Existence and stability for a nonlinear hybrid differential equation of fractional order via regular mittag–leffler kernel

S P E C I A L I S S U E P A P E RExistence and stability for a nonlinear hybrid differential kernel 1 Faculty of Exact Sciences and Informatics, UMAB Abdelhamid Ibn Badis, University of

S P E C I A L I S S U E P A P E R

Existence and stability for a nonlinear hybrid differential

kernel

1 Faculty of Exact Sciences and

Informatics, UMAB Abdelhamid Ibn

Badis, University of Mostaganem,

Mostaganem, Algeria

2 Department of Statistics, Mathematical

Analysis and Optimization, Institute of

Mathematics, University of Santiago de

Compostela, Santiago de Compostela,

Spain

3 Department of Mathematics and General

Sciences, Prince Sultan University,

Riyadh, Saudi Arabia

4 Department of Medical Research, China

Medical University, Taichung, Taiwan

5 Department of Computer Science and

Information Engineering, Asia University,

Taichung, Taiwan

Correspondence

Thabet Abdeljawad, Department of

Mathematics and General Sciences, Prince

Sultan University, PO Box 66833, Riyadh

11586, Saudi Arabia.

Email: tabdeljawad@psu.edu.sa

Communicated by: D Baleanu

Funding information

Agencia Estatal de Investigación, Grant/

Award Number: MTM2016-75140-P;

Xunta de Galicia, Grant/Award Number:

ED431C 2019/02; Prince Sultan

University, Grant/Award Number:

RG-DES-2017-01-17

This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem Then, by means of the Dhage fixed-point principle, we discuss the existence of mild solutions Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem

K E Y W O R D S hybrid differential equations, Atangana –Baleanu derivative, Ulam–Hyers stability

M S C C L A S S I F I C A T I O N 34A38; 32A65; 26A33; 34K20

Since its first appearance, fractional calculus has become a major part of pure and applied mathematics, where it has occupied researchers in the scientific world especially due to the huge use of fractional derivatives for modeling phe-nomena in many different areas such as medicine, dynamical systems, quantum mechanics, thermodynamics, neural networks, economics, demography, and geophysics Among the many articles and scientific volumes devoted to the applications of fractional calculus, we mention for the reader1as a general overview,2,3for applications to dynamical

DOI: 10.1002/mma.7349

Math Meth Appl Sci 2021;1–11 wileyonlinelibrary.com/journal/mma © 2021 John Wiley & Sons, Ltd 1

systems,4,5 for applications to viscoelasticity,6 for applications to geohydrology,7 for applications to nanotechnology, and recent papers8–10for some other applications

In particular, the Caputo derivative reserves a special place in fractional calculus because of its initial value proper-ties.11 It has helped to describe nonautonomous systems and various problems with nonlocal properties But this derivative is not always able to well describe all problems in nature, because of its singular kernel In the 2010s, novel operators have been constructed with nonsingular kernels, for the purpose of getting more suitable and conve-nient operators for modeling certain problems Here, we pay particular attention to the definition of Atangana and Baleanu,12 which brings into play the regular Mittag–Leffler kernel Their derivative of Caputo type, called ABC for short, will be used in this paper For a standard source for the value of the Caputo derivative in applications, we may refer to Diethelm11and for applications of fractional calculus to Ionescu et al and Sun et al.13,14For further recent theoretical and numerical results in the field of fractional calculus and its applications, we recommend several works.15–19

Using the ABC derivative, motivated and inspired by several works,8,20–28this paper will deal with the following:

ABC

0 Dγ yðtÞ gðt,yðtÞÞ

= hðt,yðtÞÞ 8t  J = ½0,T,0 < γ < 1,

a yð0Þ gð0,yð0ÞÞ+ b

yðTÞ gðT,yðTÞÞ= c, where a, b, c ℝ hð0,yð0ÞÞ = 0,

8

>

>

>

>

ð1Þ

where g CðJ × ℝ,ℝ∖f0gÞ,h  CðJ × ℝ,ℝÞ and a + b ≠ 0

The rest of this article will be organized as follows: Section 2 will be devoted to some essential preliminary tools and concepts that we need in order to proceed Section 3 will be for the integral representation of the solution and its related existence result via fixed-point theory The Ulam–Hyer stability of the investigated system and illustrative example will be given in this section of the main results Finally, our conclusions will be summarized in Section 4

