Non-periodic solutions to fractional differential equations: a review

Transcription

1 217 9 Õ 31 3 Sept. 217 Communication on Applied Mathematics and Computation Vol.31 No.3 DOI /j.issn Non-periodic solutions to fractional differential equations: a review SARWAR S 1,2 (1. College of Sciences, Shanghai University, Shanghai 2444, China; 2. Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57, Pakistan) Abstract The existence of periodic solution is a hot topic in the field of fractional dynamical system. The fractional derivatives are different from classical integer order derivatives, and the basic difference between them is originated from the inherent difference. Namely, the fractional derivatives are non-local operators with weak singular kernels, whereas the integer order derivative operators are local operators. In this paper, a brief overview on the recent periodicity results of fractional order (Grunwald-Letnikov, Reimann-Liouville, Caputo) derivatives and fractional differential equations are provided. It has been proved that the classical integer order derivative (if it exists) of periodic function is also a periodic function of the same period. However, the fractional order derivative of periodic function is not a periodic function of the same period. At the same time, the non-existence of periodicity of fractional dynamical system of non-constant periodic functions and the existence of periodicity of the long-time solution to the fractional dynamical system are also reviewed. Key words fractional differential equation; Riemann-Liouville derivative; Caputo derivative; non existence; periodic solution 21 Mathematics Subject Classification 26A33; 34A8; 34C25 Chinese Library Classification O175.1 Æ Ð «SARWAR S 1,2 (1. ò Ô Ã² 2444, Ü 2. COMSATS ºÆ ÔÈ Â 57, ÉÌ ) Received ; Revised Project supported by the National Natural Science Foundation of China ( ) Corresponding author SARWAR S, research interests are applied theory and computation for bifurcation, chaos and fractional dynamics. shahzadppn@gmail.com

2 25 Communication on Applied Mathematics and Computation Vol. 31 Ý ÖŠȽ ÓÜ µî Ƚ È ÏÑ ÙȽ È Ë Û» ÖÑÅ Ö ÆÍ ¹ Ƚ ÈÅ ÐÁ ÅÊÞ ÙȽ È ÅÊÞ ¼ Ç È½ È (À Grunwald-Letnikov È Reimann-Liouville È Caputo È) Ƚ Å ÖÝ ¾± Ý ³È ÙȽ ÈÒ Ý ³È Ð ÏÝ Ø ¾± Ú Ý ³È Ƚ È Å Ð Ï Ý Ý ³È Ò ÏÄ «ÈÝ ³ È È½ Ý Ö È½ Ä Ý Ö Đ È½ Reimann-Liouville È Caputo È Ö Ý 21 26A33; 34A8; 34C25 O175.1 Å A ß (217) Introduction Fractional calculus have a long history of more than 3 years, but its applications to science and engineering have been reported in recent years. Fractional differential equations (FDEs) [1-2] involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been center of attentions of many studies. It has been found that many physical phenomena can be more adequately described by the fractional differential equations, for instance, modeling anomalous diffusion [3], time dependent materials and process with long range dependence [4], dielectric relaxation phenomena in polymeric materials [5], transport of passive tracers carried by fluid flow in a porous medium in groundwater hydrology [6], viscoelastic behavior [7], transport dynamics in systems governed by anomalous diffusion [8], self-similar processes such as protein dynamics [9], long-time memory in financial time series [1] using fractional Langevin equations [11], etc. Recently, fractionalorder models of happiness [12] and love [13] have been derived, and the authors claimed that these models give a better representation than the integer-order dynamical systems. One of the areas of interest in these fields is the dynamical system and chaos which is investigated by many researchers [14-17]. In chaos theory, the idea of chaos control is based on the fact that chaotic attractors have a skeleton made of an infinite number of unstable periodic orbits which are subjected for stabilization. Thus, for chaos control in fractional dynamical systems, the existence or non-existence of periodic solutions are very important. Therefore, the importance of periodic functions or periodic solutions in physics, mathematics, engineering, economy and other scientific areas is unquestionable. It is evidently proved that the integer order derivative (if it exists) of periodic function is also a periodic function of the same period. However, the following two natural questions remain open problems till now. i) Can any fractional derivative of a periodic function also be a periodic function of the same period? ii) Can any linear or nonlinear fractional differential system have periodic solutions of the same period?

