Elsevier

Fuzzy Sets and Systems

Volume 404, 1 February 2021, Pages 111-140
Fuzzy Sets and Systems

Non-instantaneous impulses interval-valued fractional differential equations with Caputo-Katugampola fractional derivative concept

https://doi.org/10.1016/j.fss.2020.05.004Get rights and content

Abstract

In this paper, by using the Caputo-Katugampola fractional derivative concept for the interval functions, a non-instantaneous impulsive value problem of interval differential equations is investigated. The first purpose is to study the existence and uniqueness results of the solution of the given problem, and the second purpose is to present the Ulam-Hyers-Mittag-Leffler's stability results of the above problem. Finally, some examples are given to illustrate our main results.

Introduction

During the past decades, the theory of impulsive differential equations has received great attention due to their extensive applications in realistic mathematical modeling of a wide variety of practical situations and of in epidemic, optimal control [57], [65], population ecology [1], mechanical and engineering [26] and etc. Nowadays, the theory of impulsive fractional differential equations has attracted many authors and it has recently proved to be a valuable tool for the description of memory and hereditary properties of various processes that are characterized by rapid changes in their state and in which sudden, discontinuous jumps occur. Formally, there are two main types of impulsive effects to fractional differential equations such as instantaneous impulses and non-instantaneous impulses. In the first one, instantaneous impulsive fractional differential equations provide a significant instrument for the illustration of processes in the system involving the short-term perturbations whose duration is negligible in comparison with the overall duration of the whole process. For more recent contributions on this topic, we refer the interested reader to see the interesting papers [14], [18], [59], [60] and references therein. Meanwhile, non-instantaneous impulsive fractional differential equations are introduced to replace the theory of instantaneous impulses for handling the dynamics of evolutionary processes involving the disturbance which starts at an arbitrary fixed point and remains active on a finite time interval. The first literature on abstract semilinear differential equations with not instantaneous impulses in the space of piecewise continuous functions with PC-norm in Banach space is introduced by Hernández and O'Regan [19]. And these results were extended and developed by Pierri, O'Regan and Rolnik [37] in the space of piecewise continuous functions with PCα-norm in Banach space. Up to now, extensive works have been done in the field of non-instantaneous impulses, and there has been a significant development in the theory of the existence and the stability of fractional differential equations. For more recent development on this topic, one can see the interesting results of Sousa et al. [54], Wang et al. [62] and Yang et al. [64].

On the other hand, the theory of fuzzy calculus and fuzzy differential equations has been taken up by a large number of mathematicians and the study of this area has grown to be one of the most important subjects in the mathematical analysis area. Nowadays, the theory of fuzzy analysis has been attracted by many famous researchers since the great use of this theory in describing uncertainty that appears in many mathematical models expressed by dynamical systems or computer models of some deterministic real-world phenomena due to incomplete information. For more recent development on this theory, the readers can refer to the monographs and the interesting results of Ahmad et al. [9], Stefanini and Bede [10], [56], Fard et al. [13], Khastan, Nieto and Rodríguez-López [29], [30], Mazandarani et al. [32], [33], [34], Pedrycz et al. [35], [36], Gomes and Barros [17] and the references therein. It is well-known that there is an extremely important connection between fuzzy analysis and interval-valued analysis (Pedrycz and Gomide [35]), and the theory of interval differential equations was also proposed to handle the interval uncertainty that appears in many mathematical models. This theory has been developed in several theoretical directions, and a large number of applications have been considered for many different real problems (e.g., see Lupulescu [42], Malinowski [44], [45], Stefanini and Bede [55], [56], Hoa [20], Gasilov et al. [15], [16] and the references cited therein). In recent years, because of applications of fractional calculus and fractional differential equations in real-world systems, fuzzy fractional calculus, and fuzzy fractional differential equations have emerged as the significant topic, and the topic of the fractional calculus and fractional dynamic systems in a fuzzy environment can be applied as an important mathematical tool for modeling of practical systems with the effects of uncertainties. Therefore, the consideration and analysis of fractional order uncertain dynamical systems are essential in both research and practice, and so this field has attracted significant interests among the researchers in the last years [4], [5], [6], [7], [8] and the references therein. One can see that most of the current research results of fuzzy differential equations involving Caputo fractional derivative and Riemann-Liouville fractional derivative have been paid more and more attention by a large number of mathematicians because of the interesting and their applications. Very recently the approaches of Hadamard fractional derivative are also proposed as a new concept of fuzzy fractional derivative, which is useful in many problems related to mechanics and engineering, e.g., the crack problems of elasticities. We notice that the Hadamard fractional derivative is different from the Riemann-Liouville fractional derivative and Caputo fractional derivative in that the kernel function of the fractional integral in its definition contains the logarithmic function of an arbitrary exponent. Both of the concepts of the Riemann-Liouville fractional derivative and Hadamard fractional derivative have their drawbacks that the derivative of a constant is not zero. To overcome the above disadvantage of the Riemann-Liouville fractional derivative, the Caputo fractional derivative was proposed. Meanwhile, Caputo-Hadamard fractional derivative was introduced by modifying the Hadamard derivative to fix the above drawbacks. For some fundamental results in the theory of fuzzy fractional differential equations involving fuzzy-type Riemann-Liouville fractional derivative and fuzzy-type Caputo fractional derivative, the existence and uniqueness results of solution to fuzzy fractional differential equations and the method to find analytical, numerical solutions for initial value problems of fuzzy fractional differential equations, we refer the reader to the recent papers [21], [22], [23], [24], [39], [40], [41], [38], [43], [46], [48], [49] and the references therein. In addition, motivated by the definition of Caputo-Katugampola fractional derivative, which is introduced by Katugampola [27], [28], in [25] authors proposed a new definition of fuzzy fractional derivative the so-called Caputo-Katugampola fuzzy fractional derivative that generalizes the Caputo and Caputo-Hadamard fractional derivatives in the fuzzy setting into a single form. The authors also presented the existence and uniqueness results of the solutions to the initial value problem of fractional fuzzy differential equations with the above fractional derivative concept.

