ON THE OSCILLATION OF FRACTIONAL DIFFERENTIAL EQUATIONS

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1 ON HE OSCILLAION OF FRACIONAL DIFFERENIAL EQUAIONS S.R. Grce 1, R.P. Agrwl 2, P.J.Y. Wong 3, A. Zfer 4 Abstrct In this pper we initite the oscilltion theory for frctionl differentil equtions. Oscilltion criteri re obtined for clss of nonliner frctionl differentil equtions of the form D q x + f 1 (t, x) = v(t) + f 2 (t, x), lim J 1 q t + x(t) = b 1, D q denotes the Riemnn-Liouville differentil opertor of order q, 0 < q 1. he results re lso stted when the Riemnn-Liouville differentil opertor is replced by Cputo s differentil opertor. MSC 2010 : Primry 34A08: Secondry 34C10, 26A33 Key Words nd Phrses: frctionl differentil eqution, oscilltion, Riemnn-Liouville opertors, Cputo derivtive 1. Introduction Frctionl differentil equtions (FDE) hve gined considerble importnce due to their vrious pplictions in viscoelsticity, electronlyticl chemistry, control theory, mny physicl problems, etc, see e.g. [2, 5, 7, 10, 11]. A rigorous theory of FDE hs been strted quite recently, see for exmple - the books [4, 9, 11] nd to mention only few ppers relted to ordinry FDE s [3, 6, 8]. Seprtely, there re mny recent works relted to prtil FDE. Differentil equtions involving the Riemnn Liouville nd Cputo differentil opertors of frctionl order 0 < q < 1 pper to be importnt in modeling severl physicl phenomen, nd therefore deserve n independent study prllel to the well-developed theory of the ordinry differentil equtions of integer order. c Yer Diogenes Co., Sofi pp. xxx xxx

2 2 S.R. Grce, R.P. Agrwl, P.J.Y. Wong, A. Zfer In this pper we initite the study of the oscilltion theory for frctionl differentil equtions by considering equtions of the form D q x + f 1 (t, x) = v(t) + f 2 (t, x), lim J 1 q t + x(t) = b 1, (1.1) D q denotes the Riemnn-Liouville differentil opertor of order q with 0 < q 1. he opertor J p defined by J p x(t) = 1 Γ(p) (t s) p 1 x(s) ds, J 0 x := x is clled the Riemn-Liouville frctionl integrl opertor. By using the integrl opertor J p, the Riemnn-Liouville differentil opertor D q of order q for 0 < q 1 is defined by Df(t) q := d dt J 1 q f(t), nd more generlly if m 1 is n integer nd m 1 < q m, then Df(t) q := dm dt m J m q f(t). For more detils on the Riemnn-Liouville type opertors of frctionl clculus, see for exmple [12]. We ssume in this pper tht the functions f 1, f 2, nd v re continuous. It is known tht in this cse, in fct under much weker ssumptions, the initil vlue problem (1.1) is equivlent to the Volterr frctionl integrl eqution x(t) = b 1(t ) q (t s) q 1 [v(s) + f 2 (s, x(s)) f 1 (s, x(s))]ds. (1.2) ht is, every solution of (1.2) is lso solution of (1.1) nd vice-vers, see [4, Lemm 5.3]. We only consider those solutions which re continuous nd continuble to (, ), nd re not identiclly zero on ny hlf-line (b, ) for some b. he term solution henceforth pplies to such solutions of eqution (1.1) or (1.2). For n introduction to the theory of frctionl differentil equtions, we refer to [4, 11, 12]. A solution is sid to be oscilltory if it hs rbitrrily lrge zeros on (0, ); otherwise, it is clled nonoscilltory. nd 2. Min Results We will mke use of the conditions: xf i (t, x) > 0 (i = 1, 2), x 0, t (2.1)

3 ON HE OSCILLAION OF... 3 f 1 (t, x) p 1 (t) x β nd f 2 (t, x) p 2 (t) x γ, x 0, t. (2.2) p 1, p 2 C([, ), R + ) nd β, γ > 0 re rel numbers. nd We will employ the following lemm [1, Lemm 3.2.1]. Lemm 2.1. For X 0 nd Y > 0, we hve X λ + (λ 1)Y λ λxy λ 1 0, λ > 1 (2.3) X λ (1 λ)y λ λxy λ 1 0, λ < 1, (2.4) equlity holds if nd only if X = Y. nd Now we my present our first theorem when f 2 = 0. heorem 2.1. Let f 2 = 0 nd condition (2.1) hold. If t1 q (t s) q 1 v(s)ds =, (2.5) t 1 q (t s) q 1 v(s)ds =, (2.6) then every solution of eqution (1.1) is oscilltory. P r o o f. Let x(t) be nonoscilltory solution of eqution (1.1) with f 2 = 0. Suppose tht > is lrge enough so tht x(t) > 0 for t. Let F (t) = v(t) + f 2 (t, x(t)) f 1 (t, x(t)), then we see from (1.2) tht nd hence x(t) (t )q 1 b (t s) q 1 F (s) ds (t s) q 1 v(s)ds, t, t 1 q x(t) c( ) + t 1 q (t s) q 1 v(s)ds, t (2.7) nd in the proof of the following theorems ( ) 1 q ( ) 1 q c( ) = b 1 + F (s) ds. s

