Fixed point approach to the Mittag-Leffler kernel-related fractional differential equations

http://www.aimspress.com/journal/Math AIMS Mathematics, 8(4): 8633–8649. DOI:10.3934/math.2023433 Received: 07 November 2022 Revised: 28 January 2023 Accepted: 01 February 2023 Published: 06 February 2023 Research article Fixed point approach to the Mittag-Leffler kernel-related fractional differential equations Hasanen A. Hammad1,2,∗, H¨ useyin Is¸ık3, Hassen Aydi4,5,6,∗and Manuel De la Sen7 1Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia 2Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt 3Department of Engineering Science, Bandırma Onyedi Eyl¨ ul University, 10200 Bandırma, Balıkesir, Turkey 4Institut Sup´ erieur d’Informatique et des Techniques de Communication, Universit´ e de Sousse, H. Sousse 4000, Tunisia 5China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 6Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa 7Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940-Leioa ( Bizkaia), Spain *Correspondence: Email: h.abdelw[email protected], [email protected]. Abstract: The goal of this paper is to present a new class of contraction mappings, so-called η` θcontractions. Also, in the context of partially ordered metric spaces, some coupled fixed-point results for η` θ-contraction mappings are introduced. Furthermore, to support our results, two examples are provided. Finally, the theoretical results are applied to obtain the existence of solutions to coupled fractional differential equations with a Mittag-Leffler kernel. Keywords: fractional differential equation; Atangana-Baleanu fractional operator; fixed point methodology; Riemann-Liouville fractional integral Mathematics Subject Classification: 34A08, 34A12, 47H10, 54H25 8634 1. Introduction and basic facts Fractional differential equations are thought to be the most effective models for a variety of pertinent events. This makes it possible to investigate the existence, uniqueness, controllability, stability, and other properties of analytical solutions. For example, applying conservation laws to the fractional Black-Scholes equation in Lie symmetry analysis, finding existence solutions for some conformable differential equations, and finding existence solutions for some classical and fractional differential equations on the basis of discrete symmetry analysis, for more details, see [1–5]. Atangana and Baleanu unified and extended the definition of Caputo-Fabrizio [5] by introducing exciting derivatives without singular kernel. Also, the same authors presented the derivative containing Mittag-Leffler function as a nonlocal and nonsingular kernel. Many researchers showed their interest in this definition because it opens many and sober directions and carries Riemann-Liouville and Caputo derivatives [6–13]. A variety of problems in economic theory, control theory, global analysis, fractional analysis, and nonlinear analysis have been treated by fixed point (FP) theory. The FP method contributes greatly to the fractional differential/integral equations, through which it is possible to study the existence and uniqueness of the solution to such equations [14–17]. Also, this topic has been densely studied and several significant results have been recorded in [18–21]. The concepts of mixed monotone property (MMP) and a coupled fixed point (CFP) for a contractive mapping Ξ:χ×χ→χ, where χis a partially ordered metric space (POMS) have been initiated by Bhaskar and Lakshmikantham [22]. To support these ideas, they presented some CFP theorems and determined the existence and uniqueness of the solution to a periodic boundary value problem [23–25]. Many authors worked in this direction and obtained some nice results concerned with CFPs in various spaces [26–28]. Definition 1.1. [22] Consider a set χ,∅.A pair (a,b)∈χ×χis called a CFP of the mapping Ξ:χ×χ→χif a= Ξ(a,b) and b= Ξ(b,a). Definition 1.2. [22] Assume that (χ, ≤) is a partially ordered set and Ξ:χ×χ→χis a given mapping. We say that Ξhas a MMP if for any a,b∈χ, a1,a2∈χ, a1≤a2⇒Ξ(a1,b)≤Ξ(a2,b), and b1,b2∈χ, b1≤b2⇒Ξ(a,b1)≥Ξ(a,b2). Theorem 1.1. [22] Let (χ, ≤,d)be a complete POMS and Ξ:χ×χ→χbe a continuous mapping having the MMP on χ. Assume that there is a τ∈[0,1) so that d(Ξ(a,b),Ξ(k,l))≤τ 2(d(a,k)+d(b,l)), for all a ≥k and b ≤l.If there are a0,b0∈χso that a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0),then Ξhas a CFP, that is, there exist a0,b0∈χsuch that a = Ξ(a,b)and b = Ξ(b,a). The same authors