive ineger p such ha j+p = j + ω, I j+p = I j, j Z +, wihou loss of generaliy, we also assume ha, ω) T { j, j Z + } = { 1, 2,..., p }. To reach our desired end we have o ransform he sysem (1.1) ino an inegral sysem and hen use Krasnoselskii s fixed poin heorem o show he exisence of periodic soluions. The obained inegral sysem is he sum of wo mappings, one is a conracion and he oher is a compac. Also, ransforming sysem (1.1) o an inegral sysem enables us o show he uniqueness of he periodic soluion by appealing o he conracion mapping principle. The organizaion of his paper is as follows. In Secion 2, we inroduce some noaions and definiions, and sae some preliminary resuls needed in laer secions, hen we give he Green s funcion of (1.1), which plays an imporan role in his paper. In Secion 3, we esablish our main resuls for periodic soluions by applying he Krasnoselskii s fixed poin heorem and he conracion mapping principle. 2. Preliminaries In his secion, we shall recall some basic definiions and lemmas which are used in wha follows.

3 PERIODIC SOLUTIONS 71 Le T be a nonempy closed subse (ime scale) of R. The forward and backward jump operaors σ, ρ : T T and he graininess µ : T R + are defined, respecively, by σ() = inf{s T : s > }, ρ() = sup{s T : s < }, µ() = σ(). A poin T is called lef-dense if > inf T and ρ() =, lef-scaered if ρ() <, righ-dense if < sup T and σ() =, and righ-scaered if σ() >. If T has a lef-scaered maximum m, hen T k = T\{m}; oherwise T k = T. A funcion f : T R is righ-dense coninuous (rd-coninuous) provided i is coninuous a righ-dense poin in T and is lef-side limis exis a lef-dense poins in T. If f is coninuous a each righ-dense poins and each lef-dense poin, hen f is said o be a coninuous funcion on T. The se of coninuous funcions f : T R will be denoed by C(T). For x : T R and T k, we define he dela derivaive of x(), x (), o be he number (if i exiss) wih he propery ha for a given ε >, here exiss a neighborhood U T of such ha x(σ()) x(s) x ()σ() s < ε σ() s, for all s U T. If x is coninuous, hen x is righ-dense coninuous, and if x is dela differeniable a, hen x is coninuous a. Remark 2.1. x : T R n is dela derivable or righ-dense coninuous or coninuous if each enry of x is dela derivable or righ-dense coninuous or coninuous. Le x be righ-dense coninuous. If X () = x(), hen we define he dela inegral by a x(s) s = X() X(a). Definiion 2.1 (12). We say ha a ime scale T is periodic if here exiss p > such ha if T, hen ± p T. For T R, he smalles posiive p is called he period of he ime scale. Le T R be a periodic ime scale wih period p. We say ha he funcion f : T R is periodic wih period ω if here exiss a naural number n such ha ω = np, f( + ω) = f() for all T and ω is he smalles posiive number such ha f( + ω) = f(). If T = R, we say ha f is periodic wih period ω > if ω is he smalles posiive number such ha f( + ω) = f() for all T. Remark 2.2. According o 12, if T is a periodic ime scale wih period p, hen σ( + np) = σ() + np and he graininess funcion µ is a periodic funcion wih period p.

4 72 A. ARDJOUNI AND A. DJOUDI Definiion 2.2 (8). An n n-marix-valued funcion A on ime scale T is called regressive (respec o T) provided is inverible for all T k. I + µ()a(), Le A, B : T R n n be wo n n-marix-valued regressive funcions on T, we define for all T k. (A B)() := A() + B() + µ()a()b(), ( A)() := I + µ()a() 1 A() = A()I + µ()a() 1, (A()) (B()) := (A()) ( (B())), Theorem 2.1 (8). Le A be an regressive and rd-coninuous n n-marix-valued funcion on T and suppose ha f : T R n is rd-coninuous. Le T and y R n. Then he iniial value problem has a unique soluion y : T R n. y = A()y + f(), y( ) = y, Definiion 2.3 (8). Le T and assume ha A is an regressive and rd-coninuous n n-marix-valued funcion. The unique marix-valued soluion of he iniial value problem x () = A()x(), x( ) = I, where I denoes as usual he n n-ideniy marix, is called he marix exponenial funcion (a ), and i is denoed by e A (, ). Remark 2.3. Assume ha A is a consan n n-marix. If T = R, hen e A (, ) = e A( ), while if T = Z and I + A is inverible, hen e A (, ) = (I + A). In he following lemma, we give some properies of he marix exponenial funcion. Lemma 2.1 (8). Assume ha A, B : T R n n are regressive and rd-coninuous marix-valued funcions on T. Then (i) e (, s) I and e A (, ) I; (ii) e A (σ(), s) = (I + µ()a())e A (, s); (iii) e 1 A (, s) = e A (, s); (iv) e A (, s) = e 1 A (s, ) = e A (s, ); (v) e A (, s)e A (s, r) = e A (, r); (vi) e A (, s)e B (, s) = e A B (, s), if A() and B() commue, where A denoes he conjugae ranspose of A.

