Received: 25 September 2020 Revised: 21 February 2021 Accepted: 21 February 2021 DOI: 10.1002/mma.7349 SPECIAL ISSUE PAPER Existence and stability for a nonlinear hybrid differential equation of fractional order via regular Mittag–Leffler kernel Ibrahim Slimane1 | Zoubir Dahmani1 | Juan J Nieto2 | Thabet Abdeljawad3,4,5 Faculty of Exact Sciences and Informatics, UMAB Abdelhamid Ibn Badis, University of Mostaganem, Mostaganem, Algeria Department of Statistics, Mathematical Analysis and Optimization, Institute of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 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 KEYWORDS hybrid differential equations, Atangana–Baleanu derivative, Ulam–Hyers stability Department of Medical Research, China Medical University, Taichung, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan MSC CLASSIFICATION 34A38; 32A65; 26A33; 34K20 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 | INTRODUCTION 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 phenomena 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 reader1 as a general overview,2,3 for applications to dynamical Math Meth Appl Sci 2021;1–11 wileyonlinelibrary.com/journal/mma © 2021 John Wiley & Sons, Ltd SLIMANE ET AL systems,4,5 for applications to viscoelasticity,6 for applications to geohydrology,7 for applications to nanotechnology, and recent papers8–10 for some other applications In particular, the Caputo derivative reserves a special place in fractional calculus because of its initial value properties.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 convenient 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 Diethelm11 and for applications of fractional calculus to Ionescu et al and Sun et al.13,14 For 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,8,20–28 this paper will deal with the following: yðtÞ > ABC γ > D = ht, ytịị 8t J = ẵ0, T, < γ < 1, > > > gðt, yðtÞÞ < yð0Þ yðTÞ > +b = c, where a,b, c ℝ a > > gð0, yð0ÞÞ gðT, yðTÞÞ > > : hð0, yð0ÞÞ = 0, ð1Þ where g CJ ì , f0gị, h CJ ì , ℝÞ and a + b ≠ The rest of this article will be organized as follows: Section will be devoted to some essential preliminary tools and concepts that we need in order to proceed Section 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 | PRELIMINARIES We begin this section by presenting notations and main definitions that will be useful in the rest of this paper Let ~~ = CJ ì , ị is the continuous functions class, g : J × ℝ ! ℝ, and let CðJ × , ị indiJ = ẵ0, T be an interval in ℝ, S 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 > for a continuous function f on [a, b] is shown by y ða I f ÞðyÞ = ΓðγÞ ðy −σ Þα− f ðσÞdσ, γ > 0, a < y ≤ b: γ ð a 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 a Á N ðγÞ D f ðtÞ = 1−γ γ t a f ðσÞE γ ð ÀABC and in the left ABR sense (Riemann–Liouville type) is defined as ðt − σÞγ dσ −γ −γ SLIMANE ET AL a Á N ðγÞ d Dγ f ðtÞ = −γ dt t a ð ÀABR ðt −σÞγ dσ, f ðσÞE γ −γ −γ and the fractional integral relative to the above operators is ÀAB a Á −γ γ f ðtÞ + I γ f ðtÞ = ða I α f ÞðtÞ, N ðγÞ N ðγÞ where N ðγÞ > is a normalization function satisfying N ð0Þ = N ð1Þ = γ ABR γ In Atangana and Baleanu and Baleanu and Fernandez,12,30 it has been verified that ðAB a I a D f ÞðtÞ = f ðtÞ and 30 In fact, the work of Baleanu and Fernandez also contains other mathematical results on the ABC operators ÀABC γ Á the authors have established that Also in Atangana and Baleanu,12 a D f ðtÞ = ÀABR γ Á N ðγÞ γ γ a D f ðtÞ − − γ f ðaÞE γ − − γ ðt − aÞ γ AB γ ðABR a D a I f ÞðtÞ = f ðtÞ Abdeljawad31 has extended the definition to arbitrary γ > and has introduced the following result Lemma Let y(t) be well defined on [a, b] and γ (n, n + 1], then γ ABC γ ðAB a I a D yÞðtÞ = yðtÞ − n X yðkÞ ðaÞ k=0 k! ðt −aÞk : The reader may also be interested in the further extension of Fernandez32 to a complex gamma ~~ ~~ which is closed, bounded, and convex, and A : S ~~ 7! S Theorem (Dhage33) Assume that S is a non-empty subset of S, ~~ 7! S ~~ are two operators satisfying the following assumptions: and A2 : S A1 is Lipschitizian with a constant l, A2 is completely continuous, y = A1 yA2 x ) y S, 8x S, and lM < 1, where M = kA2 ðSÞk = sup kA2 ðyÞ k;y S, then the equation y = A1 yA2 y admits a solution in S | M AI N R E S U L T S 3.1 | Integral representation 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 t 7! gðt,y yÞ is a continuous function 8y ℝ and the equations in (1) are satisfied by y We denote by L1 ðJ; ℝÞ the Lebesgue integrable functions endowed with the k Á kL1 : T kxkL1 = jxðsÞjds: ð We also need to introduce the following hypotheses: SLIMANE ET AL y ðℑ1 Þ The function y 7! gðt,yÞ is injective in ℝ, 8t J ðℑ2 Þ jgðt, y1 Þ −gðt, y2 Þj ≤ w1 j y1 −y2 j 8t J and y1 , y2 ℝ and w1 > is real number ðℑ3 Þ KðtÞ L1 ðJ, ℝÞ where jhðt, yÞj ≤ KðtÞ, 8t J ðℑ4 Þ jhðt,y1 Þ −hðt, y2 Þj ≤ w2 j y1 − y2 j 8t J and y1 , y2 ℝ and w2 > is real number ðℑ5 Þ There exists η1 , η2 ℝÃ+ , where jgðt,yÞj ≤ η1 and jhðt, yÞj ≤ η2 , for all t J, for all y ℝ Lemma Let < α < 1, and assume that ðℑ1 Þ holds Then, the solution of the problem yðtÞ > ABC γ > > > D gðt, yðtÞÞ = f tị 8t J, ẵ0, 1, > < xð0Þ yðTÞ > +b = c where a, b, c ℝ and a + b ≠ a > > gð0, yð0ÞÞ gðT,yðTÞÞ > > : f ð0Þ = has the form −γ γ yðtÞ = ẵgt,ytịị f tị + I f ịtị N ðγÞ N ðγÞ ! 1 −γ γ γ b f ðTÞ + ðI f ÞðTÞ −c : − a+b N ðγÞ N ðγÞ ð2Þ yðtÞ is continuous Applying the Proof Suppose that y is solution of (1), then by the above definition, we state that gðt,yðtÞÞ ABC fractional derivative of order α, we retrieve the first equation in (1) Replacing t = 0, t = T in (2), we get ! yð0Þ 1 −γ γ γ =− b f ðTÞ + ðI f ÞðTÞ −c , gð0, yð0ÞÞ a+b N ðγÞ N ðγÞ ! yðTÞ −γ γ 1 −γ α γ γ ðI f ÞðTÞ − ðI f ÞðTÞ − c : = f ðTÞ + b f ðTÞ + gðT, yðTÞÞ N ðγÞ N ðγÞ a+b N ðγÞ N ðγÞ Since the map y 7! y gðt, yÞ in ℝ, which implies that is injective in ℝ almost everywhere