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1 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 Exisence and Uniqueness of a Periodic Soluion for hird-order Delay Differenial Equaion wih wo Deviaing Argumens A. M. A. Abou-El-Ela, A. I. Sadek and A. M. Mahmoud Absrac In his paper we use he coninuaion heorem of coincidence degree heory and analysis echniques. We esablish he exisence and uniqueness of a -periodic soluion for he hird-order delay differenial equaion wih wo deviaing argumens. A new resul on he exisence and uniqueness of a periodic soluion of his equaion is obained. In addiion an example is given o illusrae he main resul. Index erms Exisence and uniqueness, Coincidence degree, hird-order delay differenial equaions, Deviaing argumens. I. INRODUCION O our knowledge, mos exciing resuls on he exisence of periodic soluions of delay differenial equaions are usually obained by he echnique of bifurcaion, by fixedpoin heorems or by coincidence degree heory. In general i is more difficul o sudy he uniqueness of he periodic soluions. Coninuaion heorem of coincidence degree heory plays a significan role in he invesigaion of he exisence and uniqueness of periodic soluions of differenial equaions wih delay. In recen years, by using coninuaion heorem of coincidence degree heory, he exisence and uniqueness of periodic soluions for some ypes of firs and second-order delay differenial equaions wih a deviaing argumen or wo deviaing argumens were sudied, for example, [7, 2 6], ec. Bu he corresponding problem of hirdorder delay differenial equaions wih a deviaing argumen or wo deviaing argumens was discussed far less ofen, for example, [3,, 6]. In 26, Gui [3] esablished crieria for exisence of posiive periodic soluions o he following hirdorder neural delay differenial equaion wih a deviaing argumen x (3) () + aẍ() + g(ẋ( τ())) + f(x( τ())) = p(), where a is a posiive consan; g, f and p are real coninuous funcions and are defined on R; τ(), p() are periodic wih period ω. he main purpose of his paper is o provide new sufficien condiions for guaraneeing he exisence and uniqueness of a A. M. A. Abou-El-Ela is wih he Deparmen of Mahemaics, Faculy of Science, Assiu Universiy, Assiu 756, Egyp. (A El Ela@aun.edu.eg). A. I. Sadek is wih he Deparmen of Mahemaics, Faculy of Science, Assiu Universiy, Assiu 756, Egyp. (Sadeka96@homail.com). A. M. Mahmoud is wih he Deparmen of Science and Mahemaics, Faculy of Educaion, Assiu Universiy, New Valley, El-khargah 72, Egyp. (mah ayman27@yahoo.com). -periodic soluion o hird-order delay differenial equaion wih wo deviaing argumens x (3) () + ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ())) = p(), where ψ, f, τ, τ 2, p : R R and g, g 2 : R R R are coninuous funcions; τ, τ 2 and p are -periodic, g and g 2 are -periodic in heir firs argumen and >. II. PRELIMINARIES For convenience we define x k = ( x() k d) k, k, x = max [, ] x(), p = max [, ] p() and p = p()d. Le X = {x x C 2 (R, R), x( + ) = x(), forall R} and Y = {y y C(R, R), y( + ) = y(), forall R} be wo Banach spaces wih he norms x X = max{ x, ẋ, ẍ } and y Y = y. Define a linear operaor L : D(L) X Y by seing D(L) = {x x X, x (3) () C(R, R)}, and for x D(L), () Lx = x (3) (). (2) We