2 | P R E L I M I N A R I E S

We begin this section by presenting notations and main definitions that will be useful in the rest of this paper Let

J = ½0,T be an interval in ℝ, ~~S = CðJ × ℝ,ℝÞ is the continuous functions class, g : J × ℝ ! ℝ, and let CðJ × ℝ,ℝÞ indi-cate the class of Carathéodorian functions h: J × ℝ ! ℝ, where h is the Lebesgue integrable on J, such that

? t 7! h(t, y) is a measurable map for each y  ℝ

?? t 7! h(t, y) is a continuous map for each t  J

Definition (Gorenflo and Mainardi29) The fractional integral in the sense of Riemann–Liouville with order γ > 0 for

a continuous function f on [a, b] is shown by

aIγ

ð ÞðyÞ = 1

ΓðγÞ ð

y

a

y−σ

ð Þα−1fðσÞdσ, γ > 0, a < y ≤ b:

Definition (Atangana and Baleanu and Baleanu and Fernandez12,30) Let f be differentiable on (a, b), such that

f0 L1

(a, b), a < b,γ  [0, 1] Then, the derivative with a fractional order in the left ABC sense is defined as

ABC

a Dγ

 

ðtÞ =N ðγÞ

1−γ ð

t a

f0ðσÞEγ −γðt −σÞγ

1−γ

dσ

and in the left ABR sense (Riemann–Liouville type) is defined as

a Dγ

 

ðtÞ =N ðγÞ

1−γ

d

dt ð

t a

fðσÞEγ −γðt −σÞγ

1−γ

dσ,

and the fractional integral relative to the above operators is

AB

a Iγ

  ðtÞ = 1−γ

N ðγÞ fðtÞ + γN ðγÞðaIαfÞðtÞ,

whereN ðγÞ > 0 is a normalization function satisfying N ð0Þ = N ð1Þ = 1

In Atangana and Baleanu and Baleanu and Fernandez,12,30 it has been verified that ðAB

a IγABRa Dγ ÞðtÞ = f ðtÞ and

ðABR

a DγABa Iγ ÞðtÞ = f ðtÞ In fact, the work of Baleanu and Fernandez30

also contains other mathematical results on the ABC operators

Also in Atangana and Baleanu,12 the authors have established that ABC

a Dγ

 

ðtÞ =

ABR

a Dγ

 

ðtÞ−N ðγÞ1−γ fðaÞEγ − γ

1−γðt −aÞγ

Abdeljawad31has extended the definition to arbitraryγ > 0 and has introduced the following result

Lemma Let y(t) be well defined on [a, b] andγ  (n, n + 1], then

ðAB

a IγABCa DγyÞðtÞ = yðtÞ−Xn

k = 0

yðkÞðaÞ

k! ðt −aÞ

k:

The reader may also be interested in the further extension of Fernandez32to a complex gamma

Theorem (Dhage33) Assume that S is a non-empty subset of ~~S, which is closed, bounded, and convex, and A1: ~~S 7! ~~S and A2: ~~S 7! ~~S are two operators satisfying the following assumptions:

1 A1is Lipschitizian with a constant l,

2 A2is completely continuous,

3 y = A1 yA2x) y  S,8x  S, and

4 lM < 1, where M =kA2ðSÞk = sup kA2ðyÞ k;y  S,

then the equation y = A1yA2yadmits a solution in S

Let us start this section by defining what we mean by a solution of problem (1)

Definition The function y CðJ,ℝÞ is said to be a solution of problem (1) if

1 t7! y

gðt,yÞis a continuous function8y  ℝ and

2 the equations in (1) are satisfied by y

We denote by L1ðJ;ℝÞ the Lebesgue integrable functions endowed with the k  kL 1:

kxkL1=

ð

T 0

xðsÞ

j jds:

We also need to introduce the following hypotheses:

