Abstract
It is known that, under certain conditions, solutions of some ordinary differential equations of first, second or even higher order are asymptotic to polynomials as time goes to infinity. We generalize and extend some of the existing results to differential equations of non-integer order. Reasonable conditions and appropriate underlying spaces are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach power type functions as time goes to infinity. The case of fractional differential problems with fractional damping is also considered. Our results are obtained by using generalized versions of GronwallBellman inequality and appropriate desingularization techniques.
Highlights
In this paper, we consider the following fractional differential equationD01+αy (t) = f (t, y (t), D0βy (t)) t > 0, (1.1)with initial conditionsD0αy (t) |t=0 = b2 and I01−αy (t) |t=0 = b1, b1, b2 ∈ R, (1.2)where D0σ is the Riemann-Liouville fractional derivative of order σ > 0 and 0 ≤ β ≤ α ≤ 1
We would like to shed some light on this issue
We shall establish some conditions under which all solutions of the fractional differential problem (1.1)–(1.2) have the following property: limt→∞ y (t) /tα = a, for some real number a
Summary
Where D0σ is the Riemann-Liouville fractional derivative of order σ > 0 and 0 ≤ β ≤ α ≤ 1. They proved that the linear fractional equation D0αy + a (t) y = 0, 0 < α < 1 t > 0 has a solution y ∈ C ([0, ∞), R) enjoying the asymptotic property y (t) = b + ctα + O tα−1 , as t → ∞,. We shall establish some conditions under which all solutions of the fractional differential problem (1.1)–(1.2) have the following property: limt→∞ y (t) /tα = a, for some real number a The proof of this result is based on the GronwallBellman inequality and its generalization due to Bihari [2].
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