key: cord-0064641-ylg12hxi
authors: Sutar, Sagar T.; Kucche, Kishor D.
title: Existence and data dependence results for fractional differential equations involving atangana-baleanu derivative
date: 2021-06-17
journal: Rend
DOI: 10.1007/s12215-021-00622-w
sha: 4c1261b5d5af8f5d97e3ad0bfa3a047d82300163
doc_id: 64641
cord_uid: ylg12hxi

In the current paper, we consider multi-derivative nonlinear fractional differential equations involving Atangana-Baleanu fractional derivative. We investigate the fundamental results about the existence, uniqueness, boundedness and dependence of the solution on various data. The analysis is based on a fractional integral operator due to T. R. Prabhakar involving generalized Mittag-Leffler function, the Krasnoselskii’s fixed point theorem and Gronwall-Bellman inequality with continuous functions.

Fractional differential equations (FDEs) [1] [2] [3] [4] [5] appeared as an excellent mathematical tool for modeling of many physical phenomena appearing in various branches of science and engineering such as viscoelasticity, self-similar protein dynamics, continuum and statistical mechanics, dynamics of particles etc. For more details, one can refer [6] [7] [8] [9] [10] [11] and furthermore articles referred in that. Crucial development about existence and uniqueness theory, various sorts of stabilities, data dependency and the controllability results for a different class of FDEs can be found in [12] [13] [14] [15] [16] and the references cited therein.

To avoid the singularity appearing in the classical fractional differential operators many researchers are attempting to build up the theory of fractional calculus by constructing different kinds of fractional derivative operators with the nonsingular kernel. In this sense, Caputo and Fabrizio [17] constructed a new fractional derivative which a variant of Caputo derivative with the singular kernel replaced by the exponential function as its kernel. Atangana and Baleanu in [18] introduced non singular Caputo and Riemann-Liouville version of fractional differential operator with Mittag-Leffler function as its kernel.

Taking advantage of the non-singular Mittag Leffler kernel present in the Atangana-Baleanu (AB)-fractional derivative operators, recently authors from various branches of applied mathematics developed and studied mathematical models involving AB-fractional derivative. Bonyah et al. [19] considered a mathematical model involving AB-fractional derivative for co-infection of cancer and hepatitis diseases. They examined existence and uniqueness, stability analysis, and reproductive number. In [20] , Ahmad et al. investigated the fractional-order tumor-immune-vitamin model with AB-fractional derivative for existence, uniqueness, and Hyres-Ulam stability. Authors in [21] did a comparative and chaotic study of tumor and effector cells through the fractional tumor-immune dynamical model with AB-fractional derivative. In [22] , the authors utilized the fractional AB derivative to study the numerical solution of the fractional immunogenetic tumor model. Study of transmission dynamics of COVID-19 mathematical model under ABC-fractional-order derivative has been dealt in [23] . Logeswari et al. [24] explored the mathematical model for spreading of COVID-19 infection in the world with AB-fractional derivative. Further, they created a framework that generates numerical outcomes to predict the outcome of the infection spreading all over India. Other few important works that attempted to handle the issue of diverse ailment modeled in the form of FDEs involving AB-fractional derivative are [25] [26] [27] [28] [29] [30] . For additional point by point concentrates on various qualitative and quantitative properties of solutions to FDEs with AB-fractional derivative, the interested reader can refer to [31] [32] [33] [34] [35] [36] [37] [38] [39] .

On the other hand, Mohamed et al. [40] , considered multi-derivative initial value problem for Caputo FDEs and studied the existence and uniqueness of the solution and obtained numerical solution through Adomian, Picard and predictor-corrector technique. Kucche et al. in [41] extended the work of [40] to the system of multi-derivative FDEs involving the Caputo fractional derivative and studied existence, uniqueness and continuous dependence of solution. Further, they have discussed validity, convergence, and error estimation for Picard's method.