for t J, thus, the maps y 7! a gð0,xð0Þ yð0ÞÞ yðTÞ + b gðT,yðTÞÞ In the reverse sense, we have ABC Dα yðtÞ yð0Þ AB α gðt,yðtÞÞ = gð0, yð0ÞÞ + I f ðtÞ Therefore, we can easily verify that a = c yðtÞ gðt, yðtÞÞ = f ðtÞ applying AB α I y 7! y gðT,yÞ are injective on both sides, and thanks to Lemma 3, we get yð0Þ yðTÞ yð0Þ +b = ða + bÞ + AB I γ f Tị ì b, g0, y0ịị gT, yTịị g0, y0ịị which implies that Á yð0Þ À AB γ = c I f Tị ì b : g0, y0ịị a + b Consequently, the proof is complete y gð0, yÞ , SLIMANE ET AL 3.2 | Existence results ~~ and we define a Now, we prove the existence results for problem (1), to reach that we place the problem in the space S, ~~ supremum norm k Á k such that jyj = sup jyðtÞj and the multiplication by ðwzÞðtÞ = wðtÞzðtÞ 8w, z S tJ ~~ is a Banach space Thus, it is clear that S ~~ 7! S ~~ Consider the following operator to transform problem (1) into a fixed-point problem Z : S Zyịtị = ẵgt, ytịị ht, ytịị + I hðt, yðtÞÞ N ðγÞ N ðγÞ ! 1 −γ γ γ b hðT, yðTÞÞ + ðI hðT, yðTÞÞÞ −c : − a+b N ðγÞ N ðγÞ Afterwards, by applying the Banach fixed point, we prove that Z has a fixed point So we prove what follows Theorem Assume that hypotheses ðℑ1 Þ, ðℑ2 Þ, ðℑ4 Þ, and ðℑ5 Þ Then, the considered problem admits a unique solution on J if ! ð1 −γÞw2 T γ w2 jbjð1 −γÞw2 jbjT γ w2 + + + + jcj X = η1 N ðγÞ N ðγÞΓðαÞ ja + bj N ðγÞ N ðγÞΓðγÞ ð1 − γÞη2 T γ η2 + + w1 N ðγÞ N ðγÞΓðαÞ ! jbjð1 − γÞη2 jbjT γ η2 + + + jcj < 1: N ðγÞ N ðγÞΓðγÞ ja + bj ~~ so 8t J Proof Let x, y S, jZðxÞðtÞ−ZðyÞðtÞj ≤ jgðt, yðtÞÞj −γ jhðt,xðtÞÞ− hðt, yðtÞÞj N ðγÞ t γ ðt −sÞ γ − jhðs, xðsÞÞ −hðs, yðsÞÞjds N ðγÞΓðγÞ jbjð1 −γÞ + jhðT,xðTÞÞ −hðT, yðTÞÞj ja + bj N ðγÞ !# T jbjγ γ−1 ðT −sÞ jhðs,xðsÞÞ −hðs, yðsÞÞjds + jcj + N ðγÞΓðγÞ 1−γ + jgðt, xðtÞÞ −gðt, yðtÞÞj jhðt, yðtÞÞj N ðγÞ t γ ðt −sÞ γ − jhðs, yðsÞÞjds + N ðγÞΓðγÞ jbjð1 −γÞ + jhðT,yðTÞÞj ja + bj N ðγÞ !# T jbjγ γ−1 ðT −sÞ jhðs,yðsÞÞjds + jcj + N ðγÞΓðγÞ ! ð1 −γÞw2 T γ w2 jbjð1 −γÞw2 jbjT γ w2 + + + + jcj kx −yk ≤ η1 N ðγÞ N ðγÞΓðγÞ ja + bj N ðγÞ N ðγÞΓðγÞ α ð1 −γÞη2 T η2 + w1 + NðγÞ N ðγÞΓðγÞ ! jbjð1 −γÞη2 jbjT γ η2 + + jcj kx −yk: + N ðγÞ N ðγÞΓðγÞ ja + bj + ð ð ð ð SLIMANE ET AL Thus, the proof is achieved Now we are in a position to prove the next existence theorem for the studied problem Theorem Suppose that ðℑ1 Þ, ðℑ2 Þ, and ðℑ3 Þ are valid If w1ξ < 1, then (1) admits one solution at least defined on J, where ! jbj ð1 −γÞ Tγ jcj kKkL1 + + : ξ= 1+ ja + bj N ðγÞ N ðγÞΓðγÞ ja + bj n o ~~ , which satisfies the hypothesis of Theorem by S : = y S= ~~ k y k ≤ θ , where θ : Proof We define a subset S of S = −ηξw1 ξ and η = sup jgðt, 0Þj tJ Then in order to transform problem (1) into the operator equation y = A1 yA2 y, we need to define A1, A2 as A1 ðyÞðtÞ = ẵ gt, ytịị and A2 yịtị = −γ γ γ hðt, yðtÞÞ + I hðt, yðtÞÞ N ðγÞ N ðγÞ ! 