On the Oscillation of Nonlinear Fractional Differential Systems

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1 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. o Periyar Universiy),Rasipuram, Namakkal D. Tamil Nadu, India. ABSTRACT: In his aricle, we obain he oscillaory behavior of he soluions of fracional order differenial sysem of he form u() = a f(v()), v() = b g s u s, where < <. By using generalized Riccai ransformaion and inequaliy echnique, we will esablish sufficien condiions for he oscillaion of soluions of given sysem. Examples are given o illusrae he main resuls. MATHEMATICS SUBJECT CLASSIFICATION: 34A8, 34A34, 34K. KEYWORDS: Oscillaion, Soluion, Coupled sysem, Fracional differenial equaion. I. INTRODUCTION As he qualiaive heory of nonlinear sysems of differenial equaions originaed by Henri Poincare a he end of nineeenh cenury. The sudy of oscillaion of differenial equaions is one of he radiional ren in he qualiaive heory of differenial equaions. In he recen years, he number of invesigaions devoed o he oscillaion heory of he funcional differenial equaions has increased considerably, see he monographs [, 6, 7, 4, ], ordinary differenial sysem have been sudied by many auhors [, 5, 6, 8, 5] by differen meho. During he las wo decades, fracional differenial equaions have progressively fascinaed by many auhors. The budding ineresing in fracional differenial equaions is due o is effeciveness and applicabiliy o miscellaneous branches of science and engineering. A rigorous and encyclopedic sudy of fracional differenial equaions can be found in [, 4, 5,,,, 3, 4, 6] and in papers [3, 8, 9, 7, ]. In [9], Lomaidze e al. sudied he oscillaion and nonoscillaion of wo-dimensional linear differenial sysems of he form u = q v, v = p u,. In [3], Philos e al. invesigaed he oscillaion of nonlinear wo-dimensional differenial sysems of he form x = b g(y ), y = a f(x ),. To he bes of he auhors knowledge, no sudy has concerned wih he oscillaion of linear sysem of fracional order. In paricular, i seems ha here has been no work done on nonlinear fracional differenial sysem. Moivaed by he above observaion, we propose sysem (.). In he presen paper, we consider he oscillaory behavior of he soluions of fracional order differenial sysem of he form Copyrigh o IJARSET 486

2 u() = a f(v()), v() = b g s u s, (.) where < <, denoes he Riemann-Liouville fracional derivaive of order wih respec o. Throughou his paper, we assume ha he following condiions: (A ) a C,,,, a >, wih he condiion a()d = ; (A ) b C,,, and b() is no idenically zero on any inerval of he form τ,, where τ ; A 3 g C R, R, ug u >, g K M > for u, where K = s u s ; (A 4 ) f C R, R, vf(v) > and f v m > for v ; A soluion (u(), v()) o he sysem (.) is oscillaory if i has arbirarily large zeros, and is nonoscillaory oherwise. Sysem (.) is said o be oscillaory if all is soluions are oscillaory oherwise i is nonoscillaory. The main aim of his paper is o presen some new oscillaion crieria for he sysem (.) by using generalized Riccai ransformaion and inequaliy echnique. II. PRELIMINARIES In his secion, we give he definiions of fracional derivaives, inegrals and lemmas which are useful hroughou his paper. The following noaions will be used for he convenience. = Γ( + ), i i = Γ( + ), i =,, p() = p(), w() = w(), a() = a(), b() = b(), q = q, ρ = ρ, ϕ = ϕ, δ() = δ(). Definiion:. [] The Riemann-Liouville fracional inegral of order > of a funcion y: R + R on he half-axis R + is given by I + y : = ( Γ v) y v dv for > (.) provided he righ hand side is poinwise defined on R +, where Γ is he gamma funcion. Definiion:. [] The Riemann-Liouville fracional derivaive of order > of a funcion y: R + R on he halfaxis R + is given by d D + y(): = I d + y () > for > (.) provided he righ hand side is poinwise defined on R +, where is he ceiling funcion of. Lemma:. [] Le K = s y s for (,) and >. (.3) Then K = Γ D + y. III. MAIN RESULTS In his secion, we sudy oscillaory behaviour of (.) under cerain condiions. We begin wih he following lemma. Lemma: 3.. If (u(),v()) is a nonoscillaory soluion of (.), hen u() is always nonoscillaory. Proof. The proof follows from he line of Lemma.in [5]. Theorem: 3. Suppose ha he assumpions (A ) (A 4 ) hold. Assume also ha here exiss a posiive funcion δ C ([, ); R + )) such ha Copyrigh o IJARSET 486