5 PERIODIC SOLUTIONS 73 Lemma 2.2 (8). Suppose A and B are regressive marix-valued funcions, hen (i) A is regressive; (ii) A = ( A) ; (iii) (A ) = (A ) holds for any differenial marix-valued funcion A. Nex, we sae Krasnoselskii s fixed poin heorem which enables us o prove he exisence of a periodic soluion of (1.1). For is proof we refer he reader o 19. Theorem 2.2 (Krasnoselskii). Le M be a closed convex nonempy subse of Banach space (B, ). Suppose ha Φ and Ψ map M ino B such ha (i) x, y M imply Φx + Ψy M; (ii) Ψ is compac and coninuous; (iii) Φ is a conracion mapping. Then here exiss z M wih z = Φz + Ψz. Lemma 2.3. A funcion x is an ω-periodic soluion of (1.1) if and only if x is an ω-periodic soluion of he equaion +ω x() = g(, x( h())) + G(, s) D (s, u) f(x(u)) u A(s)g σ (s, x(s h(s))) s + G(, j )I j (x( j )), where j: j,+ω) G(, s) = diag(g i (, s)) n n, G i (, s) = (1 e ai (ω, )) 1 e ai ( + ω, s), 1 A() = diag(a i ()) n n, e ai (, s) = e ai (, s), a i () a i () = 1 + µ () a i (), gσ (, x( h())) = g(σ (), x σ ( h())). Proof. If x is an ω-periodic soluion of (1.1). For any T, here exiss j Z such ha j is he firs impulsive poin afer. Then for i = 1, 2,..., n, x i is an ω-periodic soluion of he equaion (2.1) x i () + a i ()x σ i () = g i (, x i ( h())) + D i (, u) f i (x(u)) u. Muliply boh sides of (2.1) by e ai (, ) and hen inegrae from o s, j T, we obain = e ai (τ, )x i (τ) τ e ai (τ, ) g i (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ,

6 74 A. ARDJOUNI AND A. DJOUDI or hen hence (2.2) e ai (s, )x i (s) = e ai (, )x i () + e ai (τ, ) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ, x i (s) = e ai (s, )x i () + e ai (s, τ) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ, i = 1, 2,..., n, j x i ( j ) = e ai ( j, )x i () + e ai ( j, τ) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ, i = 1, 2,..., n. Similarly, for s ( j, j+1, we have x i (s) = e ai (s, j )x i ( + j ) + e ai (s, τ) j + D i (τ, u) f i (x i (u)) u = e ai (s, j )x i ( j ) + + D i (τ, u) f i (x i (u)) u = e ai (s, j )x i ( j ) + e ai (s, τ) j g i (τ, x i (τ h(τ))) τ e ai (s, τ) gi (τ, x i (τ h(τ))) j τ + e ai (s, j )I (i) j (x i ( j )) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ + e ai (s, j )I (i) j (x i ( j )), for i = 1, 2,..., n. Subsiuing (2.2) in he above equaliy, we obain x i (s) = e ai (s, )x i () + e ai (s, τ) + D i (τ, u) f i (x i (u)) u Repeaing he above process for s, + ω T, we have x i (s) = e ai (s, )x i () + e ai (s, τ) g i (τ, x i (τ h(τ))) τ + e ai (s, j )I (i) j (x i ( j )). g i (τ, x i (τ h(τ)))

