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