4 4 S.R. Grce, R.P. Agrwl, P.J.Y. Wong, A. Zfer Note tht the improper integrl on the right is convergent. king the limit inferior of both sides of inequlity (2.7) s t, we get contrdiction to condition (2.5). In cse x(t) is eventully negtive, similr rgument leds to contrdiction with (2.6). Next we hve the following results. heorem 2.2. γ = 1. If nd Let conditions (2.1) nd (2.2) hold with β > 1 nd t1 q (t s) q 1 [v(s) + H β (s)]ds =, (2.8) t 1 q (t s) q 1 [v(s) + H β (s)]ds =, (2.9) H β (s) = (β 1)β β/(1 β) p 1/(1 β) 1 (s)p β/(β 1) 2 (s), then every solution of eqution (1.1) is oscilltory. P r o o f. Let x(t) be nonoscilltory solution of eqution (1.2), sy, x(t) > 0 for t >. Using (2.2) in eqution (1.2) with γ = 1 nd β > 1 nd t, we find t 1 q x(t) c( ) + t 1 q[ (t s) q 1 v(s)ds + We pply (2.3) in Lemm 2.1 with to obtin λ = β, X = p 1/β 1 x nd Y = ] (t s) q 1 [p 2 (s)x(s) p 1 (s)x β (s)]ds. (2.10) ( ) p 2 p 1/β 1/(β 1) 1 /β p 2 (t)x(t) p 1 (t)x β (t) (β 1)β β/(1 β) p 1/(1 β) 1 (t)p β/(β 1) 2 (t). (2.11) Using (2.11) in (2.10), we hve t 1 q x(t) c( ) + t 1 q (t s) q 1 [v(s) + H β (s)]ds, t. he rest of the proof is the sme s in tht of heorem 2.1.

5 heorem 2.3. γ < 1. If nd ON HE OSCILLAION OF... 5 Let conditions (2.1) nd (2.2) hold with β = 1 nd t1 q (t s) q 1 [v(s) + H γ (s)]ds =, (2.12) t 1 q (t s) q 1 [v(s) + H γ (s)]ds =, (2.13) H γ (s) = (1 γ)γ γ/(γ 1) p γ/(γ 1) 1 (s)p 1/(1 γ) 2 (s), then every solution of eqution (1.1) is oscilltory. P r o o f. Let x(t) be nonoscilltory solution of eqution (1.2), sy, x(t) > 0 for t > 1. Using condition (2.2) in eqution (1.2) with β = 1 nd γ < 1, we obtin t 1 q x(t) c( ) + t 1 q[ (t s) q 1 v(s)ds + Now we use (2.4) in Lemm 2.1 with to get λ = γ, X = p 1/γ 2 x nd Y = ] (t s) q 1 [p 2 (s)x γ (s) p 1 (s)x(s)]ds. (2.14) ( ) p 1 p 1/γ 1/(γ 1) 2 /γ p 2 (t)x γ (t) p 1 (t)x(t) (1 γ)γ γ/(1 γ) p γ/(γ 1) 1 (t)p 1/(1 γ) 2 (t). (2.15) Using (2.15) in (2.14) then yields t 1 q x(t) c( ) + t 1 q (t s) q 1 [v(s) + H γ (s)]ds, t. he rest of the proof is the sme s in tht of heorem 2.1. Finlly we present the following more generl result. heorem 2.4. γ < 1. If nd t1 q Let conditions (2.1) nd (2.2) hold with β > 1 nd (t s) q 1 [v(s) + H β,γ (s)]ds =, (2.16) t 1 q (t s) q 1 [v(s) + H β,γ (s)]ds =, (2.17)

