Journal of Function Spaces

Volume 2019, Article ID 6091602, 9 pages

https://doi.org/10.1155/2019/6091602

## Some Remarks on Estimate of Mittag-Leffler Function

^{1}College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China^{2}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China^{3}School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Correspondence should be addressed to Zhen Wang; nc.ude.tsuds@nehzgnaw

Received 16 July 2018; Revised 1 November 2018; Accepted 18 December 2018; Published 1 April 2019

Academic Editor: Maria Alessandra Ragusa

Copyright © 2019 Jia Jia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The estimate of Mittag-Leffler function has been widely applied in the dynamic analysis of fractional-order systems in some recently published papers. In this paper, we show that the estimate for Mittag-Leffler function is not correct. First, we point out the mistakes made in the estimation process of Mittag-Leffler function and provide a counterexample. Then, we propose some sufficient conditions to guarantee that part of the estimate for Mittag-Leffler function is correct. Meanwhile, numerical examples are given to illustrate the validity of the two newly established estimates.

#### 1. Introduction

Fractional calculus can date back to the seventeenth century, and now it has attracted considerable research interests due to its widespread applications in many fields. There are mainly two types of methods in the dynamic analysis of fractional-order nonlinear systems, that is, Lyapunov function based method and estimation based method. When estimation based method is employed, the solution of the fractional-order system being studied is usually expressed in terms of the Mittag-Leffler function. Obviously, the correctness of the estimate of Mittag-Leffler function is crucial to the whole estimation process and plays an important role if the estimation based method is adopted. Recently, estimation based method has been widely applied to the study of finite-time stability and synchronization of fractional-order memristor-based neural networks [1–7], stability and stabilization of nonlinear fractional-order systems [8–13], finite-time stability of fractional-order neural networks [14, 15], synchronization of fractional-order chaotic systems [16], consensus analysis of fractional-order multiagent systems [17–19], etc. The estimate on Mittag-Leffler function was first proposed in [20]. The definition of Mittag-Leffler function and the estimate of Mittag-Leffler function can be described by Definition 1 and Lemma 2, respectively, as follows.

*Definition 1 (see [21]). *The Mittag-Leffler function with one parameter is defined as where and .

The Mittag-Leffler function with two parameters is defined aswhere and . When , one has , and when and , one further has .

Lemma 2 (see [20]). *For Mittag-Leffler function, the following properties hold.**(i) There exist constants such that, for any ,where denotes a matrix and denotes any vector or induced matrix norm.**(ii) If , then for If is a diagonal stability matrix, then there exists a constant such that for where is the largest eigenvalue of the diagonal matrix .*

However, we have to point out that Lemma 2 is incorrect.

In [20], inequalities (3) and (4) are proved as follows:Actually, there are two problems in the above-mentioned proof. First,does not necessarily hold. In fact, (9) holds if all the elements of matrix are nonnegative, because matrix norms, such as -norm, -norm, and -norm, have the property of weak monotony. In other words, (9) may not hold when there exist negative elements in or . Second, does not exist when and . To confirm this point, let ; when and , the behavior of with and is shown in Figures 1(a) and 1(b), respectively. It can be obviously observed from Figure 1 that goes to infinity as goes to infinity, so has no supremum when or as goes to infinity for a fixed value of .