TO our knowledge, most exciting results on the existence
|
|
- Matilda Fowler
- 6 years ago
- Views:
Transcription
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)
CONTRIBUTION TO IMPULSIVE EQUATIONS
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria
More informationEXISTENCE OF NON-OSCILLATORY SOLUTIONS TO FIRST-ORDER NEUTRAL DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 206 (206, No. 39, pp.. ISSN: 072-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE OF NON-OSCILLATORY SOLUTIONS TO
More informationExistence of non-oscillatory solutions of a kind of first-order neutral differential equation
MATHEMATICA COMMUNICATIONS 151 Mah. Commun. 22(2017), 151 164 Exisence of non-oscillaory soluions of a kind of firs-order neural differenial equaion Fanchao Kong Deparmen of Mahemaics, Hunan Normal Universiy,
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationSTABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS
Elecronic Journal of Differenial Equaions, Vol. 217 217, No. 118, pp. 1 14. ISSN: 172-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS
More informationExistence of positive solution for a third-order three-point BVP with sign-changing Green s function
Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationResearch Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations
Hindawi Publishing Corporaion Boundary Value Problems Volume 11, Aricle ID 19156, 11 pages doi:1.1155/11/19156 Research Aricle Exisence and Uniqueness of Periodic Soluion for Nonlinear Second-Order Ordinary
More informationMonotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type
In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria
More informationExistence Theory of Second Order Random Differential Equations
Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 974-32 Volume 4, Number 3 (22), pp. 33-3 Inernaional Research Publicaion House hp://www.irphouse.com Exisence Theory of Second Order Random
More informationPositive continuous solution of a quadratic integral equation of fractional orders
Mah. Sci. Le., No., 9-7 (3) 9 Mahemaical Sciences Leers An Inernaional Journal @ 3 NSP Naural Sciences Publishing Cor. Posiive coninuous soluion of a quadraic inegral equaion of fracional orders A. M.
More informationOn Oscillation of a Generalized Logistic Equation with Several Delays
Journal of Mahemaical Analysis and Applicaions 253, 389 45 (21) doi:1.16/jmaa.2.714, available online a hp://www.idealibrary.com on On Oscillaion of a Generalized Logisic Equaion wih Several Delays Leonid
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationEssential Maps and Coincidence Principles for General Classes of Maps
Filoma 31:11 (2017), 3553 3558 hps://doi.org/10.2298/fil1711553o Published by Faculy of Sciences Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma Essenial Maps Coincidence
More informationBoundedness and Stability of Solutions of Some Nonlinear Differential Equations of the Third-Order.
Boundedness Sabili of Soluions of Some Nonlinear Differenial Equaions of he Third-Order. A.T. Ademola, M.Sc. * P.O. Arawomo, Ph.D. Deparmen of Mahemaics Saisics, Bowen Universi, Iwo, Nigeria. Deparmen
More informationExistence of positive solutions for second order m-point boundary value problems
ANNALES POLONICI MATHEMATICI LXXIX.3 (22 Exisence of posiive soluions for second order m-poin boundary value problems by Ruyun Ma (Lanzhou Absrac. Le α, β, γ, δ and ϱ := γβ + αγ + αδ >. Le ψ( = β + α,
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationExistence of multiple positive periodic solutions for functional differential equations
J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationA Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero
JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, 7887 1997 ARICLE NO. AY965143 A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationCERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, 67 76 DOI: 10.5644/SJM.10.1.09 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien
More informationEXISTENCE OF TRIPLE POSITIVE PERIODIC SOLUTIONS OF A FUNCTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER
EXISTENCE OF TRIPLE POSITIVE PERIODIC SOLUTIONS OF A FUNCTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER XI-LAN LIU, GUANG ZHANG, AND SUI SUN CHENG Received 15 Ocober 2002 We esablish he exisence
More informationAn Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation
Commun Theor Phys Beijing, China 43 2005 pp 591 596 c Inernaional Academic Publishers Vol 43, No 4, April 15, 2005 An Invariance for 2+1-Eension of Burgers Equaion Formulae o Obain Soluions of KP Equaion
More informationThe Existence, Uniqueness and Stability of Almost Periodic Solutions for Riccati Differential Equation
ISSN 1749-3889 (prin), 1749-3897 (online) Inernaional Journal of Nonlinear Science Vol.5(2008) No.1,pp.58-64 The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for Riccai Differenial Equaion
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationDIFFERENTIAL EQUATIONS
ne. J. Ma. Mah. Vo1. {1978)1-1 BEHAVOR OF SECOND ORDER NONLNEAR DFFERENTAL EQUATONS RNA LNG Deparmen of Mahemaics California Sae Universiy Los Angeles, California 93 (Received November 9, 1977 and in revised
More informationANTIPERIODIC SOLUTIONS FOR nth-order FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY
Elecronic Journal of Differenial Equaions, Vol. 216 (216), No. 44, pp. 1 8. ISSN: 172-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu ANTIPERIODIC SOLUTIONS FOR nth-order
More informationPOSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION
Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial
More informationSolution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
More informationOn the Positive Periodic Solutions of the Nonlinear Duffing Equations with Delay and Variable Coefficients
On he Posiive Periodic Soluions of he Nonlinear Duffing Equaions wih Delay Variable Coefficiens Yuji Liu Weigao Ge Absrac We consider he exisence nonexisence of he posiive periodic soluions of he non-auonomous
More informationJUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k
Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS
More informationL p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity
ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.
More informationBoundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms
Advances in Dynamical Sysems and Applicaions. ISSN 0973-531 Volume Number 1 007, pp. 107 11 Research India Publicaions hp://www.ripublicaion.com/adsa.hm Boundedness and Exponenial Asympoic Sabiliy in Dynamical
More informationSome New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations
Annals of Pure and Applied Mahemaics Vol. 6, No. 2, 28, 345-352 ISSN: 2279-87X (P), 2279-888(online) Published on 22 February 28 www.researchmahsci.org DOI: hp://dx.doi.org/.22457/apam.v6n2a Annals of
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
More informationOn the probabilistic stability of the monomial functional equation
Available online a www.jnsa.com J. Nonlinear Sci. Appl. 6 (013), 51 59 Research Aricle On he probabilisic sabiliy of he monomial funcional equaion Claudia Zaharia Wes Universiy of Timişoara, Deparmen of
More informationEIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES
Elecronic Journal of Differenial Equaions, Vol. 27 (27, No. 37, pp. 3. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON
More informationSTABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES
Novi Sad J. Mah. Vol. 46, No. 1, 2016, 15-25 STABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES N. Eghbali 1 Absrac. We deermine some sabiliy resuls concerning
More informationA remark on the H -calculus
A remark on he H -calculus Nigel J. Kalon Absrac If A, B are secorial operaors on a Hilber space wih he same domain range, if Ax Bx A 1 x B 1 x, hen i is a resul of Auscher, McInosh Nahmod ha if A has
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationNonlinear Fuzzy Stability of a Functional Equation Related to a Characterization of Inner Product Spaces via Fixed Point Technique
Filoma 29:5 (2015), 1067 1080 DOI 10.2298/FI1505067W Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma Nonlinear Fuzzy Sabiliy of a Funcional
More informationOlaru Ion Marian. In 1968, Vasilios A. Staikos [6] studied the equation:
ACTA UNIVERSITATIS APULENSIS No 11/2006 Proceedings of he Inernaional Conference on Theory and Applicaion of Mahemaics and Informaics ICTAMI 2005 - Alba Iulia, Romania THE ASYMPTOTIC EQUIVALENCE OF THE
More informationEngineering Letter, 16:4, EL_16_4_03
3 Exisence In his secion we reduce he problem (5)-(8) o an equivalen problem of solving a linear inegral equaion of Volerra ype for C(s). For his purpose we firs consider following free boundary problem:
More informationPOSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER
POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER GUANG ZHANG AND SUI SUN CHENG Received 5 November 21 This aricle invesigaes he exisence of posiive
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationEXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 29(29), No. 49, pp. 2. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN
More informationOn Gronwall s Type Integral Inequalities with Singular Kernels
Filoma 31:4 (217), 141 149 DOI 1.2298/FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Gronwall s Type Inegral Inequaliies
More informationAsymptotic instability of nonlinear differential equations
Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy
More informationOmega-limit sets and bounded solutions
arxiv:3.369v [mah.gm] 3 May 6 Omega-limi ses and bounded soluions Dang Vu Giang Hanoi Insiue of Mahemaics Vienam Academy of Science and Technology 8 Hoang Quoc Vie, 37 Hanoi, Vienam e-mail: dangvugiang@yahoo.com
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
More informationL 1 -Solutions for Implicit Fractional Order Differential Equations with Nonlocal Conditions
Filoma 3:6 (26), 485 492 DOI.2298/FIL66485B Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma L -Soluions for Implici Fracional Order Differenial
More informationOscillation of solutions to delay differential equations with positive and negative coefficients
Elecronic Journal of Differenial Equaions, Vol. 2000(2000), No. 13, pp. 1 13. ISSN: 1072-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp ejde.mah.sw.edu fp ejde.mah.un.edu (login: fp) Oscillaion
More informationOn the Solutions of First and Second Order Nonlinear Initial Value Problems
Proceedings of he World Congress on Engineering 13 Vol I, WCE 13, July 3-5, 13, London, U.K. On he Soluions of Firs and Second Order Nonlinear Iniial Value Problems Sia Charkri Absrac In his paper, we
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationImproved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method