3 δ (s) mb(s)δ(s) 4MΓ a(s) δ(s) =. (3.) Then every soluion of sysem (.) is oscillaory. Proof. Suppose ha (.) has a nonoscillaory soluion of (u(),v()) on [, ). From Lemma 3., u() is always nonoscillaory. Wihou loss of generaliy, we may assume ha u() > for and K() > for. Since he similar argumen hol for he case u() < evenually. Then (.) can be reduced o he following inequaliy, a u + mb g K,, (3.) which implies ha a u <,. (3.3) Thus u or u <,. We claim ha u for. Suppose no, here exiss T such ha u T <. Since by Lemma., we have a u <,. I is clear ha a u < a T u T for T. Therefore, K () = D Γ + u < a() D a T + u T. Inegraing he above inequaliy from T o, we have K = K T + Γ u T a s. a T T Leing, we ge lim K. Which conradics he fac ha D + u for. Define he funcion w() by he generalized Riccai subsiiuion D + u w = δ a g K,. (3.4) Then w() > for. Differeniaing (3.4) wih respec o, and using (3.), (A 3 ), we have D + w δ w mb δ Mδ u D + K. δ a g K Le δ = δ, a = a, b = b and w = w. Then D + w() = w (), D + δ() = δ (), D + K = K, D + u() = u (), and g(k()) = g(k ). Using hese ransformaions in he las inequaliy, by Lemma., we have w δ a w mb δ MΓ δ δ w. (3.5) Then, wih he inequaliy, (λ ) λab λ A λ (λ )B λ. (3.6) We ge, w δ (s) mb δ +, 4MΓ a(s). δ(s) Inegraing boh sides from o, we have δ s w w mb s δ s +. 4MΓ a s δ s Leing, we ge lim w, which conradics he hypohesis (3.). The proof is complee. Corollary: 3. Le he hypohesis of Theorem 3. hold, (3.) can be replaced by Copyrigh o IJARSET

4 and m b s δ s =, (3.7) Then every soluion of (.) is oscillaory. MΓ a(s) δ (s) δ(s) <. (3.8) For he following heorem, we inroduce a class of funcions P. Le D =, s : > s, D =, s : s. The funcion H C(D, R) is said o belong o he class P, if (T ) H (,) = for, H (,s) > for (,s) D. (T ) H has a coninuous and non-posiive parial derivaive H s (, s) and h(,s) = H s, s + H(, s) δ s δ s. Theorem: 3. Suppose ha (A ) (A 4 ) hold and assume also ha here exis H P such ha h (, s) δ s mb(s)h(, s) H(, ) 4MΓ a(s) H(, s) =. (3.9) Then every soluion of (.) is oscillaory. Proof. Proceeding as in he proof of Theorem 3., we ge (3.5). Muliplying (3.5) by H(, s) on boh sides and an inegraion from o, for, using (T ) and (3.6), we obain mb(s)δ s H(, s) H, 4MΓ a(s) H(,s) w( ). Taking limi supremum as, H(, ) and he laer inequaliy conradics o (3.9). δ s h (,s) h (,s) δ s mb(s)h(, s) mb(s)δ s + w( ) <, 4MΓ a(s) H(,s) Corollary: 3. Le he condiions of Theorem 3. hold, equaion (3.9) can be replaced by and H(, ) Then every soluion of (.) is oscillaory. m b s δ s H(, s) =, (3.) H(, ) h (,s) 4MΓ a(s) H(,s) <. (3.) Le H, s = ( s) n,, s D for some ineger n >. Then Theorem 3. gives he following resul. Corollary: 3.3 Le he assumpions of Theorem 3. hold, equaion (3.9) can be wrien as ( ) n δ s s n mb(s) =. (3.) for some ineger n >. Then every soluion of (.) is oscillaory. 4MΓ a s δ s n δ s s Le H, s = (R R s ) λ, λ is a consan, R = following resul. r(s) and lim R =. Then Theorem 3. gives he Corollary: 3.4 Le he assumpions of Theorem 3. hold, equaion (3.9) can be replaced by Copyrigh o IJARSET

