Asymptotic Integration of Some Classes of Fractional Differential Equations

Abstract

In this paper we deal with the problem of asymptotic integration of nonlinear higher order fractional differential equations of the Caputo's type. We give some conditions under which all global solutions of these equations behave like linear functions as t → ∞.

On the oscillatory behavior of solutions of nonlinear fractional differential equations

“…Oscillation results for fractional differential equations are scarce; some results can be found in [9][10][11][12]14,15]. It seems that there are no such results for fractional differential equations.…”

Asymptotic Behavior of Solutions to Nonlinear Fractional Differential Equations

“…He proved that the solution y(t) of the problem (1.12) is asymptotic to b + ct as t → ∞, for some b, c ∈ R. Also, in 2013, the same author [21] considered the following generalization…”

“…Our results will also extend the results obtained for equations (1.3) and (1.4) with α = β = 1. In particular, the problems treated in [20,21] become special cases of (1.1) corresponding to y (0) = y (0) = 0. Moreover, by the same occasion, we improve several results found in the literature, concerning the integer order, where the authors have been forced to work away from zero.…”

On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

“…, and c 0 is a real constant. Oscillation and asymptotic properties of solutions of integro-differential equations and fractional differential equations are relatively scarce in the literature; some results can be found in [2,7,9,10,12,[17][18][19][20][21]. The only results to date for forced fractional differential equations with positive and negative terms of the type (1.1) appear to be those in [11] where sufficient conditions for boundedness of nonoscillatory solutions are obtained.…”

“…The idea to transform a fractional differential equation into a Volterra integral equation is not new. For example, Medved' did this for a much simpler equation in [20,Lemma 2.5]; in this regard also see [21,Lemma 1]. We make use of Young's inequality which is not the case for example in [20] and [21].…”