Journal of Applied Mahemaics & Bioinformaics, vol., no., 01, 1-14 ISSN: 179-660 (prin), 179-699 (online) Scienpress Ld, 01 Improved Approimae Soluions for Nonlinear Evoluions Equaions in Mahemaical Physics
More informationEXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2 ISSN 223-727 EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS Yuji Liu By applying monoone ieraive meho,
More information11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu
ON EQUATIONS WITH SETS AS UNKNOWNS BY PAUL ERDŐS AND S. ULAM DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER Communicaed May 27, 1968 We shall presen here a number of resuls in se heory concerning
More informationLINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS
LINEAR INVARIANCE AND INTEGRAL OPERATORS OF UNIVALENT FUNCTIONS MICHAEL DORFF AND J. SZYNAL Absrac. Differen mehods have been used in sudying he univalence of he inegral ) α ) f) ) J α, f)z) = f ) d, α,
More informationarxiv: v1 [math.gm] 4 Nov 2018
Unpredicable Soluions of Linear Differenial Equaions Mara Akhme 1,, Mehme Onur Fen 2, Madina Tleubergenova 3,4, Akylbek Zhamanshin 3,4 1 Deparmen of Mahemaics, Middle Eas Technical Universiy, 06800, Ankara,
More informationOn the Oscillation of Nonlinear Fractional Differential Systems
On he Oscillaion of Nonlinear Fracional Differenial Sysems Vadivel Sadhasivam, Muhusamy Deepa, Nagamanickam Nagajohi Pos Graduae and Research Deparmen of Mahemaics,Thiruvalluvar Governmen Ars College (Affli.
More informationSOLUTIONS APPROACHING POLYNOMIALS AT INFINITY TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 2005(2005, No. 79, pp. 1 25. ISSN: 1072-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp SOLUIONS APPROACHING POLYNOMIALS
More informationThe L p -Version of the Generalized Bohl Perron Principle for Vector Equations with Infinite Delay
Advances in Dynamical Sysems and Applicaions ISSN 973-5321, Volume 6, Number 2, pp. 177 184 (211) hp://campus.ms.edu/adsa The L p -Version of he Generalized Bohl Perron Principle for Vecor Equaions wih
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationSobolev-type Inequality for Spaces L p(x) (R N )
In. J. Conemp. Mah. Sciences, Vol. 2, 27, no. 9, 423-429 Sobolev-ype Inequaliy for Spaces L p(x ( R. Mashiyev and B. Çekiç Universiy of Dicle, Faculy of Sciences and Ars Deparmen of Mahemaics, 228-Diyarbakir,
More informationGRADIENT ESTIMATES FOR A SIMPLE PARABOLIC LICHNEROWICZ EQUATION. Osaka Journal of Mathematics. 51(1) P.245-P.256
Tile Auhor(s) GRADIENT ESTIMATES FOR A SIMPLE PARABOLIC LICHNEROWICZ EQUATION Zhao, Liang Ciaion Osaka Journal of Mahemaics. 51(1) P.45-P.56 Issue Dae 014-01 Tex Version publisher URL hps://doi.org/10.18910/9195
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationExistence of Solutions of Three-Dimensional Fractional Differential Systems
Applied Mahemaics 7 8 9-8 hp://wwwscirporg/journal/am ISSN Online: 5-79 ISSN rin: 5-785 Exisence of Soluions of Three-Dimensional Fracional Differenial Sysems Vadivel Sadhasivam Jayapal Kaviha Muhusamy
More informationApplication of variational iteration method for solving the nonlinear generalized Ito system
Applicaion of variaional ieraion mehod for solving he nonlinear generalized Io sysem A.M. Kawala *; Hassan A. Zedan ** *Deparmen of Mahemaics, Faculy of Science, Helwan Universiy, Cairo, Egyp **Deparmen
More informationASYMPTOTIC FORMS OF WEAKLY INCREASING POSITIVE SOLUTIONS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 2007(2007), No. 126, pp. 1 12. ISSN: 1072-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp) ASYMPTOTIC FORMS OF
More informationVanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law
Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing
More informationPERMANENCE (or persistence) is an important property
Engineering Leers, 5:, EL_5 6 Permanence and Exincion of Delayed Sage-Srucured Predaor-Prey Sysem on Time Scales Lili Wang, Pingli Xie Absrac This paper is concerned wih a delayed Leslie- Gower predaor-prey
More informationHeat kernel and Harnack inequality on Riemannian manifolds
Hea kernel and Harnack inequaliy on Riemannian manifolds Alexander Grigor yan UHK 11/02/2014 onens 1 Laplace operaor and hea kernel 1 2 Uniform Faber-Krahn inequaliy 3 3 Gaussian upper bounds 4 4 ean-value
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationOn Two Integrability Methods of Improper Integrals
Inernaional Journal of Mahemaics and Compuer Science, 13(218), no. 1, 45 5 M CS On Two Inegrabiliy Mehods of Improper Inegrals H. N. ÖZGEN Mahemaics Deparmen Faculy of Educaion Mersin Universiy, TR-33169