6 6 S.R. Grce, R.P. Agrwl, P.J.Y. Wong, A. Zfer H β,γ (s) = (β 1)β β/(1 β) ξ β/(β 1) (s)p 1/(1 β) 1 (s) +(1 γ)γ γ/(1 γ) ξ γ/(γ 1) (s)p 1/(1 γ) 2 (s) with ξ C([, ), R + ), then every solution of eqution (1.1) is oscilltory. P r o o f. Let x(t) be nonoscilltory solution of eqution (1.1), sy, x(t) > 0 for t >. Using (2.2) in eqution (1.2) one cn esily write tht t 1 q x(t) c( ) + t 1 q (t s) q 1 v(s)ds ( ) +t 1 q (t s) q 1 ξ(s)x(s) p 1 (s)x β (s) ds + t 1 q (t s) q 1 (p 2 (s)x γ (s) ξ(s)x(s)) ds, t. We my bound the terms (ξx p 1 x β ) nd (p 2 x γ ξx) by using the inequlities (2.11) (with p 2 = ξ) nd (2.15) (with p 1 = ξ) respectively, to get t 1 q x(t) c( ) + t 1 q (t s) q 1 [v(s) + H β,γ (s)]ds, t. he rest of the proof is the sme s in tht of heorem 2.1. Remrk 2.1. he results obtined for (1.1) re with different nonlinerities nd one cn observe tht the forcing term v is unbounded, nd its oscilltory chrcter is inherited by the solutions. Remrk 2.2. It is not difficult to see tht the results remin vlid for frctionl differentil equtions involving Riemnn-Liouville differentil opertor D q of order q with m 1 < q m, m 1 is n integer, of the form D q x + f 1 (t, x) = v(t) + f 2 (t, x), D q k x() = b k (k = 1, 2,..., m 1), lim t + J m q x(t) = b m. (2.18) It suffices to note tht the initil vlue problem (2.18) is equivlent to the Volterr frctionl integrl eqution m b k (t ) q k x(t) = Γ(q k + 1) + 1 (t s) q 1 [v(s)+f 2 (s, x(s)) f 1 (s, x(s))]ds. k=1 (2.19)

7 ON HE OSCILLAION OF... 7 We now give n exmple to show tht the condition (2.8) cnnot be dropped. Exmple 2.1. eqution Consider the Riemnn-Liouville frctionl differentil D q 0 x + et x 3 = t1 q Γ(2 q) + tet (t 2 1) + e t x, lim t 0 + J 1 q 0 x(t) = 0, (2.20) 0 < q < 1. Clerly, in the context of (1.1) we hve = b 1 = 0, f 1 (t, x) = e t x 3, f 2 (t, x) = e t x, nd v(t) = t1 q Γ(2 q) +tet (t 2 1). We see tht the conditions (2.1) nd (2.2) re stisfied with β = 3, γ = 1 nd p 1 (t) = p 2 (t) = e t. But the condition (2.8) is not stisfied, since v(t) 0 nd t 1 q (t s) q 1 p β/(β 1) 0 2 (s)p 1/(1 β) 1 (s) ds t 1 q 0 1 (t s) 1 q ds = t q. Indeed, one cn esily verify tht x(t) = t is nonoscilltory solution of (2.20). Note tht the initil condition is stisfied in view of J 1 q 0 x(t) = t 2 q /Γ(3 q) 0 s t Cputo s derivtive pproch We my replce the Riemnn-Liouville differentil opertor D q by the Cputo differentil opertor c D q with m 1 < q m, which is defined by (see [2, 11]) cdf(t) q := J m q f (m) (t). Notice tht the Cputo differentil opertor c D q demnds functions to be m times differentible. In this cse the initil vlue problem (2.18) should be replced by cd q x+f 1 (t, x) = v(t)+f 2 (t, x), x (k) () = b k (k = 0, 1,..., m 1). (3.1) he corresponding Volterr frctionl integrl eqution, see [4, Lemm 6.2], becomes x(t) = m 1 k=0 b k (t ) k k! + 1 (t s) q 1 [v(s)+f 2 (s, x(s)) f 1 (s, x(s))]ds. (3.2)