also define a nonlinear operaor N : X Y by seing Nx = ψ(ẋ())ẍ() f(x())ẋ() g (, x( τ ())) g 2 (, x( τ 2 ())) + p(). hen we noice ha KerL = R, dim(kerl) = ; ImL = {y y Y, y(s)ds = } is a subse of Y and dim(y/iml) =, which imply codim(iml) = dim(kerl). hus he operaor L is a Fredholm operaor wih index zero. Define he coninuous projecors P : X KerL and Q : Y Y/ImL by P x() = x() = x( ) and Qy() = y(s)ds, hence ImP = ImQ = KerL = R and KerQ = ImL. Le L P := L D(L) KerP : D(L) KerP ImL, hen L P has a coninuous inverse L P on ImL defined by (L P y)() = { G(s, )y(s)ds, s G(s, ) = ( ), s ; ( s), s. (3) (Advance online publicaion: 27 February 22)

2 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 By using Ascoli-Arzela heorem we from he above equaion, ha L P (I Q)N( Ω) is compac. On he oher hand, QN( Ω) is bounded by coninuiy of funcion QN. hus N is L-compac on Ω, where Ω is an open bounded subse in X. In view of (2) and (3) he operaor equaion Lx = λnx, λ (, ); is equivalen o he following equaion x (3) () + λ{ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ()))} = λp(). he following Mawhin s coninuaion heorem is useful in obaining he exisence of -periodic soluion of () [2]. heorem 2.: (Mawhin s coninuaion heorem) Le X and Y be wo Banach spaces. Suppose ha L : D(L) X Y is a Fredholm operaor wih index zero and N : X Y is L-compac on Ω, where Ω is an open bounded subse in X. Moreover assume ha all he following condiions are saisfied: (a) Lx λnx, forall x Ω D(L), λ (, ); (b) Nx ImL, forall x Ω KerL; (c) he Brower degree deg{qn, Ω KerL, }. hen Lx = Nx has a leas one soluion on Ω D(L). Lemma 2.: I is convenien o inroduce he following assumpions: (i) Assume ha here exis non-negaive consans a, a 2, b, b 2, c and c 2 such ha ψ(y) a, ψ(y ) ψ(y 2 ) a 2 y y 2 for all y, y, y 2 R, f(x) c, f(x ) f(x 2 ) c 2 x x 2 for all x, x, x 2 R and g i (, u) g i (, v) b i u v for all, u, v R, i =, 2. (ii) Suppose ha he following condiions are saisfied: (H ) One of he following condiions holds () (g i (, u) g i (, v))(u v) > for all, u, v R, u v, i =, 2, (2) (g i (, u) g i (, v))(u v) < for all, u, v R, u v, i =, 2; (H 2 ) here exiss d > such ha one of he following condiions holds () x{g (, x) + g 2 (, x) p} > for all R, x > d, (2) x{g (, x) + g 2 (, x) p} < for all R, x > d; If x() is a -periodic soluion of (4), hen (4) x d + 2 ẋ 2. (5) (iii) Assume ha (i) and (ii) hold such ha a 2 + c (b + b 2 ) 3 8 <. (6) If x() is a -periodic soluion of (), hen ẍ [(b +b 2 )d+max{ g (,) + g 2 (,) : }+ p ] 2{ a 2 c 2 4 (b +b 2 ) 3 := k. (7) (iv) Suppose ha (i) (iii) hold. Also le he following condiion holds ( ) a 2 + a 2 k + c 2 4 ( ) (8) + b + b 2 + c 2 k <. hen () has a mos one -periodic soluion. Proof of (ii). Le x() be an arbirary -periodic soluion of (4). hen by inegraing (4) from o we {g (, x( τ ())) + g 2 (, x( τ 2 ())) (9) p()}d =, which implies ha here exiss R such ha g (, x( τ ( ))) + g 2 (, x( τ 2 ( ))) p =. () Nex we show ha he following claim is rue. Claim. If x() is a -periodic soluion of (4), hen here exiss 2 R such ha x( 2 ) d. () Assume by he way of conradicion ha () does no hold. hen x() > d forall R, (2) which ogeher wih (H ), (H 2 ) and () imply ha one of he following four relaions holds: x( τ ( )) > x( τ 2 ( )) > d, (3) x( τ 2 ( )) > x( τ ( )) > d, (4) x( τ ( )) < x( τ 2 ( )) < d, (5) x( τ 2 ( )) < x( τ ( )) < d. (6) Suppose ha (3) holds, in view of (H )(), (H )(2), (H 2 )() and (H 2 )(2) we consider four cases as follows: Case I: If (H )() and (H 2 )() hold, according o (3) we < g (, x( τ 2 ( ))) + g 2 (, x( τ 2 ( ))) p < g (, x( τ ( ))) + g 2 (, x( τ 2 ( ))) p, which conradics (). hus () is rue. Case II: If (H )(2) and (H 2 )() hold, according o (3) we < g (, x( τ ( ))) + g 2 (, x( τ ( ))) p < g (, x( τ ( ))) + g 2 (, x( τ 2 ( ))) p, which conradics (). hus () is rue. Case III: If (H )() and (H 2 )(2) hold, according o (3) we > g (, x( τ ( ))) + g 2 (, x( τ ( ))) p > g (, x( τ ( ))) + g 2 (, x( τ 2 ( ))) p, which conradics (). hus () is rue. Case IV: If (H )(2) and (H 2 )(2) hold, according o (3) we > g (, x( τ 2 ( ))) + g 2 (, x( τ 2 ( ))) p > g (, x( τ ( ))) + g 2 (, x( τ 2 ( ))) p, (Advance online publicaion: 27 February 22)

3 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 which conradics (). hus () is rue. Suppose (4) (or(5), or(6)) holds; using he mehods similarly o hose used in Case I-IV we can show ha () is rue. his complees he proof of he above claim. Le 2 = m +, where [, ] and m is an ineger, hen from () and for any [, + ] we obain Also x() = x( ) + ẋ(s)ds < d + ẋ(s) ds. x() = x( + ) + ẋ(s)ds + d + + ẋ(s) ds. By combining he above wo inequaliies we find x() d + 2 ẋ(s) ds. Using he Cauchy-Schwarz inequaliy yields x() d + 2 ( ẋ(s) 2 ds) 2 = d + 2 ẋ 2. herefore we x = max [, ] x() d + 2 ẋ 2. his complees he proof of condiion (ii) in Lemma 2.. Proof of (iii). Le x() be a -periodic soluion of () for a cerain λ (, ). Muliplying () by x (3) () and hen inegraing i over [, ] and by using condiion (i), we find x (3) () 2 2 a ẍ() x(3) () d +c ẋ() x(3) () d + p() x(3) () d + { g (, x( τ )) g (, ) + g (, ) } x (3) () d + { g 2(, x( τ 2 )) g 2 (, ) + g 2 (, ) } x (3) () d. hen we ge x (3) () 2 2 a ẍ() x(3) () d +c ẋ() x(3) () d +b x( τ ()) x (3) () d + g (, ) x (3) () d +b 2 x( τ 2()) x (3) () d + g 2(, ) x (3) () d + p() x(3) () d a ẍ() x(3) () d +c ẋ() x(3) () d +b x x(3) () d + p x(3) () d +b 2 x x(3) () d + max{ g (, ) + g 2 (, ) : } x(3) () d. hus from (5) we obain x (3) () 2 2 a ẍ() x(3) () d +c ẋ() x(3) () d + 2 (b + b 2 ) ẋ 2 x(3) () d +[(b + b 2 )d + max{ g (, ) + g 2 (, ) : } + p ] x(3) () d. By using Cauchy-Schwarz inequaliy we find x (3) () 2 2 a ẍ 2 x (3) 2 + c ẋ 2 x (3) (b + b 2 ) ẋ 2 x (3) 2 + [(b + b 2 )d + max{ g (, ) + g 2 (, ) : } + p ] x (3) 2. (7) Since x() = x( ), here exiss a consan ξ [, ] such ha ẋ(ξ) = and Again ẋ() = ẋ(ξ) + ξ ẍ(s)ds ξ ẍ(s) ds, [ξ, + ξ]. (8) ẋ() = ẋ(ξ + ) + ξ+ ẍ(s)ds ẋ(ξ + ) + ξ+ ẍ(s) ds = ξ+ ẍ(s) ds, [, ]. he inequaliies (8) and (9) imply 2 ẋ() ξ ẍ(s) ds + ξ+ ẍ(s) ds = ẍ(s) ds. herefore by using Cauchy-Schwarz inequaliy we so (9) ẋ() 2 ( ẍ(s) 2 ds) 2, forall [, ], (2) ẋ 2 ẍ 2, (2) ẋ 2 max [, ] ẋ(s) 2 ( ẍ(s) 2 ds) 2 = 2 ẍ 2. (22) Since x() is periodic funcion for [, ] and by using he above similar echnique, we find ẍ() 2 x(3) () d. Which ogeher wih Cauchy-Schwarz inequaliy imply ẍ 2 ( x(3) (s) 2 ds) 2 = 2 x (3) 2, ẍ 2 max [, ] ẍ(s) 2 x(3) (s) ds 2 x(3) 2. By subsiuing from (24) in (22) we ge (23) (24) ẋ x (3) 