1−γ γ γ b hðT, yðTÞÞ + ðI hðT, yðTÞÞÞ −c : − a+b N ðγÞ N ðγÞ Thus, we need to clarify that A1, A2 fulfill Theorem conditions: we begin by showing that A1 is a w1 Lipschitzian oper~~ thus by using ðℑ Þ, we have jA xðtÞ −A yðtÞj = jgðt, xðtÞÞ −gðt, yðtÞÞj ≤ w jxðtÞ −yðtÞj ≤ w kx −yk, ~~ For x, y S, ator on S 1 1 so ~~ jA1 xðtÞ −A1 yðtÞj ≤ w1 jjx − yjj 8x, y S ~~ We start by ensuring the continuity of A on S Next, A2 is continuous and compact on S into S Consider the converging sequence (yn), which converges to y in S Hence, by the theorem of Lebesgue dominated convergence, t ðt −sÞγ − hðs, yn ðsÞÞds −γ γ lim hðt, yn ðsÞÞ + = N ðγÞ n!∞ N ðγÞ ΓðγÞ t ð −γ γ hðt, yn ðsÞÞ + N ðγÞ N ðγÞ ΓðγÞ ð lim n!∞ ðt −sÞγ − n!∞ lim hðs,yn ðsÞÞds: This implies that for all t J, −γ γ γ hðt,yn ðtÞÞ + I hðt, yn ðtÞÞ N ðγÞ N ðγÞ ! 1 −γ γ b hðT, yn ðTÞÞ + − ðI α hðT, yn ðTÞÞÞ −c a+b N ðγÞ N ðγÞ lim A2 yn ðtÞ = lim n!∞ t −γ γ lim hðt,y ðsÞÞ + ðt − sÞγ − n!∞ lim hðs, yn ðsÞÞds n N ðγÞ n!∞ N ðγÞ ΓðγÞ 1 −γ − b lim hðT,yn ðTÞÞ + a+b N ðγÞ n!∞ # ! T γ γ −1 ðT − sÞ n!∞ lim hðs, yn ðsÞÞds − c = A2 yðtÞ: N ðγÞ ΓðγÞ ð = ð n!∞ SLIMANE ET AL ~~ Moreover, we prove the compactness of A2 on S To that we prove the uniform boundedness of the set A2(S) in S For y S, using hypothesis ðℑ3 Þ, 8t J: t −γ γ γ −1 ðt − sÞ hðs, yðsÞÞ ds jhðt,yðtÞÞj + jA2 yðtÞj≤ N ðγÞ N ðγÞ ΓðγÞ 1 −γ b + jhðT, yðTÞÞj a+b N ðγÞ # ! T γ γ −1 + ðT −sÞ hðs, yðsÞÞ ds + jcj N ðγÞ ΓðγÞ ð1 − γÞ Tγ jbjð1 −γÞ jbjT γ kKkL1 + kKkL1 + jcj : kKkL1 + kKkL1 + ≤ N ðγÞΓðγÞ N ðγÞÞΓðγÞ N ðγÞ ja + bj N ðγÞ ð ð Consequently, jA2 yðtÞj ≤ ! jbj ð1 −γÞ Tγ jcj + = ξ: kKkL1 + 1+ ja + bj N ðγÞ N ðγÞΓðγÞ ja + bj ~~ We demonstrate that A (S) is an equicontinuous set On the other hand, let t , t J, where t1 < t2 then for any y S ~ ~ on S: jA2 yðt Þ −A2 yðt Þj ≤ −γ jhðt , yðt ÞÞ −hðt , yðt ÞÞj N ðγÞ t1 γ ½ðt −sÞγ − − ðt − sÞγ − hðs, yðsÞÞ, ds + N ðγÞ ΓðγÞ t2 γ γ −1 ðt2 − sÞ hðs, yðsÞÞds + N ðγÞ ΓðγÞ t1 ð ð knowing that |h(t2, y(t2)) − h(t1, y(t1)) |!0 when |t2 − t1|!0, and thanks to hypothesis ðℑ3 Þ, we get t1 kKkL1 ẵt2 sị t1 sị γ − , ds N ðγÞ Γðγ + 1Þ t2 γ kKkL1 ðt2 − sÞ γ − ds + N ðγÞ Γðγ + 1Þ kKkL1 ≤ ½ðt2 −t1 Þ γ + t2γ −t1γ : Γðγ + 1Þ jA2 yðt2 Þ −A2 yðt1 Þj ≤ ð ð Hence, for ϵ > 0, there exists a δ > such that jt2 − t1 j < δ ) jA2 yðt Þ −A2 yðt Þj < ϵ In consequence, A2(S) is compact by the Arzelá–Ascoli theorem; therefore, A2 is a complete continuous operator on S ~~ and x S, where y = A yA x, we get Finally, we show that the last hypothesis of Theorem is satisfied, so for y S SLIMANE ET AL jyðtÞj = jA1 yðtÞjjA2 xðtÞj t 1−γ γ ðt −sÞ γ − hðs, xðsÞÞ ds ≤ jgðt, yðtÞÞj jhðt, xðtÞÞj + N ðγÞ ΓðγÞ N ðγÞ 1 −γ b + jhðT, xðTÞÞj + a+b N ðγÞ # ! T γ γ − ðT −sÞ hðs, xsịị ds + jcj N ị ị ẵj gðt, yðtÞÞ −gðt,0Þj + jgðt, 0Þj ! ð1 −γÞ Tγ jbj ð1 −γÞ Tγ jcj + kKkL1 + + + N ðγÞ NðγÞΓðγÞ ja + bj N ðγÞ NðγÞΓðγÞ ja + bj ! ð1 −γÞ Tγ jbj ị T jcj + + + ẵw1 jytịj + η kKkL1 + : N ðγÞ N ðγÞΓðγÞ ja + bj N ðγÞ N ðγÞΓðγÞ ja + bj ð ð This implies jj yjj ≤ ηξ : −w1 ξ h i ð1 − γÞ Tγ + N ðγÞΓðγÞ kKkL1 + jajcj This leads us to conclude that M = kA2 ðSÞk = + jajbj + bj + bj = ξ N