Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

and Applied Analysis 3 The first variant of differential operator was used in 13 to study the existence of solutions x t of nonlinear fractional differential equations that obey the restrictions x t −→ 1 when t −→ ∞, x′ ∈ ( L1 ∩ L∞ ) 0, ∞ ,R . 1.5 The second variant of differential operator, see 14 , was employed to prove that, for any real numbers x0, x1, the linear fractional differential equation 0D 1 α t x a t x 0, t > 0, 1.6 possesses a solution x t with the asymptotic development x t x0 O 1 tα−1 x1t when t −→ ∞. 1.7 A recent application of the Caputo derivative can be found in 15 . All of these fractional differential operators are based upon the natural splitting of the second-order operator d2/dt2, namely, x′′ x′ ′. Here, we shall introduce a different fractionalizing of x′′ which is based on the identities tx′′ ( tx′ − x)′ [tx′ − x x 0 ]′, t > 0, 1.8 stemming from the integration technique in the Lie algebra L2, cf. 16, page 23 . In the following section, we give a positive partial answer to the preceding open question. In fact, we produce some simple conditions regarding the continuous function a : 0, ∞ → R such that, given c ∈ R − {0}, the fractional differential equation FDE below 0D α t [ tx′ − x x 0 ] a t x 0, t > 0, 1.9 possesses a solution with the asymptotic development x t ct x 0 o 1 when t → ∞. 2. Asymptotically Linear Solutions Let us start with a result regarding the case of intermediate asymptotic. Proposition 2.1. Set the numbers ε ∈ 0, 1 , c / 0, and c1 ∈ 0, 1 , A > 0, such that max { |c|, 1 1 − ε } · Γ 1 − α A ≤ c1. 2.1 Assume also that a ∈ C 0, ∞ ,R is confined to ( 1 t1−ε ) |a t | ≤ A tα , t > 0. 2.2 4 Abstract and Applied Analysis Then, the FDE 0D α t ( tx′ − x) a t x 0, t > 0, 2.3 has a solution x ∈ C 0, ∞ ,R ∩ C1 0, ∞ ,R , with limt↘0 t2−αx′ t 0, which verifies the asymptotic formula x t ct O t when t → ∞. Proof. Introduce the complete metric space M D, δ , where D {y ∈ C 0, ∞ ,R : supt>0 t −ε|y t | ≤ c1, t > 0} and the metric δ is given by the usual formula δ ( y1, y2 ) sup t>0 ∣ y1 t − y2 t ∣ ∣ tε , y1, y2 ∈ D. 2.4 In particular, limt↘0y t 0 for all y ∈ D. Introduce the function x : 0, ∞ → R via the formulas y tx′ − x, x t ct − t ∫ ∞ t y s s2 ds, t > 0. 2.5 Since limt↘0x t 0, we deduce that x can be continued backward to 0; so, its extension x belongs to C 0, ∞ ,R ∩ C1 0, ∞ ,R . Also, limt↘0 t1−αy t limt↘0 t2−αx′ t 0. Define further the integral operator T : M → M by the formula T ( y ) t − 1 Γ α ∫ t 0 a s t − s 1−α [ cs − s ∫ ∞ s y τ τ2 dτ ] ds, t > 0. 2.6