3 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 251 Recently, the above mentioned two problems are considered by some researchers and some basic results about periodicity are obtained [18-23], but the development of periodic solutions of FDEs is a bit slow. Even oscillatory behavior has been observed by numerical simulations in many systems such as a fractional-order Van der Pol system [15], fractionalorder Chua and Chen s systems [24-25], a fractional-order Rössler system [26], and a fractionalorder financial system [27]. In this paper, we present a brief overview and discuss some recent results on the existence or non-existence of periodic solutions of FDEs. The analysis on the existence of periodic solutions is more complex than integer order differential equations, since fractional derivatives are non-local and have weakly singular kernels. If some important references have been omitted, we apologize in advance for those omissions. In the earliest study on the non-existence of periodic solutions, Tavazoei and Haeri have proved that the Caputo type fractional order time invariant system does not have a non-constant periodic solution due to the existence of non-exponential aperiodic multimodes in solution of linear time invariant fractional order systems [18]. In [19], Tavazoei proved that the fractional order derivative based on each well-known Grunwald-Letnikov, Reimann-Liouville, Caputo definitions of periodic functions with specific periods cannot be a periodic function with the same periods. The existence of periodic solutions is investigated in [2] that the periodic solutions can be detected in time invariant Caputo type fractional order systems if the lower terminal of the Caputo derivative is or the difference between lower and upper terminal is very large. Recently, using the Mellin transform [21], the final value theorem of the Laplace transform [22], and the Fourier transform [23], the authors proved that the fractional-order derivative of a periodic function cannot be a function with the same period. To complement the literature, we will review the existence or the non-existence of periodicity results of FDEs. The rest of the paper is organized as follows: In Section 1, we introduce some basic definitions and previously known results that will be used in our main results. The nonexistence of periodicity of fractional order derivatives of non-constant periodic functions is given in Section 2. In Section 3, we review the non-existence of periodicity of fractional dynamical system of non-constant periodic functions. The existence of periodicity of the long-time solution to the fractional dynamical system is reviewed in Section 4. Concluding remarks and comments are included in the last section. 1 Preliminaries This section provides some basic definitions and lemmas, which have importance in the study of fractional calculus. Now, we give some definitions and properties which will be frequently used throughout this paper. As we all know, there are several definitions

4 252 Communication on Applied Mathematics and Computation Vol. 31 of fractional order derivatives and integrals such as Grunwald-Letnikov, Riemann-Liouville, Caputo, Riesz, Erdélyi-Kober and Hadamard. However, they are not equivalent. In this paper, we only focus on three of them which are most frequently applied, namely, Grunwald- Letnikov, Riemann-Liouville fractional derivative, and Caputo derivative. One can consult [1, 2, 28-3] for mathematical properties of fractional derivative and integral. Let C[a, b] be the Bannach space of all continuous functions mapping [a, b] into R where the norm x [a,b] = max t [a,b] x(t). Definition 1 The Riemann-Liouville integral of function f(t) with order α > is defined as RLD α a,t f(t) = 1 Γ(α) t a (t s) α 1 f(s)ds, t > a. (1) Definition 2 The Riemann-Liouville derivative of function f(t) with order α > is defined as d n RLD α a,t f(t) = 1 Γ(n α) dt n where n 1 < α < n Z +. t a (t s) n α 1 f(s)ds, t > a, (2) Definition 3 The Caputo derivative of function f(t) with order α > is defined as CD α a,tf(t) = where n 1 < α < n Z +. 1 Γ(n α) t a (t s) α 1 f (n) (s)ds, t > a, (3) Proposition 1 The Riemann-Liouville derivative and Caputo derivative have the following relation, if f C m 1 [a, t] and f (m) is integrable on [a, t], where n 1 < α < n Z +. m 1 RLD α a,t f(t) = CD α a,t f(t) + k= f (k) (a)(t a) k α, (4) Γ(k + 1 α) Definition 4 The Grunwald-Letnikov derivative of function f(t) with order α > is defined as where n 1 < α < n Z +. n GLD α a,tf(t) = lim h α ( 1) r Γ(α + 1) f(t rh), (5) h r!γ(α r + 1) nh=t r= Proposition 2 Suppose that f(t) is a sufficiently smooth and continuous function. Then the relation between the Riemann-Liouville, Caputo and Grunwald-Letnikov fractional order derivatives is m 1 RLD α a,tf(t) = GL D α a,tf(t) = C D α a,tf(t) + k= f (k) (a + )(t a) k α. (6) Γ(k α + 1)