2. (25) By subsiuing from (24) in (2) we From (5) and (25) we obain ẋ x (3) 2. (26) x d x (3) 2. (27) hen by subsiuing from (24) and (25) in (7) we find x (3) 2 2 a 2 x(3) c 2 4 x(3) (b + b 2 ) 3 8 x(3) 2 2 +[(b + b 2 )d + max{ g (, ) + g 2 (, ) : } + p ] x (3) 2. herefore we obain x (3) 2 [(b+b2)d+max{ g(,) + g2(,) : }+ p ] a 2 c 2 4 (b+b2) 3 8 By subsiuing from (28) in (23) and (26) we find ẍ [(b +b 2 )d+max{ g (,) + g 2 (,) : }+ p ] := k, 2{ a 2 c 2 4 (b +b 2 ) 3. (28) (29) (Advance online publicaion: 27 February 22)

4 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 ẋ [(b +b 2 )d+max{ g (,) + g 2 (,) : }+ p ] 2 4{ a 2 c 2 4 (b +b 2 ) 3 := 2 k. (3) his complees he proof of condiion (iii) in Lemma 2.. Proof of (iv). Suppose ha x () and x 2 () are wo - periodic soluions of (), hen we x (3) Se () x(3) 2 () + ψ(ẋ ())ẍ () ψ(ẋ 2 ())ẍ 2 () +f(x ())ẋ () f(x 2 ())ẋ 2 () +g (, x ( τ ())) g (, x 2 ( τ ())) +g 2 (, x ( τ 2 ())) g 2 (, x 2 ( τ 2 ())) =. hen we find z() = x () x 2 (). z (3) () + {ψ(ẋ ())ẍ () ψ(ẋ 2 ())ẍ 2 ()} +{f(x ())ẋ () f(x 2 ())ẋ 2 ()} +{g (, x ( τ ())) g (, x 2 ( τ ()))} +{g 2 (, x ( τ 2 ())) g 2 (, x 2 ( τ 2 ()))} =. (3) Since x () and x 2 () are wo -periodic soluions, by inegraing (3) over [, ] we obain {g (, x ( τ ())) g (, x 2 ( τ ())) +g 2 (, x ( τ 2 ())) g 2 (, x 2 ( τ 2 ()))}d =. By using he inegral mean-value heorem, i follows ha here exiss a consan γ [, ] such ha g (γ, x (γ τ (γ))) g (γ, x 2 (γ τ (γ))) +g 2 (γ, x (γ τ 2 (γ))) g 2 (γ, x 2 (γ τ 2 (γ))) =. Bu (H ) and (32) imply z(γ τ (γ))z(γ τ 2 (γ)) = (x (γ τ (γ)) x 2 (γ τ (γ)))(x (γ τ 2 (γ)) x 2 (γ τ 2 (γ))). (32) Since z() = x () x 2 () is a coninuous funcion in R, i follows ha here exiss ξ R such ha z(ξ) =. Se ξ = n + γ, where γ [, ] and n is an ineger hen we ge z( γ) = z(n + γ) = z(ξ) =. (33) hus for any [ γ, γ + ] we obain and z() = z( γ) + γ ż(s)ds γ ż(s) ds, z() = z( γ + ) + γ+ ż(s)ds γ+ ż(s) ds. Combining hese wo inequaliies and using Cauchy-Schwarz inequaliy yield 2 z() γ+ γ ż(s) ds = ż(s) ds ( ż(s) 2 ds) 2 = ż 2. herefore z 2 ż 2. (34) Muliplying (3) by z (3) () and hen by inegraing i over [, ] i follows z (3) () 2 2 = {ψ(ẋ ())ẍ () ψ(ẋ 2 ())ẍ 2 ()}z (3) ()d {f(x ())ẋ () f(x 2 ())ẋ 2 ()}z (3) ()d {g (, x ( τ ())) g (, x 2 ( τ ()))}z (3) ()d {g 2(, x ( τ ())) g 2 (, x 2 ( τ ()))}z (3) ()d. From (i) we ge z (3) () 2 2 ψ(ẋ ()) ẍ () ẍ 2 ()) z (3) () d + ψ(ẋ ()) ψ(ẋ 2 ()) ẍ 2 ()) z (3) () d + f(x ()) ẋ () ẋ 2 ()) z (3) () d + f(x ()) f(x 2 ()) ẋ 2 ()) z (3) () d +b x ( τ ()) x 2 ( τ ()) z (3) () d +b 2 x ( τ 2 ()) x 2 ( τ 2 ()) z (3) () d a z() z(3) () d + a 2 ż() ẍ 2() z (3) () d +c ż() z(3) () d + c 2 z() ẋ 2() z (3) () d +b z( τ ()) z (3) () d +b 2 z( τ 2()) z (3) () d. By using Cauchy-Schwarz inequaliy we z (3) 2 2 a z 2 z (3) 2 + a 2 ż 2 ẍ 2 z (3) 2 +c ż 2 z (3) 2 + c 2 z ẋ 2 z (3) 2 +b z z (3) 2 + b 2 z z (3) 2. From (25), (26), (27), (29) and (3) we obain z (3) a z (3) a 2k 2 z (3) c 2 z (3) c 2k 4 z (3) (b + b 2 ) 3 z (3) 2 2 {a 2 + (a 2k + c ) (b + b 2 + c 2 k 2 ) 3 z(3) 2 2. Since z(), ż(), z() and z (3) () are coninuous -periodic funcions, by (iv), (22), (33), (34) and he above inequaliy we z() ż() z() z (3) () =, forall R. hus x () x 2 (), for all R. Hence () has a mos one -periodic