7 + D i (τ, u) f i (x i (u)) u τ + PERIODIC SOLUTIONS 75 j: j,+ω) e ai (s, j )I (i) j (x i ( j )), for i = 1, 2,..., n. Le s = + ω in he above equaliy, we have +ω x i ( + ω) = e ai ( + ω, )x i () + e ai ( + ω, τ) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ + e ai ( + ω, j )I (i) j (x i ( j )), j: j,+ω) i = 1, 2,..., n. Noicing ha x i ( + ω) = x i () and e ai ( + ω, ) = e ai (ω, ), we obain +ω (2.3) (1 e ai (ω, ))x i () = e ai ( + ω, τ) gi (τ, x i (τ h(τ))) + D i (τ, u) f i (x i (u)) u τ for i = 1, 2,..., n. Noice ha (2.4) +ω + j: j,+ω) e ai ( + ω, τ)g i (τ, x i (τ h(τ))) τ = e ai ( + ω, + ω)g i ( + ω, x i ( + ω h( + ω))) e ai ( + ω, )g i (, x i ( h())) +ω e ai ( + ω, j )I (i) j (x i ( j )), e ai ( + ω, τ)a i (τ)g σ i (τ, x i (τ h(τ))) τ = 1 e ai (ω, )g i (, x i ( h())) +ω I follows from (2.3) and (2.4) ha e ai ( + ω, τ)a i (τ)g σ i (τ, x i (τ h(τ))) τ, i = 1, 2,..., n. +ω x i () = g i (, x i ( h())) + 1 e ai (ω, ) 1 e ai ( + ω, τ) D i (τ, u) f i (x i (u)) u a i (τ)gi σ (τ, x i (τ h(τ))) τ + 1 e ai (ω, ) 1 e ai ( + ω, j )I (i) j (x i ( j )) j: j,+ω) = g i (, x i ( h())) + +ω G i (, τ) D i (τ, u) f i (x i (u)) u

8 76 A. ARDJOUNI AND A. DJOUDI a i (τ)gi σ (τ, x i (τ h(τ))) τ + j: j,+ω) G i (, j )I (i) j (x i ( j )), for i = 1, 2,..., n. Nex, we prove he converse. Le +ω x i () = g i (, x i ( h())) + G i (, s) D i (s, u) f i (x i (u)) u a i (s)gi σ (s, x i (s h(s))) s + G i (, j )I (i) j (x i ( j )), where j: j,+ω) G i (, s) = (1 e ai (ω, )) 1 e ai ( + ω, s), i = 1, 2,..., n. Then if i, i Z +, we have x i () = gi (, x i ( h())) +ω { + G i (, s) D i (s, u) f i (x i (u)) u a i (s)gi σ (s, x i (s h(s)))} s +ω + G i (, + ω) D i ( + ω, u) f i (x i (u)) u a i ( + ω)gi σ ( + ω, x i ( + ω h( + ω))) G i (, ) D i (, u) f i (x i (u)) u a i ()gi σ (, x i ( h())) = g i (, x i ( h())) + + +ω { G i (, s) = g i (, x i ( h())) + D i (, u) f i (x i (u)) u a i ()g σ i (, x i ( h())) D i (s, u) f i (x i (u)) u a i (s)gi σ (s, x i (s h(s)))} s = a i ()x σ i () + g i (, x i ( h())) + D i (, u) f i (x i (u)) u a i ()x σ i () If = i, i Z +, we obain x i ( + i ) x i( i ) = G i ( i, j )I (i) j (x i ( j )) j: j + i,+ i +ω) D i (, u) f i (x i (u)) u, i = 1, 2,..., n. = G i ( i, i + ω)i (i) i (x i ( i + ω)) G i ( i, i )I (i) i (x i ( i )) = I (i) i (x i ( i )), i = 1, 2,..., n. G i ( i, j )I (i) j (x i ( j )) j: j i, i +ω)

9 PERIODIC SOLUTIONS 77 So we know ha, x is also an ω-periodic soluion of (1.1). proof. Throughou his paper, we make he following assumpions. This complees he (H1) The funcion g = (g 1, g 2,..., g n ) saisfies a Lipschiz condiion in x. Tha is, for i {1, 2,..., n}, here exiss a posiive consan L i such ha g i (, x) g i (, y) L i x y, for all T, x, y R n. (H2) The funcion f = (f 1, f 2,..., f n ) saisfies a Lipschiz condiion in x. Tha is, for i {1, 2,..., n}, here exiss a posiive consans M i such ha f i (x) f i (y) M i x y, for all T, x, y R n. (H3) For j Z, I j = (I (1) j, I (2) j,..., I (n) j ) saisfies Lipschiz condiion. Tha is, for i {1, 2,..., n} here exiss a posiive consan P (i) j such ha I (i) (x) I (i) j (y) P (i) j x y, for all x, y R n. j (H4) There exiss a posiive consan N i such ha To apply Theorem 2.2 o (1.1), we define D i (, u) u N i. P C(T) = {x : T R n : x (j, j+1 ) T C( j, j+1 ) T, x( j ) = x( j), x( + j ), j Z+ }. Consider he Banach space X = {x P C(T) : x( + ω) = x()}, wih he norm x = max,ωt x(), where x() = max 1 i n x i (). Lemma 2.4 (12). Le x X. Then here exiss x σ, and x σ = x. Noicing ha G i (, s) (1 e ai (ω, )) 1 := η i, for convenience, we inroduce he noaion η := max 1 i n η i, N := max 1 i n N i, γ := max max a i (), 1 i n,ω T L := max 1 i n L i, P j := max P (i) j, P := max P j. 1 i n 1 j p M := max 1 i n M i, Define he mapping H : X X by +ω (2.5) (Hϕ)() = g(, ϕ( h())) + G(, s) D (s, u) f(ϕ(u)) u A(s)g σ (s, ϕ(s h(s))) s + G(, j )I j (x( j )). j: j,+ω)