To the best of our knowledge, the theory of instantaneous impulses in the area of fuzzy differential equations is very limited, and there are only a few remarkable works which provide the main theoretical tools for the qualitative analysis and the solving method to find analysis solutions of the impulsive problems in the fuzzy concept. For more recent contributions on this topic, one can refer to the works of Rodríguez-López and Nieto [46], [47], An et al. [7], Benchohra et al. [11], Vu et al. [58], Wang et al. [63] and the references cited therein. Meanwhile, there is no literature on non-instantaneous impulsive in the area of fractional differential equations in the fuzzy and interval setting. So, research in this paper is strongly motivated by the works presented in [19], [37], [60], [62]. We continue to study fractional differential equations with not-instantaneous impulsive effects subjected to the interval uncertainty. In this work, by considering the concept of Caputo-Katugampola fractional derivative in the interval environment, we consider a new class of interval-valued differential equations with non-instantaneous impulsive effects of the form as follows:{Da+α,ρCX(t)=F(t,X(t)),t(tk,sk],k{0,1,2,...,m}X(t)=Ik(t,X(t)),t(sk1,tk],k{1,2,...,m}X(a)=X0, where ρ>0,α(0,1); F:[a,b]×I(R)I(R) is jointly continuous; Ik:[sk1,tk]×I(R)I(R) is also jointly continuous for all k=1,2,...,m, which is called non-instantaneous impulses, and tk satisfy a=t0<s0<t1s1t2<<tmsmtm+1=b; I(R) is called the space of all nonempty compact intervals of the real line R; Da+α,ρCX is called the Caputo-Katugampola fractional derivative of the interval function X given by Definition 2.5 in Section 2. The advantage of the fractional operator proposed here is the freedom of choice of the Caputo and Caputo-Hadamard fractional differential operators with the values of the parameter ρ. This yields that the problem (1.1) allows one to interpolate many different types of the non-instantaneous impulsive problem for differential equations. Indeed, by basing on the problem (1.1) we observe that:

  • 1)

    Let ρ tend to 1 and α=1. Then, if the interval-valued functions F and Ik become the crisp functions and X0 is a crisp number in R, the problem (1.1) turns into the non-instantaneous impulsive problem for real-valued differential equations investigated by Wang et al. [60].

  • 2)

    Let α(0,1). Then, if the interval-valued functions F and Ik become the crisp functions and X0 is a crisp number in R, the problem (1.1) turns into the non-instantaneous impulsive problem of real-valued fractional differential equations with the concept of Caputo fractional derivative when ρ1 investigated by Wang et al. [62].

