Attractivity of solutions of Riemann–Liouville fractional differential equations

. Some new weakly singular integral inequalities are established by a new method, which generalize some results of this type in some previous papers. By these new integral inequalities, we present the attractivity of solutions for Riemann-Liouville fractional differential equations. Finally, several examples are given to illustrate our main results


Introduction
The study of fractional differential equations has been of great interest in the past three decades. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications in various sciences. In particular, the existence, uniqueness and stability results of fractional differential equations have been studied by many papers and books. In recent years, many researchers have began to investigate the attractivity of solutions of fractional differential equations. For example, Furati and Tatar [4] investigated the asymptotic behavior for solutions of a weighted Cauchy-type nonlinear fractional problem. Kassim, Furati and Tatar [8] studied the asymptotic behavior of solutions for a class of nonlinear fractional differential equations involving two Riemann-Liouville fractional derivatives of different orders. Zhou et al. [13] studied the attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. Gallegos and Duarte-Mermoud [5] studied the asymptotic behavior of solutions to Riemann-Liouville fractional systems. Tuan et al. [11] presented some results for existence of global solutions and attractivity for multidimensional fractional differential equations involving Riemann-Liouville derivative. Cong, Tuan and Trinh [2] presented some distinct asymptotic properties of solutions to Caputo fractional differential equations.
In 1981, Henry [7, p. 190] studied the following weakly singular integral inequality where α, β, γ are positive with β + γ > 1 and α + γ > 1. Webb [12] also studied the following weakly singular Gronwall inequality where 0 < α, β, γ < 1 with α + γ < 1 and β + γ < 1. Recently, Zhu [14] considered the following inequality where α > δ ≥ 0 and 0 < β < 1. Zhu [15] also considered the following weakly singular integral inequality where 1 > α ≥ δ ≥ 0, 0 < µ < 1 and 0 < β < 1. Some results of this type are also proved by Denton and Vatsala [3], Haraux [6], Kong and Ding [9]. Applying weakly singular integral inequality (1.1), we begin to investigate the attractivity of solutions of fractional differential equation where β ∈ (0, 1) and t ∈ (0, +∞). As far as I know, there have been few papers to study the attractivity of fractional differential equation (1.6) by weakly singular integral inequalities. The conclusion and the method of the proof in this paper seem to be new. The outline of this paper is as follows. In Section 2, we introduce some notations, definitions and theorems needed in our proofs. In Section 3, we obtain some new results concerning weakly singular integral inequalities. In the last Section, we give some sufficient conditions on the attractivity of solutions of fractional differential equation (1.6). Finally, some examples are given to illustrate our main results.

Preliminaries
In this section, we introduce some notations, definitions and theorems which will be needed later.
is called the Riemann-Liouville fractional integral operator of order β.
is an absolutely continuous function, is called the Riemann-Liouville fractional differential operator of order β.
Using the Hölder inequality, Zhu [15] obtained the following inequality.
Recently, Zhu [15,Corollary 4.5] obtained the following result which is very useful for the study of the main purpose of this paper. Theorem 2.5. Let 0 < β < 1 and 0 < µ ≤ 1. Suppose f : (0, +∞) × R → R is a continuous function, and there exist nonnegative functions l(t) and k(t) such that β . Then the fractional differential equation (1.6) has at least one global solution in C 1−β (0, +∞).

Weakly singular integral inequalities
In this section, we are now to prove some results concerning weakly singular integral inequalities, which can be used to study the attractivity of solutions for fractional differential equation (1.6). We first study the weakly singular integral inequality (1.1) for the case µ = 1.
Proof. Applying Lemma 2.4, we have In (3.6), we know that A(t) is a nondecreasing function on [0, +∞) and using the Gronwall integral inequality [1, Corollary 1.2], we obtain From (3.3) and (3.7), we get Thus, we complete the proof.