ðℑ1Þ The function y 7! y

gðt,yÞis injective inℝ, 8t  J

ðℑ2Þ gðt,yj 1Þ−gðt,y2Þj≤ w1j y1−y2j

8t  J and y1, y2 ℝ and w1> 0 is real number

ðℑ3Þ KðtÞ  L1ðJ,ℝÞ where hðt,yÞj j≤ KðtÞ, 8t  J

ðℑ4Þ hðt,yj 1Þ−hðt,y2Þj≤ w2j y1− y2j

8t  J and y1, y2 ℝ and w2> 0 is real number

ðℑ5Þ There exists η1,η2 ℝ

+, where gðt,yÞj j≤ η1and hðt,yÞj j ≤ η2, for all t J, for all y  ℝ

Lemma Let 0 <α < 1, and assume that ðℑ1Þ holds Then, the solution of the problem

ABC

0 Dγ yðtÞ gðt,yðtÞÞ

= fðtÞ 8t  J,γ  ½0,1,

a xð0Þ gð0,yð0ÞÞ+ b

yðTÞ gðT,yðTÞÞ= c where a, b, c ℝ and a + b ≠ 0

fð0Þ = 0

8

>

>

>

>

has the form

yðtÞ = g t,yðtÞ½ ð Þ 1−γ

N ðγÞ fðtÞ + γN ðγÞðIγ ÞðtÞ

− 1

a+ b b

1−γ

N ðγÞ fðTÞ + γN ðγÞðIγ ÞðTÞ

−c

:

ð2Þ

Proof Suppose that y is solution of (1), then by the above definition, we state thatgðt,yðtÞÞyðtÞ is continuous Applying the ABC fractional derivative of orderα, we retrieve the first equation in (1) Replacing t = 0, t = T in (2), we get

yð0Þ gð0,yð0ÞÞ=− 1

a+ b b

1−γ

N ðγÞfðTÞ + γN ðγÞð ÞðTÞIγ

−c

, yðTÞ

gðT,yðTÞÞ=

1−γ

N ðγÞfðTÞ + γN ðγÞð ÞðTÞ−Iγ

1

a+ b b

1−γ

N ðγÞfðTÞ + αN ðγÞð ÞðTÞIγ

−c

:

Since the map y7! y

gðt,yÞis injective inℝ almost everywhere for t  J, thus, the maps y 7! y

gð0,yÞ, y7! y

gðT,yÞare injective

inℝ, which implies that agð0,yð0ÞÞxð0Þ + bgðT,yðTÞÞyðTÞ = c

In the reverse sense, we have ABC

0 Dαgðt,yðtÞÞyðtÞ 

= fðtÞ applying AB

0 Iα on both sides, and thanks to Lemma 3, we get

yðtÞ

gðt,yðtÞÞ=gð0,yð0ÞÞyð0Þ +AB

0 IαfðtÞ

Therefore, we can easily verify that

a yð0Þ gð0,yð0ÞÞ+ b

yðTÞ gðT,yðTÞÞ=ða + bÞ

yð0Þ gð0,yð0ÞÞ+AB0 Iγ ðTÞ × b, which implies that

yð0Þ gð0,yð0ÞÞ=

1

a+ b c−AB

0 Iγ ðTÞ × b

:

Consequently, the proof is complete

3.2 | Existence results

Now, we prove the existence results for problem (1), to reach that we place the problem in the space ~~S, and we define a supremum normk  k such that jyj = sup

t  JjyðtÞj and the multiplication by ðwzÞðtÞ = wðtÞzðtÞ 8w,z ~~S

Thus, it is clear that ~~Sis a Banach space

Consider the following operator to transform problem (1) into a fixed-point problem Z: ~~S 7! ~~S

ZðyÞðtÞ = g t,yðtÞ½ ð Þ 1−γ

N ðγÞhðt,yðtÞÞ + γN ðγÞIγhðt,yðtÞÞ

−a+ b1 b 1−γ

N ðγÞhðT,yðTÞÞ + γN ðγÞðIγhðT,yðTÞÞÞ

−c

:

Afterwards, by applying the Banach fixed point, we prove that Z has a fixed point So we prove what follows

Theorem Assume that hypothesesðℑ1Þ, ðℑ2Þ, ðℑ4Þ, and ðℑ5Þ Then, the considered problem admits a unique solution

on J if

X = η1

ð1−γÞw2

N ðγÞ +

Tγw2

N ðγÞΓðαÞ+

1

ja + bj

jbjð1−γÞw2

N ðγÞ +

jbjTγw 2

N ðγÞΓðγÞ+jcj

+ w1 ð1−γÞη2

N ðγÞ +

Tγη2

N ðγÞΓðαÞ



+ 1

ja + bj jbjð1−γÞη2

N ðγÞ +

jbjTγη2

N ðγÞΓðγÞ+jcj

< 1:

Proof Let x, y ~~S, so 8t  J

ZðxÞðtÞ−ZðyÞðtÞ

j j≤ g t,yðtÞj ð Þj 1−γ

N ðγÞjh t, xðtÞð Þ−h t,yðtÞð Þj



+ γ

N ðγÞΓðγÞ ð

t

0 ðt −sÞγ −1jh s, xðsÞð Þ−h s,yðsÞð Þjds + 1

ja + bj

jbjð1−γÞ

N ðγÞ jh T, xðTÞð Þ−h T,yðTÞð Þj

+ jbjγ

N ðγÞΓðγÞ ð

T

0 ðT −sÞγ −1jh s, xðsÞð Þ−h s,yðsÞð Þjds + jcj

!#

+ g t, xðtÞj ð Þ−g t,yðtÞð Þj 1−γ

N ðγÞjh t, yðtÞð Þj



+ γ

N ðγÞΓðγÞ ð

t

0 ðt −sÞγ −1jh s, yðsÞð Þjds + 1

ja + bj

jbjð1−γÞ

N ðγÞ jh T, yðTÞð Þj

+ jbjγ

N ðγÞΓðγÞ ð

T

0 ðT −sÞγ −1jh s, yðsÞð Þjds + jcj

!#

≤ η1 ð1−γÞw2

N ðγÞ +

Tγw2

N ðγÞΓðγÞ+

1

ja + bj jbjð1−γÞw2

N ðγÞ +

jbjTγw 2

N ðγÞΓðγÞ+jcj

kx −yk + w1 ð1−γÞη2

N ðγÞ +

Tαη2

N ðγÞΓðγÞ



+ 1

ja + bj

jbjð1−γÞη2

N ðγÞ +

jbjTγη2

N ðγÞΓðγÞ+jcj

kx −yk:

Thus, the proof is achieved.

Now we are in a position to prove the next existence theorem for the studied problem

Theorem Suppose thatðℑ1Þ, ðℑ2Þ, and ðℑ3Þ are valid If w1ξ < 1, then (1) admits one solution at least defined on J, where

ξ = 1 + jbj

ja + bj

ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ

kKkL1+ jcj

ja + bj:

Proof We define a subset S of ~~S, which satisfies the hypothesis of Theorem 4 by S: = y  ~~S= k y k ≤ θn o, where θ :

=1−wηξ

1 ξandη = sup

t  Jjgðt,0Þj

Then in order to transform problem (1) into the operator equation y = A1yA2y, we need to define A1, A2as

A1ðyÞðtÞ = g t,yðtÞ½ ð Þ and

A2ðyÞðtÞ = 1−γ

N ðγÞhðt,yðtÞÞ + γN ðγÞIγhðt,yðtÞÞ

− 1

a+ b b

1−γ

N ðγÞhðT,yðTÞÞ + γN ðγÞðIγhðT,yðTÞÞÞ

−c

:

Thus, we need to clarify that A1, A2fulfill Theorem 4 conditions: we begin by showing that A1is a w1Lipschitzian oper-ator on ~~S For x, y ~~S, thus by using ðℑ2Þ, we have Aj 1xðtÞ−A1yðtÞj = g t,xðtÞj ð Þ−g t,yðtÞð Þj ≤ w1jxðtÞ−yðtÞj ≤ w1kx −yk, so

A1xðtÞ−A1yðtÞ

j j≤ w1jjx −yjj 8x,y  ~~S

Next, A2is continuous and compact on S into ~~S We start by ensuring the continuity of A2on S

Consider the converging sequence (yn), which converges to y in S Hence, by the theorem of Lebesgue dominated convergence,

lim

n!∞

1−γ

N ðγÞhðt,ynðsÞÞ + γ

N ðγÞ

1 ΓðγÞ ð

t

0 ðt −sÞγ −1h s, yð nðsÞÞds

= 1−γ

N ðγÞn!∞limhðt,ynðsÞÞ + γ

N ðγÞ

1 ΓðγÞ ð

t

0 ðt −sÞγ −1lim

n!∞h s, yð nðsÞÞds:

This implies that for all t J,

lim

n!∞A2ynðtÞ = lim

n!∞

1−γ

N ðγÞhðt,ynðtÞÞ + γ

N ðγÞIγhðt,ynðtÞÞ

− 1

a+ b b

1−γ

N ðγÞhðT,ynðTÞÞ + γ

N ðγÞðIαhðT,ynðTÞÞÞ

−c

= 1−γ

N ðγÞnlim!∞hðt,ynðsÞÞ + γ

N ðγÞ

1 ΓðγÞ ð

t

0 ðt −sÞγ −1lim

n !∞h s, yð nðsÞÞds

− 1

a+ b b

1−γ

N ðγÞn!∞lim hðT,ynðTÞÞ +



γ

N ðγÞ

1 ΓðγÞ ð

T

0 ðT −sÞγ −1lim

n !∞h s, yð nðsÞÞds

#

−c

!

= A2yðtÞ:

Moreover, we prove the compactness of A2on S To do that we prove the uniform boundedness of the set A2(S) in ~~S For y S, using hypothesis ðℑ3Þ, 8t  J:

A2yðtÞ

j j≤ N ðγÞ1−γjhðt,yðtÞÞj + γ

N ðγÞ

1 ΓðγÞ ð

t

0 ðt −sÞγ −1h s, yðsÞð Þ ds

+ 1

a+ b b

1−γ

N ðγÞjhðT,yðTÞÞj



+ γ

N ðγÞ

1 ΓðγÞ ð

T

0 ðT −sÞγ −1h s, yðsÞð Þds

# +jcj

!

≤ ð1−γÞN ðγÞkKkL1+ T

γ

N ðγÞΓðγÞkKkL 1+ 1

ja + bj jbjð1−γÞ

N ðγÞ kKkL 1+ jbjTγ

N ðγÞÞΓðγÞkKkL 1+jcj

:

Consequently,

A2yðtÞ

j j≤ 1 + jbj

ja + bj

ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ

kKkL1+ jcj

ja + bj=ξ:

On the other hand, let t1, t2 J, where t1< t2then for any y ~~S We demonstrate that A2(S) is an equicontinuous set

on ~~S:

A2yðt2Þ−A2yðt1Þ

j j ≤ 1−γ

N ðγÞjhðt2, yðt2ÞÞ−hðt1, yðt1ÞÞj + γ

N ðγÞ

1 ΓðγÞ ð

t 1

0

½ðt2−sÞγ −1−ðt1−sÞγ −1h s,yðsÞð Þ,ds

+ γ

N ðγÞ

1 ΓðγÞ ð

t 2

t 1

ðt2−sÞγ −1h s, yðsÞð Þds

knowing that |h(t2, y(t2))− h(t1, y(t1)) |!0 when |t2− t1|!0, and thanks to hypothesis ðℑ3Þ, we get

A2yðt2Þ−A2yðt1Þ

j j ≤ N ðγÞγ kKkL 1

Γðγ + 1Þ ð

t 1

0

½ðt2−sÞγ −1−ðt1−sÞγ −1,ds

+ γ

N ðγÞ

kKkL1

Γðγ + 1Þ ð

t 2

0

ðt2−sÞγ −1ds

≤ kKkL 1

Γðγ + 1Þ½ðt2−t1Þγ+ t2γ−tγ1:

Hence, forϵ > 0, there exists a δ > 0 such that jt2−t1j < δ ) Aj 2yðt2Þ−A2yðt1Þj <ϵ

In consequence, A2(S) is compact by the Arzelá–Ascoli theorem; therefore, A2is a complete continuous operator

on S

Finally, we show that the last hypothesis of Theorem 4 is satisfied, so for y ~~S and x  S, where y = A1yA2x, we get

j j = Aj 1yðtÞj Aj 2xðtÞj

≤ g t,yðtÞj ð Þj 1−γ

N ðγÞjhðt,xðtÞÞj + γN ðγÞ

1 ΓðγÞ ð

t

0 ðt −sÞγ −1h s, xðsÞð Þ ds

+ 1

a+ b b

1−γ

N ðγÞjhðT,xðTÞÞj +



γ

N ðγÞ

1 ΓðγÞ ð

T

0 ðT −sÞγ −1h s, xðsÞð Þ ds

# +jcj

!