Introduction
Consider the ordinary differential equation Here, the functions h : 1, ∞ → 0, ∞ and g : 0, ∞ → 0, ∞ are continuous, and there exists ε ∈ 0, 1 with Abstract and Applied Analysis Then, given c, d ∈ R, 1.1 has a solution x t , defined in a neighborhood of ∞, which is expressible as ct o t for ε 0, as ct o t 1−ε for ε ∈ 0, 1 and, finally, as ct d o 1 for ε 1 when t → ∞.Such a solution is called asymptotically linear in the literature.In particular, these developments apply to the homogeneous linear differential equation x a t x 0.
A unifying technique of proof for such estimates can be read in 1 and is based on the next reformulation of the differential equation 1.1 for some t 0 ≥ 1 large enough.For a different approach, the so-called Riccatian method, in the case of intermediate asymptotic ε ∈ 0, 1 , c 0 , see the technique from 2, 3 .
The study of asymptotically linear solutions to linear and nonlinear ordinary differential equations is of importance in fluid mechanics, differential geometry Jacobi fields, e.g., 4, page 239 , bidimensional gravity the geodesics of the Euclidean planar spray x 0 being the asymptotically linear solutions x t ct d , and others.In this note, we are interested in the existence of a fractional variant for the problem of asymptotically linear solutions which can be formulated as follows: are there any nontrivial fractional differential equations which have only asymptotically linear solutions and also their solution sets contain solutions (asymptotically linear) for all the prescribed values of numbers c, d, and ε?To the best of our knowledge, this is an open problem in the theory of fractional differential equations.
Fractional differential equations have been of great interest during the last few years.This follows from the intensive development of the theory of fractional calculus 5, 6 followed by the applications of its methods in various sciences and engineering 7 .We can mention that the fractional differential equations are playing an important role in fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, control theory, economy, nuclear magnetic resonance, mechanics, optics, signal processing, and so on.Basically, the fractional differential equations are used to investigate the dynamics of the complex systems; the models based on these derivatives have given superior results as those based on the classical derivatives, see 8, page 305 , 9-11 .
To introduce a fractional differential operator of order 1 α, there are three options.The first two consist of a mixed ordinary differential-Caputo fractional differential operator, namely, C 0 D α t x t 0D α t x t , and, respectively, a Riemann-Liouville fractional differential operator 0D The first variant of differential operator was used in 13 to study the existence of solutions x t of nonlinear fractional differential equations that obey the restrictions The second variant of differential operator, see 14 , was employed to prove that, for any real numbers x 0 , x 1 , the linear fractional differential equation possesses a solution x t with the asymptotic development A recent application of the Caputo derivative can be found in 15 .
All of these fractional differential operators are based upon the natural splitting of the second-order operator d 2 /dt 2 , namely, x x .Here, we shall introduce a different fractionalizing of x which is based on the identities stemming from the integration technique in the Lie algebra L 2 , cf. 16, page 23 .
In the following section, we give a positive partial answer to the preceding open question.In fact, we produce some simple conditions regarding the continuous function a : 0, ∞ → R such that, given c ∈ R − {0}, the fractional differential equation FDE below 0D α t tx − x x 0 a t x 0, t > 0, 1.9 possesses a solution with the asymptotic development x t ct x 0 o 1 when t → ∞.

Asymptotically Linear Solutions
Let us start with a result regarding the case of intermediate asymptotic.

2.1
Assume also that a ∈ C 0, ∞ , R is confined to

Abstract and Applied Analysis
Then, the FDE , with lim t 0 t 2−α x t 0, which verifies the asymptotic formula x t ct O t ε when t → ∞.
Proof.Introduce the complete metric space M D, δ , where D {y ∈ C 0, ∞ , R : sup t>0 t −ε |y t | ≤ c 1 , t > 0} and the metric δ is given by the usual formula In particular, lim t 0 y t 0 for all y ∈ D. Introduce the function x : 0, ∞ → R via the formulas Since lim t 0 x t 0, we deduce that x can be continued backward to 0; so, its extension x belongs to C 0, ∞ , R ∩ C 1 0, ∞ , R .Also, lim t 0 t 1−α y t lim t 0 t 2−α x t 0. Define further the integral operator T : M → M by the formula The estimate shows that T is well defined by taking into account 2.1 , 2.2 .

Abstract and Applied Analysis 5
Now, given y 1 , y 2 ∈ D, we have and so δ T y 1 , T y 2 ≤ c 1 δ y 1 , y 2 .The operator T being a contraction, it has a unique fixed point y 0 ∈ D. Since t ∞ t y 0 s /s 2 ds O t ε when t → ∞, the proof is complete.Theorem 2.2.Assume that 2.1 holds true and a ∈ C 0, ∞ , R verifies the sharper restriction where 1 > β > α ε.Then, the solution x of FDE 2.3 from Proposition 2.1 has the asymptotic development x t ct o 1 when t → ∞.
Proof.Notice that where B is the Beta function, cf. 8, page 6 .
Via 2.9 , we have the estimate

2.11
By means of L'H ôpital's rule, we conclude that recall 2.5 The proof is complete.
Our main contribution is given next.

2.13
Assume also that a ∈ C 0, ∞ , R satisfies the inequality Then the FDE 1.9 has a solution 2.17 As before, we have the estimates for all y, y 1 , y 2 ∈ D.
Finally, for the fixed point y 0 of the operator T , we have that

2.19
The proof is complete.

Conclusion
A particular case of Theorem 2.3 is when c 0, d 1, that is, when the solution of 1.9 reads as x t 1 o 1 for t → ∞.Notice from 2.14 that the behavior of the functional coefficient a t is confined to lim t → ∞ a t 0. However, there is no restriction with respect to the eventual zeros of a t .On the other hand, in the recent contribution 13, Section 3 , we were forced to request that the functional coefficient of the FDE has a unique zero in 0, ∞ .In conclusion, the fractional differential operators proposed in 1.9 , 2.3 allow more freedom for the functional coefficient.