10 78 A. ARDJOUNI AND A. DJOUDI To apply Theorem 2.2, we need o consruc wo mappings: one is a conracion and he oher is coninuous and compac. We express (2.5) as where (Hϕ)() = (Φϕ)() + (Ψϕ)(), (2.6) (Φϕ)() = g(, ϕ( h())), and (2.7) (Ψϕ) () +ω = G(, s) D (s, u) f(ϕ(u)) u A(s)g σ (s, ϕ(s h(s))) s + G(, j )I j (ϕ( j )). j: j,+ω) Lemma 2.5. Suppose (H1) holds and L < 1, hen Φ : X X, as defined by (2.6), is a conracion. Proof. Trivially, Φ : X X. For ϕ, ψ X, we have (2.8) Φ(ϕ) Φ(ψ) = max,ω T max 1 i n g i(, ϕ i ( h())) g i (, ψ i ( h())) L ϕ ψ. Hence Φ defines a conracion mapping wih conracion consan L. Lemma 2.6. Suppose (H1) (H4) hold, hen Ψ : X X, as defined by (2.7), is coninuous and compac. Proof. Evaluaing (2.7) a + ω gives (Ψϕ)( + ω) +2ω = G( + ω, s) D (s, u) f(ϕ(u)) u A(s)g σ (s, ϕ(s h(s))) s +ω + G( + ω, j )I j (ϕ( j )). = = j: j +ω,+2ω) +ω +ω G( + ω, v + ω) v+ω D (v+ω, u) f(ϕ(u)) u A(v + ω)g σ (v + ω, ϕ(v + ω h(v + ω))) v + v G(, v) D (v, u) f(ϕ(u)) u A(v)g σ (v, ϕ(v h(v))) v + k: k,+ω) k: k,+ω) G(, k )I j (ϕ( k )) G(, k )I j (ϕ( k ))

11 PERIODIC SOLUTIONS 79 = (Ψϕ)(). Tha is, Ψ : X X. Now, we show ha Ψ is coninuous. Le ϕ, ψ X, given ε >, ake ε δ = ηω(mn + Lγ) + P, such ha for ϕ ψ δ. By using he Lipschiz condiion, we obain Ψϕ Ψψ +ω max,ω T G(, s) D (s, u) f(ϕ(u)) u D (s, u) f(ψ(u)) u s +ω + max,ω T G(, s)a(s)g σ (s, ϕ(s h(s))) g σ (s, ψ(s h(s))) s + max G(, j )I j (ϕ( j )) I j (ψ( j )) ω η,ω T j: j,+ω) + ηγ ω D (s, u) f(ϕ(u)) f(ψ(u)) u s g σ (s, ϕ(s h(s))) g σ (s, ψ(s h(s))) s + η max 1 j p I j(ϕ( j )) I j (ψ( j )) ηω(mn + Lγ) + P ϕ ψ < ε. This proves Ψ is coninuous. Nex, we need o show ha Ψ is compac. Consider he sequence of periodic funcions {ϕ n } X and assume ha he sequence is uniformly bounded. Le Θ > be such ha ϕ n Θ, for all n N. In view of (H1) (H3), we arrive a (2.9) (2.1) (2.11) g σ (, x) g σ (, x) g σ (, ) + g σ (, ) = max,ω T max 1 i n gσ i (, x) g σ i (, ) + α g L x + α g, f(x) f(x) f() + f() = max 1 i n f i(x) f i () + α f M x + α f, I j (x) I j (x) I j () + I j () = max 1 i n n I(i) j (x) I (i) j () + α Ij P j x + α Ij, for j Z +, where α g = g σ (, ), α f = f() and α Ij = I j (). Hence,

