Abstract
We establish here that under some simple restrictions on the functional coefficienta(t)the fractional differential equationD0tα[tx′−x+x(0)]+a(t)x=0, t>0, has a solution expressible asct+d+o(1)fort→+∞, whereD0tαdesignates the Riemann-Liouville derivative of orderα∈(0,1)andc,d∈ℝ.
Highlights
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/dt[2], namely, x x
The FDE 1.9 has a solution x ∈ C 0, ∞, R ∩ C1 0, ∞, R, with x 0 limt 0 t2−αx t 0, which has the asymptotic development x t ct d o 1 when t −→ ∞
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, ∞
Summary
Consider the ordinary differential equation x f t, x 0, t ≥ 1, 1.1 where the function f : 1, ∞ × R → R is continuous such that f t, x. 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 t1−ε for ε ∈ 0, 1 and, as ct d o 1 for ε 1 when t → ∞ Such a solution is called asymptotically linear in the literature. 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. 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/dt[2], namely, x x. We produce some simple conditions regarding the continuous function a : 0, ∞ → R such that, given c ∈ R − {0}, the fractional differential equation FDE below
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