≤ g t,yðtÞ½j ð Þ−gðt,0Þj + gðt,0Þj j ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ+

jbj

ja + bj

ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ

kKkL1+ jcj

ja + bj

≤ w½ 1jyðtÞj + η ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ+

jbj

ja + bj

ð1−γÞ

N ðγÞ +

Tγ

N ðγÞΓðγÞ

kKkL1+ jcj

ja + bj

:

This implies

jy

j jj≤1−wηξ

1ξ:

This leads us to conclude that M = Ak 2ðSÞk = 1 +ja + bjjbj 

ð1−γÞ

N ðγÞ + T

γ

N ðγÞΓðγÞ

kKkL1+ja + bjjcj

=ξ

Therefore: lM = w1ξ < 1

Thus, the hypotheses of Theorem 4 are satisfied: y = A1yA2y admits a solution in S; in consequence, problem (1) admits a solution on J

In this part, the stability in Ulam–Hyers sense will be analyzed Motivated by Rus and Wang et al.,25,27

let us provide first the following definition

Definition The equation in (1) is stable in Ulam–Hyers sense if for all ϑ > 0 and for all solution y  CðJ,ℝÞ of (3) there is a real numberλ > 0 and x  CðJ,ℝÞ a solution of the equation in problem (1) where

ABC

0 Dγ yðtÞ gðt,yðtÞÞ

such that

yðtÞ−xðtÞ

Remark 10 y CðJ,ℝÞ is said to be a solution of (3) iff there is a function u  CðJ,ℝÞ depending on y such that

♠ uðtÞj j≤ ϑ, 8t  J and

♠♠ ABC

0 Dγgðt,yðtÞÞyðtÞ 

= hðt,yðtÞÞ + uðtÞ, 8t  J

Lemma For y CðJ,ℝÞ solution of (3), the following inequality holds:

yðtÞ gðt,yðtÞÞ −φð0Þ−AB

0 Iγhðt,yðtÞÞ AB

0 Iγu

  ðtÞ

≤ ϑ N ðγÞ1−γ + T

γ

N ðγÞΓðγÞ

,

whereφð0Þ =gð0,yð0ÞÞyð0Þ

Theorem Suppose that the hypothesis of Theorem 7 and w2≤N ðγÞ1−γ are fulfilled Thus, (1) is Ulam–Hyers stable Proof Suppose that assumptionsðℑ4Þ, w2≤N ðγÞ1−γ, and gðt,yÞj j≤ η1 8t  J, 8y  ℝ are valid, and suppose that y(t) be a solution of (3) and x(t) be a solution of problem (1) satisfying gðt,xð0ÞÞxð0Þ =φð0Þ = 1

a + b −b AB

0 IγhðT,xðTÞÞ

+ c

Thanks to Definition 9 and Lemma 11, we have

yðtÞ−xðtÞ

j j≤ η1

yðtÞ gðt,yðtÞÞ −

xðtÞ gðt,xðtÞÞ

≤ η1

yðtÞ gðt,yðtÞÞ −AB0 Iγhðt,yðtÞÞ +AB

0 Iγhðt,yðtÞÞ−φð0Þ−AB

0 Iγhðt,xðtÞÞ

≤ η1

yðtÞ gðt,yðtÞÞ −AB0 Iγhðt,yðtÞÞ−φð0Þ 1 AB0 Iγhðt,yðtÞÞ−AB

0 Iγhðt,xðtÞÞ

≤ ϑη1

1−γ

N ðγÞ+

Tγ

N ðγÞΓðγÞ

+ w2η1AB0 IγjyðtÞ−xðtÞj:

Therefore, by the inequality of Gronwall mentioned in Theorem 2.1,22we can state that

ky−xk ≤ ϑη1

1−γ

N ðγÞ+

Tγ

N ðγÞΓðγÞ

N ðγÞ

N ðγÞ−ð1−γÞw2

Eγ γw2Tγ

N ðγÞ−ð1−γÞw2

,

≤ ϑλ,

whereλ = η1

1 −γ

N ðγÞ+ T

γ

N ðγÞΓðγÞ

N ðγÞ−ð1−γÞw 2Eγ γw2 Tγ

N ðγÞ−ð1−γÞw 2

Hence, (1) is stable in Ulam–Hyers sense

Take into consideration the following problem:

ABC

1 +ðsinðtÞ=16ÞsinjyðtÞj

= tyðtÞ

1 +jyðtÞj for all t J = ½0,π:

1

2yð0Þ +1

2yðπÞ = 0

8

>

>

Obviously, for w1= 1=16,KðtÞ = t,η = 1 and taking N ðγÞ = 1 as a normalization function, all hypotheses of Theorem 8 are satisfied