12 8 A. ARDJOUNI AND A. DJOUDI (2.12) Ψϕ n max,ω T ω η + η +ω G(, s) D (s, u) f(ϕ n (u)) u s + max A(s)g σ (s, ϕ n (s h(s))) D (s, u) f(ϕ n (u)) u s + ηγ p I j (ϕ n ( j )) j=1,ω T j: j,+ω) ω G(, j )I j (ϕ n ( j )) g σ (s, ϕ n (s h(s))) s ηωn(m ϕ n + α f ) + ηγω(l ϕ n + α g ) + η( max 1 j p (P j ϕ n + α Ij )) ηωθ(mn + γl) + η(ωnα f + γωα g + P Θ + α) := D, where α = max 1 j p α Ij. Thus he sequence {Ψϕ n } is uniformly bounded. Now, i can be easily checked ha (Ψϕ n ) () = A()(Ψϕ n ) σ () + A()g σ (, ϕ n ( h())). D (, u) f(ϕ n (u)) u Consequenly, i follows from (2.1), (2.11), (2.12) and Lemma 2.4 ha (Ψϕ n ) () A (Ψϕ n ) σ + D (, u) f(ϕ n (u)) u + A()g σ (, ϕ n ( h())) A (Ψϕ n ) + N (M ϕ n + α f ) + A (L ϕ n + α g ) A (D + LΘ + α g ) + N (MΘ + α f ), for all n. Tha is, (Ψϕ n ) A (D + LΘ + α g )+N (MΘ + α f ), hus he sequence {Ψϕ n } is uniformly bounded and equi-coninuous. The Arzela-Ascoli heorem implies ha Ψ is compac. Our main resuls reads as follows. 3. Main Resuls Theorem 3.1. Assume ha (H1) (H4) hold and L < 1. Suppose ha here is a posiive consan G such ha all soluions x of (1.1), x X, saisfy x G, and he inequaliy γωα g + ωnα f + α 1/η ω(γl + MN) L/η P G, holds. Then (1.1) has an ω-periodic soluion.

13 PERIODIC SOLUTIONS 81 Proof. Define M = {ϕ X : ϕ G}. Then Lemma 2.6 implies Ψ : X X and Ψ is compac and coninuous. Also, from Lemma 2.5, he mapping Φ is a conracion and Φ : X X. We need o show ha if ϕ, ψ M, hen Φϕ + Ψψ G. Le ϕ, ψ M wih ϕ, ψ G, from (2.9) (2.11), we have Φϕ + Ψψ Φϕ + Ψψ LG + ηωg(γl + MN) + η(γωα g + ωnα f + GP + α) G. Thus Φϕ + Ψψ M. We see ha all he condiions of Krasnoselskii heorem are saisfied on he se M. Hence here exiss a fixed poin z in M such ha z = Φz + Ψz. By Lemma 2.3, his fixed poin is a soluion of (1.1). Theorem 3.2. Suppose ha (H1) (H4) hold. If Υ := ηω(γl + MN) + P < 1, hen (1.1) has an unique ω-periodic soluion. Proof. For ϕ, ψ X, we have Hϕ Hψ η ω + ηγ + η This complees he proof. ω D (s, u) f(ϕ(u)) f(ψ(u)) u s g σ (s, ϕ(s h(s))) g σ (s, ψ(s h(s))) s p I j (ϕ( j )) I j (ψ( j )) j=1 ηωmn ϕ ψ + ηγωl ϕ ψ + ηp ϕ ψ < ηω(γl + MN) + P ϕ ψ = Υ ϕ ψ. Acknowledgemens. The auhors graefully acknowledge he reviewers for heir helpful commens. References 1 M. Adivar and Y. N. Raffoul, Exisence of periodic soluions in oally nonlinear delay dynamic equaions, Elecron. J. Qual. Theory Differ. Equ. 29 (29), S. Alhubii, H. A. Makhzoum and Y. N. Raffoul, Periodic soluion and sabiliy in nonlinear neural sysem wih infinie delay, Appl. Mah. Sci. 7(136) (213), A. Ardjouni and A. Djoudi, Exisence of periodic soluions for nonlinear neural dynamic equaions wih funcional delay on a ime scale, Aca Univ. Palack. Olomuc. Fac. Rerum Naur. Mah. 52(1) (213), A. Ardjouni and A. Djoudi, Exisence of periodic soluions for nonlinear neural dynamic equaions wih variable delay on a ime scale, Commun. Nonlinear Sci. Numer. Simul. 17 (212),

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