Inspired by the work of [42] [43] [44] , on the line of [40, 41] , we consider multi-derivative nonlinear FDEs involving Riemann-Liouville version of AB-fractional derivative (ABR derivative) of the from:

where J = [0, T], T > 0 , 0 < < 1 , * 0 D denotes the ABR-fractional differential operator of order and f ∈ C(J × ℝ, ℝ) is a non-linear function.

We derive an equivalent fractional integral equation to ABR-FDEs (1.1)-(1.2) analytically and via Laplace transform. Using the properties of fractional integral operator E , , ;a+ , we derive some supplementary results. The existence of solution is obtained by using Krasnoselskii's fixed point theorem. We obtain uniqueness of solution via Gronwall-Bellman inequality as well as using the properties of fractional integral operator E , , ;a+ . The boundedness and the continuous dependence of the solution is obtained through Gronwall-Bellman inequality for continuous function.

We organize our work as follows: In sect. 2, we recall basic definitions and results about AB-fractional derivative and the generalised Mittag-Leffler function. In sect. 3, we derive an equivalent fractional integral equation to ABR-FDEs (1.1)-(1.2) analytically as well as using the Laplace transform. In sect. 4, we derive supplementary results and existence and uniqueness of solution. In sect. 5, we derive boundedness and data dependence of solution. In sect. 6, an example is provided to illustrate the existence results.

In this section, we recall basic definitions and results about AB-fractional derivative and the generalised Mittag-Leffler function. Definition 2.4 [45, 46] The generalized Mittag-Leffler function , (z) for the complex , , with Re( ) > 0 is defined by where ( ) k is the Pochhammer symbol given by Note that

We need the following results related with Laplace transformation.

Definition 2.5 [46, 47] Let , , , ∈ ℂ (Re( ), Re( ) > 0), b > a . Then, the fractional integral operator E , , ;a+ on a class L(a, b) is defined by

where ∈ ℂ, (Re( ) > 0) and D + a+ is the Riemann-Liouville fractional differential operator of order + with lower terminal a. Lemma 2.5 [46, 47] Let , , , ∈ ℂ (Re( ), Re( ) > 0) . If the integral equation is solvable in the space L(a, b), then its unique solution ( ) is given by where ∈ ℂ, (Re( ) > 0) and D + a+ is the Riemann-Liouville fractional differential operator of order + with lower terminal a. Lemma 2.6 (Krasnoselskii's Fixed Point Theorem [15] ) Let Ω be a Banach space. Let S be a bounded, closed, convex subset of Ω and let F 1 , F 2 be maps of S into Ω such that F 1 + F 2 ∈ S for every pair , ∈ S . If F 1 is contraction and F 2 is completely continuous, then the equation has a solution on S. 

Proof Proof follows by taking h( ) = f ( , ( )), ∈ J, in the Theorem 3.1. ◻

The proof of following theorem is based on the properties of fractional integral operator E , , ;a+ studied in [46, 47] . Then for any ∈ S and any 1 , 2 ∈ J with 1 < 2 , we find ∈ (a, b) , ∈ (a, b) . ∈ (a, b) . Consider the set,

One can verify that S is closed, convex and bounded subset of Banach space Ω . Consider the operators F 1 ∶ S → Ω and F 2 ∶ S → Ω defined by,

where we take F as defined in the Eq. We prove that the operators F 1 and F 2 satisfies conditions of Lemma 2.6. The proof of the same have been given in following steps.

Step 1) F 1 is contraction.

Using Lipschitz condition on f, for any , ∈ C(J) and ∈ J we obtain, This gives,

Step 2) F 2 is completely continuous. Using Ascoli-Arzela theorem and Theorem 4.1, one can easily verify that the operator F 2 = −F is completely continuous.

Step 3) F 1 + F 2 ∈ S , for , ∈ S. For any , ∈ S , using Theorem 4.1, we obtain By definition of R i.e. condition(4.4), we get

We write from inequalities (4.5) and (4.6) This gives

(4.5)

This shows that F 1 + F 2 ∈ S , for , ∈ S.