5 δ s R R s λ δ mb(s) s λ =. (R R ) λ 4MΓ a s δ s r s (R R s ) Then every soluion of (.) is oscillaory. (3.3) Le H, s = (log( s ))n, > s >, n > is an ineger. Then Theorem 3. gives he following resul. Corollary: 3.5 Le he assumpions of Theorem 3. hold, equaion (3.9) can be replaced by δ s log( ) n δ mb(s) s n =. (3.4) (log ( ))n s 4MΓ a s δ s s log ( s ) Then every soluion of (.) is oscillaory. Le H, s = ( lim du θ(u) s du θ(u) ) n, > s >, n > is an ineger, θ:, R + is a coninuous funcion such ha =. Then Theorem 3. yiel he following resul. Corollary: 3.6 Le he assumpions of Theorem 3. hold, equaion (3.9) can be replaced by ( du s θ (u ) Then every soluion of (.) is oscillaory. δ s ( du ) n δ mb(s) s n ) n s θ(u) 4MΓ a s δ s =. (3.5) θ(s)( du ) In his secion, we give examples o illusrae our main resuls. IV. EXAMPLES Example: 4.. Consider he coupled sysem of fracional nonlinear differenial equaions u = cos + cos, v = Here =, a = sin, b = C x = cos π d, S x = x sin cos π(sin C x cos S(x)) cos π sin C x cos S x x s θ(u ) s u s,,, where C(x) and S(x) are he Fresnel inegrals, ha is sin π d wih C(x) π, S(x) π, f v = v + v and g(u) = u. I is easy o see ha f v = v ( v ) > ε = m >. v If we ake δ = hen δ =. Consider δ (s) mb(s)δ(s) 4MΓ a(s) δ(s) sin s cos s = ε π sin s C x cos s S x ε sin s cos s > as. π π sin s cos s All he condiions of Theorem 3. are saisfied. Hence every soluion of (4.) is oscillaory. Infac (u, v()) = (sin, cos ) is one such soluion of (4.). (4.) (4.) Copyrigh o IJARSET

6 Example: 4.. Consider he fracional coupled sysem of nonlinear differenial equaions u = sin sin, v = +cos sin +cos π(cos C x sin S(x)) s u s,, Here =, a = sin, b =, where C(x) and S(x) are given in (4.) π(cos C x sin S(x)) wih he condiion C(x) π, S(x) π, f v = v + v and g(u) = u. I is clear ha f v = v ( v v) > ε = m >. Assume δ = hen δ =. Consider δ (s) mb(s)δ(s) 4MΓ a(s) δ(s) sin s + cos s = ε π(cos s C x + sin s S(x)) ε sin s + cos s > as. π π sin s + cos s All he condiions of Theorem 3. are saisfied. Hence every soluion of (4.3) is oscillaory. Infac, (u, v()) = (cos, sin ) is one such soluion of (4.3). Example: 4.3. Consider he coupled sysem of fracional differenial equaions u = e f(v ), v = πe γ (,) s u s,, Here =, a = e, b = πe γ (, f v =,) v, g(u) = u and g (u) = = M >. Now, o see ha vf(v) <, f v = = m, hen A 4 fails o hold. If we choose δ = e hen δ = e. Consider δ s mb s δ s 4MΓ a s δ s = 4Γ e s < as. All condiions of Theorem 3. are no saisfied, since A 4 violaes. Hence here is a nonoscillaory soluion, (u, v()) = (e, ) is such soluion of (4.4). V. CONCLUSION In his paper, we have esablished some oscillaion resuls for he fracional order differenial sysem using differenial inequaliy mehod and Riccai ransformaion. The resuls obained are essenially new, have improved and exended some of he resuls already prevailing in he exising lieraure. The main resuls are illusraed wih he suiable examples. (4.3) (4.4) REFERENCES [] S. Abbas, M. Benchohra, and G. M. N Guerekaa, Topics in fracional differenial equaions, Springer, Newyork,. [] D. Bainov and D. Mishev, Oscillaory heory of operaor-differenial equaions, World scienific, Singapore, 987. Copyrigh o IJARSET

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