More information1 Solutions to selected problems
1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen
More informationON THE OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS. Cairo University, Orman, Giza 12221, Egypt
a 1/α s)ds < Indian J. pre appl. Mah., 396): 491-507, December 2008 c Prined in India. ON THE OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS SAID R. GRACE 1, RAVI P. AGARWAL 2 AND MUSTAFA
More informationOn Carlsson type orthogonality and characterization of inner product spaces
Filoma 26:4 (212), 859 87 DOI 1.2298/FIL124859K Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Carlsson ype orhogonaliy and characerizaion
More informationHomotopy Perturbation Method for Solving Some Initial Boundary Value Problems with Non Local Conditions
Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, 23-25 Ocober, 23, San Francisco, USA Homoopy Perurbaion Mehod for Solving Some Iniial Boundary Value Problems wih
More informationHILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS
HILLE AND NEHARI TYPE CRITERIA FOR THIRD-ORDER DYNAMIC EQUATIONS L. ERBE, A. PETERSON AND S. H. SAKER Absrac. In his paper, we exend he oscillaion crieria ha have been esablished by Hille [15] and Nehari
More informationOSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES
Dynamic Sysems and Applicaions 6 (2007) 345-360 OSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES S. H. SAKER Deparmen of Mahemaics and Saisics, Universiy of Calgary,
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationDifferential Harnack Estimates for Parabolic Equations
Differenial Harnack Esimaes for Parabolic Equaions Xiaodong Cao and Zhou Zhang Absrac Le M,g be a soluion o he Ricci flow on a closed Riemannian manifold In his paper, we prove differenial Harnack inequaliies
More informationSUFFICIENT CONDITIONS FOR EXISTENCE SOLUTION OF LINEAR TWO-POINT BOUNDARY PROBLEM IN MINIMIZATION OF QUADRATIC FUNCTIONAL
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, Series A, OF HE ROMANIAN ACADEMY Volume, Number 4/200, pp 287 293 SUFFICIEN CONDIIONS FOR EXISENCE SOLUION OF LINEAR WO-POIN BOUNDARY PROBLEM IN
More informationOSCILLATION OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS
Elecronic Journal of Qualiaive Theory of Differenial Equaions 2010, No. 43, 1-10; hp://www.mah.u-szeged.hu/ejqde/ OSCILLATION OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS B. BACULÍKOVÁ AND J. DŽURINA
More informationCHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR
Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 31, 2006, 39 46 CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Joaquim Marín and Javier
More informationOn two general nonlocal differential equations problems of fractional orders
Malaya Journal of Maemaik, Vol. 6, No. 3, 478-482, 28 ps://doi.org/.26637/mjm63/3 On wo general nonlocal differenial equaions problems of fracional orders Abd El-Salam S. A. * and Gaafar F. M.2 Absrac
More informationApproximating positive solutions of nonlinear first order ordinary quadratic differential equations
Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/233835.25.2367 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Approximaing posiive soluions of nonlinear firs order ordinary quadraic
More informationTHE FINITE HAUSDORFF AND FRACTAL DIMENSIONS OF THE GLOBAL ATTRACTOR FOR A CLASS KIRCHHOFF-TYPE EQUATIONS
European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN 59-995 HE FINIE HAUSDORFF AND FRACAL DIMENSIONS OF HE GLOBAL ARACOR FOR A CLASS KIRCHHOFF-YPE EQUAIONS Guoguang Lin & Xiangshuang Xia Deparen
More informationA Sharp Existence and Uniqueness Theorem for Linear Fuchsian Partial Differential Equations
A Sharp Exisence and Uniqueness Theorem for Linear Fuchsian Parial Differenial Equaions Jose Ernie C. LOPE Absrac This paper considers he equaion Pu = f, where P is he linear Fuchsian parial differenial
More informationEfficient Solution of Fractional Initial Value Problems Using Expanding Perturbation Approach
Journal of mahemaics and compuer Science 8 (214) 359-366 Efficien Soluion of Fracional Iniial Value Problems Using Expanding Perurbaion Approach Khosro Sayevand Deparmen of Mahemaics, Faculy of Science,
More informationarxiv: v1 [math.pr] 19 Feb 2011
A NOTE ON FELLER SEMIGROUPS AND RESOLVENTS VADIM KOSTRYKIN, JÜRGEN POTTHOFF, AND ROBERT SCHRADER ABSTRACT. Various equivalen condiions for a semigroup or a resolven generaed by a Markov process o be of
More informationA New Perturbative Approach in Nonlinear Singularity Analysis
Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The
More information