5 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 253 Definition 5 The Mellin transform of a locally Lebesgue integrable function g : [, ) C is defined by M(g)(z) = g(t)t z 1 dt. The largest open vertical strip of the complex plane, of the form S g = {z C : a < Re(z) < b} in which the integral converges, is called the fundamental strip of the Mellin transform. Proposition 3 (Mellin transform of the convolution) Considering the Mellin convolution g h of two functions g, h : [, ) C defined by (g h)(t) = the following equality holds g(ts)h(s)ds, M(g h)(z) = M(g)(z) M(h)(1 z) for any z from the fundamental strip of M(g) such that 1 z belongs to the fundamental strip of M(h). Proposition 4 (inverse of the Mellin transform) Let g : [, ) C be an integrable function such that its Mellin transform M(g) has the fundamental strip S g = {z C : a < Re(z) < b}. If c (a, b) is such that M(g)(c + is) is integrable, then the following equality holds 1 2iπ almost everywhere on [, ). c+i c i M(g)(z)t z dz = g(t), We refer to [32-33] for an overview of the Mellin transform and its applications. Definition 6 [34] A non-constant solution x(t) of any system is said to be a periodic solution if there exists T > such that x(t) = x(t + T) for all t R. Lemma 1 [21] Let n N and T >. If x : (, ) R is a non-constant T-periodic function of class C m on (, ), then for any k N, k n, the k-th order derivative x (k) is also a non-constant T-periodic function. Proof We can prove this lemma by reductio ad absurdum in the following steps. i) Let k N, k n. If x : (, ) R is a non-constant T-periodic function, then x (k) (t + T) = x (k) (t) for all t (, ). ii) For the second part, we will assume that there exists k N, k n such that x (k) (t) is a constant function.

6 254 Communication on Applied Mathematics and Computation Vol Non-existence of periodicity of fractional order derivatives of non-constant periodic functions A series of papers were devoted to the study of the non-existence of periodicity of fractional order derivatives such as the Grunwald-Letnikov, Riemann-Liouville and the Caputo derivative and concluded that the fractional order derivative of periodic function is not periodic function of the same period as integer order derivative. Tavazoei firstly, gave a well known results on the non-existence of periodicity of fractional order derivative (Grunwald- Letnikov, Riemann-Liouville and Caputo derivatives) of a non-constant periodic function. Now, we consider the fractional order initial value problem (IVP) in the following form: { CD αi,t x i(t) = f i (x 1 (t), x 2 (t),, x n (t)), < α i < 1, i = 1, 2,, n, (7) x i (t) t= = x i (), x i R, i = 1, 2,, n, where C D α,t is the Caputo derivative, f i are smooth functions for i = 1, 2,, n in the IVP (7). Lemma 2 Suppose that g i (t) = f i (x 1 (t), x 2 (t),, x n (t)) is a continuous function. Then, the IVP (7) is equivalent to the nonlinear Volterra integral equation of the second kind x i (t) = x i () + 1 Γ(α) t (t s) αi 1 g i (s)ds, i = 1, 2,, n. (8) In other words, every solution of the Volterra integral equation (8) is also the solution of our original IVP (7) and vise versa. Suppose that x i (t) = x i (t + T) is a non-constant continuous periodic solution of the IVP (7). Then (8) can be written as where g i (t) = f i ( x 1 (t), x 2 (t),, x n (t)). (t s) αi 1 g i (s)ds =, i = 1, 2,, n, (9) Lemma 3 If x(t) = ( x 1 (t), x 2 (t),, x n (t)) is a periodic solution of the IVP (7) with period T, then (pt s) αi 1 g i (s)ds =, i = 1, 2,, n for all p N. (1) Lemma 4 If (1) is true for all p N, then the following relation can be deduced: g i (s)ds =, i = 1, 2,, n. (11) Lemma 5 If (1) is true for all p N, then the following relation can be obtained: (pt s) βi 1 g i (s)ds =, i = 1, 2,, n, (12)

