Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

We establish here that under some simple restrictions on the functional coefficient the fractional differential equation , has a solution expressible as for , where designates the Riemann-Liouville derivative of order and . 1. Introduction Consider the ordinary differential equation where the function is continuous such that Here, the functions and are continuous, and there exists with Then, given , , (1.1) has a solution , defined in a neighborhood of , which is expressible as for , as for and, finally, as for when . Such a solution is called asymptotically linear in the literature. In particular, these developments apply to the homogeneous linear differential equation . 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 large enough. For a different approach, the so-called Riccatian method, in the case of intermediate asymptotic ( , ), 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 being the asymptotically linear solutions ), 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 , , and ? 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