In fact, sincektkL1=π22, we can find thatξw1=9

8π2× 1

16=1289π2< 1 Moreover, the value ofθ used in the definition of

Sin the proof of Theorem 8 can be chosen asθ =18π 2

5

4 | C O N C L U S I O N

The fractional operators with Mittag–Leffler law have been investigated theoretically more deeply The presence of the nonsingular Mittag–Leffler kernel in the ABC fractional derivative produces fractional integral operators consisting of a linear combination of the function and its Reiamann–Liouville fractional integral This representation is reflected in the calculations and analysis applied to the nonlinear hybrid differential systems studied in this manuscript Indeed, the existence and uniqueness via the contraction principle point tool, existence of mild solutions via Dhage fixed-point principle, and the Ulam–Hyers stability have been all discussed The applied techniques and concepts investigated

in this work can motivate researchers to study theoretically and numerically other different types of fractional differen-tial systems under different types of kernels so that they become effective and useful to model certain real-world problems

A C K N O W L E D G E M E N T S

This research of J J Nieto was partially supported by Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community Fund FEDER, and Xunta de Galicia under Grant ED431C 2019/02 T Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) Group Number RG-DES-2017-01-17

C O N F L I C T O F I N T E R E S T S

This work does not have any conflicts of interest

O R C I D

Juan J Nieto https://orcid.org/0000-0001-8202-6578

Thabet Abdeljawad https://orcid.org/0000-0002-8889-3768

R E F E R E N C E S

1 Hilfer R Applications of Fractional Calculus in Physics Singapore: World Scientific; 2000.

2 Jiao Z, Chen Y, Podlubny I Distributed Order Dynamic Systems, Modeling, Analysis and Simulation London, Heidelberg, New York: Springer; 2012.

3 Kaslik E, Sivasundaram S Nonlinear dynamics and chaos in fractional-order neural networks Neural Netw 2012;32:245-256.

4 Koeller RC Applications of fractional calculus to the theory of viscoelasticity ASME J Appl Mech 1984;51(2):299-307 https://doi.org/10 1115/1.3167616

5 Mainardi F Fractional Calculus and Waves in Linear Viscoelatiity: An Introduction to Mathematical Models Bologna/Italy: World Scien-tific Publishing; 2010.

6 Atangana A Fractional Operators with Constant and Variable Order with Application to Geo-hydrology New York: Academic Press; 2017.

7 Baleanu D, Guvenc ZB, Machado JAT New Trends in Nanotechnology and Fractional Calculus Applications, XI, 531 1st Edition Heidelberg: ISBN Springer; 2010.

8 Baleanu D, Etemad S, Rezapour S A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions Bound Value Probl 2020;2020:64 https://doi.org/10.1186/s13661-020-01361-0

9 Momani S, Odibat Z Analytical approach to linear fractional partial differential equations arising in fluid mechanics Phys Lett A 2006; 355(4-5):271-279.

10 Pinto CMA, Carvalho ARM A latency fractional order model for HIV dynamics J Comput Appl Math 2017;312:240-256.

11 Diethelm K Analysis of fractional differential equations: an application- oriented exposition using differential operators of Caputo type; 2010.

12 Atangana A, Baleanu D New fractional derivative with non-local and non-singular kernel Therm Sci 2016;20(2):757-763.

13 Ionescu C, Lopes A, Copota D, Machado JAT, Bates JHT The role of fractional calculus in modeling biological phenomena: a review; 2017.

14 Sun HG, Zhang Y, Baleanu D, Chen W, Chen YQ A new collection of real world applications of fractional calculus in science and engi-neering Commun Nonlinear Sci Numer Simul 2018;64:213-231.

15 Baleanu D, Ghanbari B, Jihad HA, Jajarmi A, Pirouz HM Planar system-masses in an equilateral triangle: numerical study within frac-tional calculus CMES-Comput Model Eng Sci 2020;124(3):953-968.

16 Jajarmi A, Baleanu D A new iterative method for the numerical solution of high-order nonlinear fractional boundary value problems Front Phys 2020;8:220.

17 Sajjadi SS, Baleanu D, Jajarmi A, Pirouz HM A new adaptive synchronization and hyperchaos control of a biological snap oscillator Chaos Solitons Fractals 2020;109919:138.