7 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 255 where β i s can be chosen in (, 1) R. Since (pt s)βi 1 g i (s)ds = is constant (zero) for all β i (, 1), hence, d k dβ k i ( (pt s) βi 1 g ) i (s)ds =, i = 1, 2,, n, (13) where k N []. By replacing the derivative and integration order and taking k successive derivative of (pt s) βi 1 with respect to β i, one can find the following relation: (ln(pt s)) k (pt s) βi 1 g i (s)ds =, i = 1, 2,, n (14) for any k N []. By considering the expansion (pt s) (r βi+1) (r βi+1) ln(pt s) = e (r β i + 1) k = (ln(pt s)) k, (15) k! k= where < s < pt and r R, and using the relation (14), the following results can be deduced: (pt s) r g i (s)ds =, i = 1, 2,, n, r R, p N. (16) Lemma 6 Relation (16) results in for every t (, T) and m N []. (s t ) m g i (s)ds =, i = 1, 2,, n Proof The proofs of Lemmas 3 6 are contained in [18]. By considering the expansion e (s t )2 b 2 = ( 1) k (s t ) 2k b 2k, (17) k! k= and using Lemma 6, it can be deduced that e (s t )2 b 2 b π g i(s)ds =, i = 1, 2,, n, (18) where t (, T) and b >. From the Gaussian function properties, we know that (s t ) 2 e b lim 2 b + b π = δ(s t ), (19) where δ(t) denotes the Dirac delta function. Hence, when b +, (18) becomes δ(s t ) g i (s)ds =, i = 1, 2,, n. (2)

8 256 Communication on Applied Mathematics and Computation Vol. 31 Theorem 1 Suppose that g(t) is a non-constant periodic function with period T, i.e., g(t) = g(t + T) for all t. If g(t) is m-times differentiable, function C D α,tg(t), where < α / N and m is the first integer greater than α, cannot be a periodic function with period T. Theorem 2 Suppose that g(t) is (m 1)-times continuously differentiable and g (m) (t) is bounded. If g(t) is a non-constant periodic function with period T, function RL D α,tg(t) and GL D α,tg(t), where < α / N and m is the first integer greater than α, cannot be periodic functions with period T. The proofs of Theorems 1 2 are in [19]. Based on Lemmas 3 6 and using (17) (2), one can easily prove these theorems. Remark 1 After careful examination, Kaslik and Sivasundaram [21] pointed out that the authors [18] used wrong arguments in the proof of Lemma 5 so the results of Theorems 1 2 have flawed since proofs of Theorems 1 2 based on Lemma 5. In the proof of this lemma, the authors suppose that S + i and S i are the areas between the curve (pt t) βi 1 g i (t) and t-axis, located above and bellow the axis in the interval [, T], respectively. S + i and S i are wrongly considered to be independent of p, but they clearly depend on p N. Thus, (31) from [18] should be (pt t) βi 1 g i (t)dt = S + i (p) + S i (p). Therefore, using (1) and (32) from [18], we can get p αi βi S + i (p) (p 1)αi βi S i (p) (p 1)αi βi S + i (p) pαi βi S i (p), where α i < β i. This can be written as ( p 1 p ) βi α i S + i (p) ( p ) βi α i. S i (p) p 1 Taking limit p, we can simply obtain that S + i lim (p) p S = 1. i (p) However, it is impossible to conclude that S + i (p) = S i (p) for any p N, which leads to the conclusion of Lemma 5. For the non-existence of periodicity of the Caputo derivative of the non-constant periodic function, Kaslik gave the same result as Theorem 1 with the use of the Mellin transform and highlights the difference between fractional and integer-order derivatives [21]. Theorem 3 Let α (, )\N and n = [α] + 1. If x : (, ) R is a non-constant T-periodic function of class C n, then its Caputo derivative C D α x cannot be a T-periodic function.