From steps 1 to 3, it follows that all the conditions of Lemma 2.6 are satisfied. Therefore by applying it, the operator equation If we choose a normalizing function B( ) satisfying above condition, then by applying Theorem 4.3, ABR-FDEs (6.1)-(6.2) has unique solution. One can verify that ABR-FDEs (6.1)-(6.2) has the unique solution one can acquire various qualitative properties of the higher class of fractional integrodifferential equations involving the AB fractional derivative in the sense of Caputo and Riemann-Liouville through the inequalities derived by B. G. Pachpatte [48] . But Pachpatte's inequalities can be applied only when the normalization function B( ) is strictly positive satisfying B(0) = B(1) = 1.

iv) Using Lemma 2.4 and Lemma 2.5, for any f ∈ C(J) , we have

An introduction to the fractional calculus and fractional differential equations

Fractional differential equations

Theory of fractional dynamic systems

Theory and applications of fractional differential equations

The analysis of fractional differential equations

On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity

A fractional calculus approach of self-similar protein dynamics

Application of Fractional Calculus in

Fractional calculus: some basic problems in continuous and statistical mechanics

Relaxation in filled polymers: A fractional calculus approach

Fractional Dynamics: Application of fractional calculus to dynamics of particles. Field and Media

Analysis of fractional differential equations

Theory of fractional functional differential equations

General uniqueness and monotone iterative technique for fractional differential equations

Existence of fractional neutral functional differential equations

Ulam stability and data dependence for fractional differential equations with Caputo derivative

A new definition of fractional derivative without singular kernel

New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model

Mathematical modeling of cancer and hepatitis co-dynamics with non-local and non-singular kernel

Analysis of the fractional tumour-immune-vitamins model with Mittag-Leffler kernel

A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment

A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative

Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative

Mathematical model for spreading of COVID-19 virus with the Mittag-Leffler kernel

A new fractional modeling and control strategy for the outbreak of dengue fever

A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator

A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence

A new features of the fractional Euler-Lagrange equation for a coupled oscillator using a nonsingular operator approach

Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleuno derivative

On comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative

On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative

On the nonlinear dynamical systems within the generalized fractinal derivative with Mittag-Leffler kernel

Fractional differential equations with Atangana-Baleanu fractional derivative: Analysis and applications

Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernel

Study of evolution problem under Mittag-Leffler type fractional order derivative

New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-differential equations

On new approach of fractional derivative by Mittag-Leffler kernel to neutral integro-differential systems with impulsive conditions. Chaos, Solitons Fract

Analysis of nonlinear fractional differential equations involving AB-Caputo derivative

On nonlinear hybrid fractional differential equations with AB-Caputo derivative

Adomian and perdictor-corrector methods for an initial value problem of arbitrary (fractional) prders differential equation

Theory of system of nonlinear fractional differential equations

On certain Volterra integro-differential equations

Theory of nonlinear implicit fractional differential equations

Some theorems on fractional semiliear evolution equations

Higher Transcendental Functions

Generalized Mittag-Leffler function and generalized fractional calculus operators

A singular integral equation with a generalized Mittag-Leffler function in the kernel

Inequalities for differential and integral equations

‖F 1 + F 2 ‖ ≤ R, for all , ∈ S.Applying Lemma 2.7, we get which shows that ( ) = ( ) , for all ∈ J . This proves the uniqueness of solution of ABR-FDEs (1.1)-(1.2). 

Applying Lemma 2.7, we get Let any , 0 ∈ ℝ and consider the following system of ABR-FDEs Proof We find for any ∈ J Applying Lemma 2.7, we get

Because of the presence of the nonsingular kernel in the equivalent fractional integral equation to FDEs involving AB derivatives, we can reasonably apply the Gronwall-Bellman inequality with continuous real valued functions to investigate the qualitative properties. Of Course, special attention needs in the case of complex valued functions. Further,