  • 3)

    Let α(0,1) and ρ0+. Then, the problem (1.1) turns into the non-instantaneous impulsive problem of real-valued differential equations with Caputo-Hadamard fractional derivative if the interval-valued functions F and Ik become the crisp functions and X0 is a crisp number in R.

  • 4)

    Let ρ tend to 1, then the problem (1.1) becomes the non-instantaneous impulsive problem for fuzzy differential equations when α=1.

  • 5)

    Let α(0,1). Then, the problem (1.1) turns into the non-instantaneous impulsive problem of fuzzy differential equations with the concept of Caputo fractional derivative when ρ1 and becomes the non-instantaneous impulsive problem of fuzzy differential equations with the concept of Caputo-Hadamard fractional derivative when ρ0+.

Instead of examining a large number of equations given by 1) - 5), in this paper we only need to consider the problem (1.1). In this sense, this study provides new results, which are valid for all possible particular cases because the properties of the more general operator are preserved in particular cases including the crisp cases. Our contributions of this paper include: (i) establishing a natural formula of solutions for the problem (1.1) by providing an appropriate condition to show the equivalence between the problem (1.1) and fractional interval integral equations with impulsive effects; (ii) by applying the fixed point approach, the results of existence and uniqueness of solution of the problem (1.1) are presented; (iii) introduce Ulam-Hyers's stability concepts for the problem (1.1) and provide the sufficient conditions to guarantee the problem (1.1) is Ulam-Hyers-Rassias stable and Ulam-Hyers-Mittag-Leffler stable.

We organize the rest of this paper as follows. Some definitions, notations, remarks, and theorems that will be used to prove our main results are presented in Section 2. In Section 3, a standard framework is established to obtain a suitable formula of solutions for the problem (1.1), and new concepts of Ulam-Hyers's stability for the problem (1.1) are given. In addition, we give three main results as follows: the first and second results based on fixed point theorem of the alternative for contractions on a generalized complete metric space to investigate the existence of solution and μ-Ulam-Hyers-Rassias stability for the problem (1.1); consequently, in the third result, the sufficient conditions are proposed to guarantee the problem (1.1) is Ulam-Hyers-Mittag-Leffler stable. In Section 4, some examples are given to illustrate the theoretical results.

Section snippets

Fundamental theorems

In this section, some basic notations and results on fractional order integral and differential calculus for interval-valued functions are presented which will be useful throughout the paper. Denote by I(R) the set of all nonempty compact intervals of the real line R. Let A=[A_,A], B=[B_,B] belong to I(R), then the usual interval operations, i.e. Minkowski addition and scalar multiplication, are defined as follows:respectively. Let λ=1, we denote the scalar multiplication A:=(1)A=(1)[A_,A

Main results

Let J=[a,b]. Throughout this paper, by PC(J,I(R)), we denote the space of the piecewise continuous functions (it means X(t) is continuous on Jk,k=0,1,2,...,m, where Jk=(tk,sk] and t0=a,sm=b) with the uniform norm XPC=sup{H[X(t),0],tJ} such that X(tk+) and X(tk) exist for any k=0,1,2,...,m. In this section, by using the concept of the Caputo-Katugampola generalized fractional derivative given as in Definition 2.5, we consider the following fractional interval differential equations with not

Numerical examples

In this section, we present the example to two classes of non-instantaneous impulsive problems of the population growth model in the interval environment with the concept of Caputo-Katugampola fractional derivative.

Example 4.1

Let ρ>0 be the fixed parameter and α(0,1). We consider the following non-instantaneous impulsive problem of the population growth mode with the concept of Caputo-Katugampola fractional derivative:{Da+α,ρCX(t)=(r(t)m(t))X(t),t(a,2](3,5],X(t)=[d1b(t),d2b(t)],t(2,3],X(a)=X0, where

Conclusion and future works

About the novelty of this paper, we emphasize that a new concept of piecewise continuous solutions of interval-valued fractional differential equations is initially given, which is involving both non-instantaneous impulsive effects and the concept of Caputo-Katugampola fractional derivative. In other words, our problem (3.1) is more general than the models in the previous results in both cases of crisp fractional differential equations with non-instantaneous impulses and uncertain fractional

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.319.

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