ðγÞ Therefore: lM = w1 ξ < Thus, the hypotheses of Theorem are satisfied: y = A1 yA2 y admits a solution in S; in consequence, problem (1) admits a solution on J 3.3 | Stability in Ulam–Hyers sense 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 > and for all solution y CðJ, ℝÞ of (3) there is a real number λ > and x CðJ, ℝÞ a solution of the equation in problem (1) where ABC γ yðtÞ D − hðt, yðtÞÞ ≤ ϑ 0 gðt, yðtÞÞ ð3Þ jyðtÞ −xðtÞj ≤ ϑλ: ð4Þ such that 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 ♠ juðtÞj ≤ ϑ, 8t J and yðtÞ γ ♠♠ ABC D gðt,yðtÞÞ = hðt, yðtÞÞ + uðtÞ, 8t J Lemma For y CðJ, ℝÞ solution of (3), the following inequality holds: SLIMANE ET AL yðtÞ ÀAB γ Á AB γ I u ðtÞ gðt, yðtÞÞ − φð0Þ− I hðt, yðtÞÞ = 1−γ Tγ , + ≤ ϑ N ðγÞ N ðγÞΓðγÞ where φð0Þ = gð0,yð0Þ yð0ÞÞ Theorem Suppose that the hypothesis of Theorem and w2 ≤ N1 −ðγÞγ are fulfilled Thus, (1) is Ulam–Hyers stable that y(t) Proof Suppose that assumptions ðℑ4 Þ, w2 ≤ N1 −ðγÞγ , and jgðt,yÞj ≤ η1 8t J, 8y ℝ are valid, and À suppose  γ à beÁa xð0Þ = φð0Þ = a +1 b −b AB I hðT, xðTÞÞ +c solution of (3) and x(t) be a solution of problem (1) satisfying gðt,xð0ÞÞ Thanks to Definition and Lemma 11, we have jyðtÞ− xðtÞj ≤ ≤ ≤ ≤ yðtÞ xðtÞ − η1 gðt, yðtÞÞ gðt, xðtÞÞ yðtÞ AB γ AB γ AB γ − I hðt, yðtÞÞ + I hðt, yðtÞÞ− φð0Þ− I hðt, xðtÞÞ η1 gðt, yðtÞÞ yðtÞ γ AB γ AB γ − I hðt, yðtÞÞ −φð0Þ + η1 AB η1 I hðt,yðtÞÞ− I hðt, xðtÞÞ gðt, yðtÞÞ −γ Tγ γ + w2 η1 AB + ϑη1 I jyðtÞ −xðtÞj: N ðγÞ N ðγÞΓðγÞ Therefore, by the inequality of Gronwall mentioned in Theorem 2.1,22 we can state that ky − xk ≤ ϑη1 ≤ ϑλ, where λ = η1 1−γ N ðγÞ 1−γ Tγ N ðγÞ γw2 T γ , + Eγ N ðγÞ− ð1 −γÞw2 N ðγÞ N ðγÞΓðγÞ N ðγÞ −ð1 − γÞw2 N ðγÞ γw2 T γ Tγ + N ðγÞΓðγÞ E N ðγÞ − ð1 − γÞw2 γ N ðγÞ − ð1 − γÞw2 Hence, (1) is stable in Ulam–Hyers sense 3.4 | Example Take into consideration the following problem: > ABC 1=2 > > : yð0Þ + yðπÞ = 2 Obviously, for w1 = 1=16,KðtÞ = t, η = and taking N ðγÞ = as a normalization function, all hypotheses of Theorem are satisfied 9π = 128 < Moreover, the value of θ used in the definition of In fact, since ktkL1 = π2 , we can find that ξw1 = 98 π × 16 S in the proof of Theorem can be chosen as θ = 18π 10 SLIMANE ET AL | C ON C L U S I ON 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 fixed-point tool, existence of mild solutions via Dhage fixedpoint 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 differential systems under different types of kernels so that they become effective and useful to model certain real-world problems ACK NO WLE DGE MEN TS 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 CONFLICT OF INTERESTS This work does not have any conflicts of interest ORCID Juan J Nieto https://orcid.org/0000-0001-8202-6578 Thabet Abdeljawad https://orcid.org/0000-0002-8889-3768 R EF E RE N C E S Hilfer R Applications of Fractional Calculus in Physics Singapore: World Scientific; 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