9 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 257 The proof of Theorem 3 based on the definition of the Caputo derivative of order α (, )\N and the properties of the Mellin transform. In this proof, the authors suppose that C D α x(t) = C D α x(t + T) for all t and x(t) is a nonzero constant periodic function. Using Lemma 2 and making the change of variables s + T = s and s = T ts with simplification leads us to Define the function t (1 + s) n α 1 x (n) (t ts)ds = for all t >. (21) h(u) = { x (n) (T u), if u [, T],, if u > T. Equation (21) can be written as (1 + s) n α 1 h(ts)ds = for all t >. The Mellin convolution h g of functions h(t) and g(t) = (1+t) n α 1 is equal to. Applying the Mellin transform to this equality, it follows that M(h g)(z) = for any z where this Mellin transform is defined. Applying the inverse Mellin transform, we find that the function h is equal to almost everywhere on (, ), and therefore, x (n) is identically null because it is continuous. Since all the derivatives of the function x, up to the order n, are T-periodic functions, it can be easily seen that the function x is constant, which contradicts the initial assumptions. The proof of the theorem is now completed. The results of the above theorem lead us to the following theorem. Theorem 4 Let α (, )\N and n = [α]+1. If x : (, ) R is a non-constant T- periodic function of class C n, then its Riemann-Liouville derivative RL D α x and its Grunwald- Letnikov derivative GL D α x cannot be T-periodic functions. Remark 2 The arguments used in the proof of Theorem 3 do not apply in the case of integer-order derivatives, i.e., α N. Indeed, if we assume that α N, then n = [α]+1 = α + 1 and the function g defined in the proof of Theorem 3 is g(t) = 1 for any t (, ). Since the integral t z 1 dt is divergent for any z C, it follows that the Mellin transform of the function g does not exist. In [23], Kang, et al. used the technique of the Fourier analysis and proved that the Caputo fractional derivative of a non-constant periodic function cannot be periodic. The authors claimed that in the section of [28], the detailed analysis of a non-existence of