8 8 S.R. Grce, R.P. Agrwl, P.J.Y. Wong, A. Zfer he oscilltion criteri obtined for the Riemnn-Liouville cse red lmost the sme. he only chnge is the ppernce of t 1 m insted of t 1 q in the conditions. his difference becomes more distinctive when m = 1. nd heorem 3.1. Let f 2 = 0 nd condition (2.1) hold. If t1 m (t s) q 1 v(s)ds =, (3.3) t 1 m (t s) q 1 v(s)ds = (3.4) then every solution of eqution (3.1) is oscilltory. heorem 3.2. γ = 1. If nd Let conditions (2.1) nd (2.2) hold with β > 1 nd t1 m (t s) q 1 [v(s) + H β (s)]ds =, (3.5) t 1 m (t s) q 1 [v(s) + H β (s)]ds = (3.6) H β (s) = (β 1)β β/(1 β) p 1/(1 β) 1 (s)p β/(β 1) 2 (s), then every solution of eqution (3.1) is oscilltory. heorem 3.3. γ < 1. If nd Let conditions (2.1) nd (2.2) hold with β = 1 nd t1 m (t s) q 1 [v(s) + H γ (s)]ds =, (3.7) t 1 m (t s) q 1 [v(s) + H γ (s)]ds = (3.8) H γ (s) = (1 γ)γ γ/(γ 1) p γ/(γ 1) 1 (s)p 1/(1 γ) 2 (s), then every solution of eqution (3.1) is oscilltory. heorem 3.4. γ < 1. If Let conditions (2.1) nd (2.2) hold with β > 1 nd t1 m (t s) q 1 [v(s) + H β,γ (s)]ds =, (3.9)

9 ON HE OSCILLAION OF... 9 nd t 1 m (t s) q 1 [v(s) + H β,γ (s)]ds = (3.10) H β,γ (s) = (β 1)β β/(1 β) ξ β/(β 1) (s)p 1/(1 β) 1 (s) +(1 γ)γ γ/(1 γ) ξ γ/(γ 1) (s)p 1/(1 γ) 2 (s) with ξ C([, ), R + ), then every solution of eqution (3.1) is oscilltory. References [1] R. P. Agrwl, S. R. Grce, nd D. O Regn, Oscilltion heory for Second Order Liner, Hlf-liner, Superliner nd Subliner Dynmic Equtions, Kluwer, Dordrecht, [2] M. Cputo, Liner models of dissiption whose Q is lmost frequency independent II, Geophys. J. R. Astr. Soc., 13 (1967), ; Reprinted in: Frct. Clc. Appl. Anl. 11, No 1 (2008), [3] K. Diethelm nd N.J. Ford, Anlysis of frctionl differentil equtions, J. Mth. Anl. Appl., 265 (2002), [4] K. Diethelm, he Anlysis of Frctionl Differentil Equtions, Springer, Berlin, [5] K. Diethelm nd A.D. Freed, On the solution of nonliner frctionl differentil equtions used in the modelling of viscoplsticity, In: F. Keil, W. Mckens, H. Vob nd J. Werther (Eds.), Scientific Computing in Chemicl Engineering II: Computtionl Fluid Dynmics, Rection Engineering nd Moleculr Properties, Springer, Heidelberg, 1999, [6] N.J. Ford, M. Luis Morgdo, Frctionl boundry vlue problems: Anlysis nd numericl methods, Frct. Clc. Appl. Anl. 14, No 4 (2011), ; DOI: /s ; t [7] W.G. Glöckle nd.f. Nonnenmcher, A frctionl clculus pproch to self similr protein dynmics, Biophys. J., 68 (1995), [8] J.R. Gref, L. Kong, B. Yng, Positive solutions for semipositone frctionl boundry vlue problem with forcing term, Frct. Clc. Appl. Anl. 15, No 1 (2012), 8 24; DOI: /s ; t: [9] A.A. Kilbs, H.M. Srivstv, J.J. rujillo, heory nd Applictions of Frctionl Differentil Equtions, North-Hollnd Mth. Studies 204, Elsevier, Amsterdm, 2006.

10 10 S.R. Grce, R.P. Agrwl, P.J.Y. Wong, A. Zfer [10] R. Metzler, S. Schick, H.G. Kilin nd.f. Nonnenmcher, Relxtion in filled polymers: A frctionl clculus pproch, J. Chem. Phys., 103 (1995), [11] I. Podlubny, Frctionl Differentil Equtions, Acdemic Press, Sn Diego, [12] S.G. Smko, A.A. Kilbs nd O.I. Mrichev, Frctionl Integrls nd Derivtives: heory nd Applictions, Gordon nd Brech, Yverdon, Dept. of Engineering Mthemtics, Fculty of Engineering Ciro University, Omn, Giz 12221, EGYP e-mil: srgrce@eng.cu.edu.eg 2 Dept. of Mthemtics, exs A & M University Kingsville Kingsville, X 78363, USA Received: November 2, 2011 e-mil: grwl@tmuk.edu 3 School of Electricl nd Electronic Engineering Nnyng echnologicl University, 50 Nnyng Avenue Singpore , SINGAPORE e-mil: ejywong@ntu.edu.sg 4 Dept. of Mthemtics, Middle Est echnicl University Ankr, URKEY zfer@metu.edu.tr