soluion. his complees he proof of condiion (iv) in Lemma 2.. So he proof of Lemma 2.2 is now complee. III. MAIN RESUL heorem 3.: Suppose ha (i) and (iv) hold, hen () has a unique -periodic soluion. Proof. Condiion (iv) of Lemma 2. saes ha () has a mos one -periodic soluion. hus o prove heorem 3. i suffices o show ha () has a leas one -periodic soluion. o do his, we shall apply heorem 2.. Firs we will claim ha he se of all possible -periodic soluions of (4) is bounded. Le x() be a -periodic soluion of (4). Muliplying (4) by x (3) () and hen by inegraing i over [, ] we obain x(3) () 2 d = λ ψ(ẋ())ẍ()x(3) ()d λ f(x())ẋ()x(3) ()d λ g (, x( τ ()))x (3) ()d λ g 2(, x( τ 2 ()))x (3) ()d + λ p()x(3) ()d. (Advance online publicaion: 27 February 22)

5 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 In view of (i), (5) and he inequaliy of Cauchy-Schwarz we x (3) 2 2 {a 2 + c (b + b ) 3 x(3) 2 2 +[(b + b )d + max{ g (, ) + g 2 (, ) : } + p ] x (3) 2, which ogeher wih condiion (iii) imply ha here exiss M > such ha x (3) 2 < M. his ogeher wih (23), (26) and (27) leads o ẍ 2 M, ẋ M, x d M. Le M = max{d M, M, 2 M }, hen we Ω = {x x X, x < M} as a non-empy open bounded subse of X. So condiion (a) in heorem 2. holds. In view of (H 2 )() and (H 2 )(2) we will consider wo cases: Case (i): If (H 2 )() holds. Since QNx = {ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ())) p}d, for any x Ω KerL = Ω R, hen x is a consan wih x() = M or x() = M. hen QN(M) = {g (, M) + g 2 (, M) p}d <, QN( M) = {g (35) (, M) + g 2 (, M) p}d >, which implies ha condiion (b) of heorem 2. is saisfied. Furhermore define a coninuous funcion H(x, µ) by seing H(x, µ) = µx + ( µ)qnx = µx ( µ) {ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ())) p}d, in view of (35) we ge xh(x, µ) < for all x Ω KerL and µ [, ]. hus H(x, µ) is a homoopic ransformaion. By using he homoopy invariance heorem we deg{qn, Ω KerL, } = deg{ x, Ω KerL, }, so condiion (c) of heorem 2. is saisfied. Case (ii): If (H 2 )(2) holds. Since QNx = {ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ())) p}d, for any x Ω KerL = Ω R, x() = M or x() = M, we obain QN(M) = {g (, M) + g 2 (, M) p}d >, QN( M) = {g (36) (, M) + g 2 (, M) p}d <, which implies ha condiion (b) of heorem 2. is saisfied. Define H(x, µ) = µx + ( µ)qnx = µx ( µ) {ψ(ẋ())ẍ() + f(x())ẋ() +g (, x( τ ())) + g 2 (, x( τ 2 ())) p}d, in view of (36) we ge xh(x, µ) > for all x Ω KerL and µ [, ]. hus H(x, µ) is a homoopic ransformaion. By using he homoopy invariance heorem we deg{qn, Ω KerL, } = deg{x, Ω KerL, } =, so condiion (c) of heorem 2. is saisfied. herefore i follows from heorem 2. ha () has a leas one -periodic soluion. his complees he proof. IV. EXAMPLE Example 4.. In his secion we apply he main resul obained in he previous secion o an example. Consider he exisence and uniqueness of a π-periodic soluion of he hird-order delay differenial equaion wih wo deviaing argumens where x (3) () + 2 (sin ẋ)ẍ() + 6 (cos x)ẋ() +g (, x( cos 2)) + g 2 (, x( sin 2)) (37) = π cos 2, = π, τ () = cos 2, τ 2 () = sin 2, g (, x) = 8π(+cos 2 ) an x, g 2 (, x) = 2π ( + sin2 ) an x and p() = π cos 2. By (37) we find noicing ha a = a 2 = 2, b = 8π, b 2 = c = c 2 = 6, p = p()d = π p = π, π 6π, π cos 2d =, we can ge d =, (d is an arbirary small posiive consan). hen we can obain [(b +b 2)d+max{ g (,) + g 2(,) : }+ p ] 2{ a 2 c 2 4 (b+b2) 3 {( = 8π + 6π ) + π }π 2{ π 2 2 π ( 8π + 6π ) π3 := k =.68, ( ) a 2 + a 2 k + c 2 ( = π =.62 <. 