10 258 Communication on Applied Mathematics and Computation Vol. 31 periodicity of the Riemann-Liouville fractional derivatives of periodic functions is absent. To complete the literature, the specific results are as follows [23]. Lemma 7 Let g(t) be a nonzero sufficiently smooth periodic function. Then, the Riemann-Loiuville fractional derivative RL D α,tg(t) is aperiodic. Theorem 5 Let g(t) be a nonzero sufficiently smooth periodic function. Then, the Caputo fractional derivative RL D α,tg(t) is aperiodic. The proofs of Lemma 7 and Theorem 5 are contained in [23]. Remark 3 The statement of the main of Theorem 1 in [23] (Theorem 5 in this paper) has a typographical mistake. There should be C D α,t g(t) instead of RLD α,tg(t). The proofs of Lemma 7 and Theorem 5 are based on the theory of the Fourier analysis. According to that the Fourier transform of a sufficiently smooth periodic function is the superposition of numerable Dirac delta pulses at multiple frequencies [35-36], it means that the Fourier transform of periodic function cannot contain any nonzero continuous segments. Now, we extend Theorem 5 to the Grunwald-Letnikov fractional order derivative. Theorem 6 Let f(t) be a nonzero sufficiently smooth periodic function. Then, the Grunwald-Letnikov fractional derivative GL D α,tf(t) is aperiodic. Proof According to (6), there holds and thus there also holds Note that m 1 GLD α,t f(t) = C D α,t f(t) + k= m 1 F{ GL D α,t f(t)}(ω) = F{ CD α,t f(t)}(ω) + k= f (k) ( + )(t) k α Γ(k α + 1), (22) f (k) ( + ) Γ(k α + 1) F{tk α }(ω). (23) F{t k α }(ω) = L{t k α }(iω) = (iω) α k 1 Γ(k α + 1), (24) and we know that from (17) of [23], we have F{ C D α,tf(t)}(ω) = (iω) α{ a 2 Substituting (24) and (25) in (23), we get F{ GL D α,tf(t)}(ω) = (iω) α{ a 2 1 iω + n=1 [ a n iω (nω) 2 ω 2 nω ] m 1 +b n (nω) 2 ω 2 f (k) ( + 1 } ) (iω) k+1. (25) k= 1 iω + + n=1 [ a n iω (nω) 2 ω 2 +b n nω (nω) 2 ω 2 ]}. (26)

11 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 259 iω nω Since the functions (nω) 2 ω and 2 (nω) 2 ω are linearly uncorrelated, the right-hand side 2 of (26) will be a nonzero continuous function at ω nω unless a = and a n = b n = for n 1. So for the same reason of Lemma 7, we conclude that the Grunwald-Letnikov fractional derivative cannot transform non-constant periodic functions into periodic ones. The proof is completed. 3 Non-existence of periodicity of fractional dynamical system of non-constant periodic functions For (7) and < α < 1, Tanvazoei firstly gave a well known result for non-existence of periodic solutions in time invariant fractional order systems, and the specific result is as follows [18]. Theorem 7 have any periodic solution. The time invariant fractional order systems represented by (7) cannot Some one can easily prove this theorem using Lemmas 3 6 and relations (17) (2). But the proof of this theorem is based on Lemma 5 and we have already discussed in Remark 1 that the proof of Lemma 5 is not correct. In conclusion, it has to be highlight that the statements of Theorems 1 2 from [19] and the main theorem from [18] are correct but the proofs are incorrect because these proofs are based on Lemma 5 from [18]. Wang, et al. [22] studied the non-existence periodic solution by using the final-value theorem of the Laplace transform of the following nonhomogeneous fractional Cauchy problem CD α,tu(t) = f(t), t J := [, ), (27) u() = u, where C D α,t is the Caputo fractional derivative of order α (, 1) with the lower limit zero and u, f are T-periodic L -functions where T >. The author proved that the IVP (27) does not have any nonzero periodic L -solutions on J. Kaslik and Sivasundaram [21] considered the following general class of fractional-order differential systems and proved that there are no constant periodic solutions. Let α = {α 1, α 2,, α p }, with α i (, )\N for any i {1, 2,, p} and m N. We consider the following general class of fractional-order differential systems in vector form D α x = f(t,x(t),x (t),,x (m) (t)), t, (28) where f : [, ) R p(m+1) R p, is a given function and D α x = (D α1 x 1 (t), D α2 x 2 (t),, D αp x p (t)) T,