4 (b + + b 2 + c 2 k 2 ) ( π π + ) 3 8 6π ) π π I is obvious ha all he assumpions (ii)-(iv) hold. Hence by heorem 3., equaion (37) has a unique π- periodic soluion. REFERENCES []. A. Buron, Sabiliy and Periodic Soluions of Ordinary and Funcional Differenial Equaions, Academic Press, 985. [2] R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differenial Equaions, Lecure Noes in Mah., Springer-Verlag, Berlin, 568, 977. [3] Z. Gui, Exisence of posiive periodic soluions o hird-order delay differenial equaions, Elec. J. of Diff. Eqs., 9, Z. Gui, Exisence of posiive periodic soluions o hird-order delay differenial equaions, Elec. J. of Diff. Eqs., 9, -7, 26, -7. [4] J. K. Hale, heory of Funcional Differenial Equaions, Springer- Verlag, New York, 977. (Advance online publicaion: 27 February 22)

6 IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 [5] J. K. Hale and S. M. Verduyn Lunel, Inroducion o Funcional Differenial Equaions, Springer-Verlag, New York, 993. [6] V. Kolmanovskii and A. Myshkis, Inroducion o he heory and applicaions of funcional differenial equaions, Kluwer Academic Publishers, Dordrech, 999 [7] B. Liu and L. Huang, Periodic soluions for a kind of Rayleigh equaion wih a deviaing argumen, J. Mah. Anal. Appl., 32, 26, [8] B. Liu and L. Huang, Exisence and uniqueness of periodic soluion for a kind of firs order neural funcional differenial equaions, J. Mah. Anal. Appl., 322,, 26, [9] B. Liu and L. Huang, Exisence and uniqueness of periodic soluions for a kind of firs order neural funcional differenial equaions wih a deviaing argumen, aiwanese Journal of Mahemaics, (2), 27, [] J. Shao, L. Wang, Y. Yu and J. Zhou, Periodic soluions for a kind of Liénard equaion wih a deviaing argumen, J. compu. Appl. Mah., 228, 29, [] H. O. ẹjumọla and B. chegnani, Sabiliy, boundedness and exisence of periodic soluions of some hird and fourh-order nonlinear delay differenial equaions, J. Nigerian Mah. Soc., 9, 2, 9-9. [2] Y. Wang and J. ian, periodic soluions for a Liénard equaion wih wo deviaing argumens, Elec. J. of Diff. Eqs., 4, 29, -2. [3] Y. Wu, B. Xiao and H. Zhang, Periodic soluions for a kind of Rayleigh equaion wih wo deviaing argumens, Elec. J. of Diff. Eqs., 7, 26, -. [4] J. Xiao and B.Liu, Exisence and uniqueness of periodic soluions for firs-order neural funcional differenial equaions wih wo deviaing argumens, Elec. J. of Diff. Eqs., 7, 26, -. [5] Q. Zhou, F. Long, Exisence and uniqueness of periodic soluions for a kind of Liénard equaion wih wo deviaing argumens, J. Compu. Appl. Mah., 26, 27, [6] Q. Zhou, B. Xiao, Y. Yu, B. Liu and L. Huang, Exisence and uniqueness of periodic soluions for a kind of Rayleigh equaion wih a deviaing argumen, J. Korean Mah. Soc., 44(3), 27, [7] Y. Zhu, On sabiliy, boundedness and exisence of periodic soluion of a kind of hird-order nonlinear delay differenial sysem, Ann. of Diff. Eqs., 8(2), 992, (Advance online publicaion: 27 February 22)

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