12 26 Communication on Applied Mathematics and Computation Vol. 31 where D αi, i {1, 2,, p}, is understood as one of the fractional-order derivatives (Riemann- Liouville, Caputo or Grunwald-Letnikov). In the following, we denote n = ([α 1 ], [α 2 ],, [α p ]) N p and we say that the function x(t) = (x 1 (t), x 2 (t),, x p (t)) T is of class C m (I, R p ) if and only if x i (t) is of class C mi (I, R) for any i {1, 2,, p}, where I denotes a real interval. Corollary 1 [21] (the non-autonomous case) Let k N and assume that the function f is T-periodic with respect to its first argument. Then, there are no non-constant kt-periodic solutions of class C m of system (28). Corollary 2 [21] (the autonomous case) If the function f is constant with respect to its first argument, then there are no non-constant periodic solutions of class C m of system (28). Corollaries 1 and 2 hold even for some of the differentiation orders α 1, α 2,, α p from system (28) are positive integers. Even though, based on Corollary 2, the exact periodic solution does not exist in the autonomous case. Similar results about the non-existence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems are studied in [23] by using the Fourier transform. The specific results are as follows. Theorem 8 Suppose that a function f(t,,,, ) is sufficiently smooth and T- periodic with respect to the first argument. Then the initial-valued problem of the following Caputo fractional-order differential equation CD α,tx(t) = f(t, x(t), x (t),, x (m) (t)) (29) with < α < N cannot have any non-constant exact periodic solution. Since the autonomous Caputo fractional-order system is a special case of (29), the above theorem leads us to the following corollary. Corollary 3 Let a function f(,,, ) be sufficiently smooth. Then, the initialvalued problem of the following autonomous Caputo fractional-order differential equation CD α,tx(t) = f(x(t), x (t),, x (m) (t)) (3) with < α < N cannot have any non-constant exact periodic solution. 4 Existence of periodicity of the long-time solution to fractional dynamical system In above Theorem 7, Tavazoei has claimed that the existence of the periodic solution of IVP (7) is impossible. However, Yazdani and Salarieh [2] proved that this is not a general claim and the periodic solutions can exist in the fractional order system (7) under some

13 No. 3 SARWAR S: Non-periodic solutions to fractional differential equations: a review 261 conditions or circumstances. The authors proved that the periodic solution of IVP (7) can be detected by considering the steady state behavior of solutions. Since the fractional derivative in IVP (7) is not local and it is different from integer order derivative, the solution of IVP (7) depends on its whole memory at any time t. In other words, the periodic solution might be detected if the difference between lower and upper terminal in the fractional derivative of IVP (7) should be chosen sufficiently large, i.e., lower terminal should approaches to infinity. This argument become much stronger after study the periodicity of the Weyl fractional-order derivative of a periodic function. In the section 19 of [31], the periodicity of fractional-order differential operator of periodic function has been discussed. From the above discussion, we have the following results. Theorem 9 Let f(x) satisfy the Lipschitz condition. Then, the solutions of the following fractional order system are memory dependent: CD α a,tx = f(x), x(a) = x a. (31) It means that the solution of (31), which is denoted by φ(t, x a ), and the solution of CD α b,ty = f(y), y(b) = y b = φ(b, xa ), b > a, (32) which is denoted by ψ(t, y b ), do not coincide for t b. The above result can be extracted from the non-constant initialization concept of fractional order system in [37]. Theorem 1 The time-invariant fractional order system presented by (31) do not have any periodic solution unless the lower terminal of the derivative is ±. Theorem 11 Let g(t) be a sufficiently smooth periodic function. Then, the Weyl fractional-order derivative RL D α,tg(t) is still a periodic function of the same period. The proofs of Theorems 8 and 9 are given in [2], and Theorem 1 is in [23]. The existence of weighted pseudo-almost periodic solutions of fractional-order differential equations has been studied in [38-4]. Moreover, recently, many results about the existence of solutions of periodic boundary problems for fractional differential equations have been investigated in [41-44]. 5 Conclusions In this paper, we present a complete review on the existence or non-existence of periodicity of fractional derivative (Riemann-Liouville, Caputo and Grunwald-Letnikov) of periodic function, periodic solution of FDEs including long time solution. This paper almost covers recent contributions in this area. If some important references happen not to be here, we do apologize for these omissions.

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