proved that Theorem 1.1 is still valid if we replace the hypothesis of continuity with the following: Assume χhas the property below: AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8635 (†) if a non-decreasing sequence {am} → a,then am≤afor all m; (‡) if a non-increasing sequence {bm} → b,then b≤bmfor all m. The following auxiliary results are taken from [29,30], which are used efficiently in the next section. Let Θrepresent a family of non-decreasing functions θ: [0,∞)→[0,∞) so that P∞ m=1θm(τ)<∞ for all τ > 0,where θnis the n-th iterate of θjustifying: (i) θ(τ)=0⇔τ=0; (ii) for all τ > 0, θ(τ)< τ; (iii) for all τ > 0,lims→τ+θ(s)< τ. Lemma 1.1. [30] If θ: [0,∞)→[0,∞)is right continuous and non-decreasing, then limm→∞ θm(τ)= 0for all τ≥0iffθ(τ)< τ for all τ > 0. Let e Lbe the set of all functions e `: [0,∞)→[0,1) which verify the condition: lim m→∞ e `(τm)=1 implies lim m→∞ τm=0. Recently, Samet et al. [29] reported exciting FP results by presenting the concept of α-θ-contractive mappings. Definition 1.3. [29] Let χbe a non empty-set, Ξ:χ→χbe a map and α:χ×χ→Rbe a given function. Then, Ξis called α-admissible if α(a,b)≥1⇒α(Ξa,Ξb))≥1,∀a,b∈χ. Definition 1.4. [29] Let (χ, d) be a metric space. Ξ:χ→χis called an α-θ-contractive mapping, if there exist two functions α:χ×χ→[0,+∞) and θ∈Θsuch that α(a,b)d(Ξ(a,b))≤θ(d(a,b)), for all a,b∈χ. Theorem 1.2. [29] Let (χ, d)be a metric space, Ξ:χ→χbe an α-ψ-contractive mapping justifying the hypotheses below: (i) Ξis α-admissible; (ii) there is a0∈χso that α(a0,Ξa0)≥1; (iii) Ξis continuous. Then Ξhas a FP. Moreover, the authors in [29] showed that Theorem 1.2 is also true if we use the following condition instead of the continuity of the mapping Ξ. •If {am}is a sequence of χso that α(am,am+1)≥1 for all mand limm→+∞am=a∈χ, then for all m,α(am,a)≥1. AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8636 The idea of an α-admissible mapping has spread widely, and the FPs obtained under this idea are not small, for example, see [31–34]. Furthermore, one of the interesting directions for obtaining FPs is to introduce the idea of Geraghty contractions [30]. The author [30] generalized the Banach contraction principle and obtained some pivotal results in a complete metric space. It is worth noting that a good number of researchers have focused their attention on this idea, for example, see [35–37]. In respect of completeness, we state Geraghty’s theorem. Theorem 1.3. [30] Let Ξ:χ→χbe an operator on a complete metric space (χ, d). Then Ξhas a unique FP if Ξsatisfies the following inequality: d(Ξa,Ξb)≤e `(d(a,b))d(a,b),for any a,b∈χ, where e `∈e L. We need the following results in the last part. Definition 1.5. [5] Let σ∈H1(s,t),s<t,and ν∈[0,1).The Atangana–Baleanu fractional derivative in the Caputo sense of σof order νis described by ABC sDνσ(ζ)=Q(ν) 1−ν ζ Zs σ0(ϑ)Mν −ν(ζ−ϑ)ν 1−ν!dϑ, where Mνis the Mittag-Leffler function given by Mν(r)= ∞ P m=0 rm Γ(mν+1) and Q(ν) is a normalizing positive function fulfilling Q(0) =Q(1) =1 (see [4]). The related fractional integral is described as AB sIνσ(ζ)=1−ν Q(ν)σ(ζ)+ν Q(ν)(sIνσ)(ζ),(1.1) where sIνis the left Riemann-Liouville fractional integral defined by (sIνσ)(ζ)=1 Γ(ν) ζ Zs (ζ−ϑ)ν−1σ(ϑ)dϑ. (1.2) Lemma 1.2. [38] For ν∈(0,1),we have AB sIνABC Dνσ(ζ)=σ(ζ)−σ(s). The outline for this paper is as follows: In Section 1, we presented some known consequences about α-admissible mappings and some useful definitions and theorems that will be used in the sequel. In Section 2, we introduce an η` θ-contraction type mapping and obtain some related CFP results in the context of POMSs. Also, we support our theoretical results with some examples. In Section 5, an application to find the existence of a solution for the Atangana-Baleanu coupled fractional differential equation (CFDE) in the Caputo sense is presented. AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8637 2. Main results Let Lbe the set of all functions `: [0,∞)→[0,1) satisfying the following condition: lim m→∞ `(τn)=1 implies lim m→∞ τn=1. We begin this part with the following definitions: Definition 2.1. Suppose that Ξ:χ×χ→χand η:χ2×χ2→[0,∞) are two mappings. The mapping Ξis called η-admissible if η((a,b),(k,l))≥1⇒η((Ξ(a,b),Ξ(b,a)),(Ξ(k,l),Ξ(l,k))) ≥1,∀a,b,k,l∈χ. Definition 2.2. Let (χ, $) be a POMS and Ξ:χ×χ→χbe a given mapping. Ξis termed as an η` θ-coupled contraction mapping if there are two functions η:χ2×χ2→[0,∞) and θ∈Θso that η((a,b),(k,l))$(Ξ(a,b),Ξ(k,l))≤` θ $(a,k)+$(b,l) 2!!θ $(a,k)+$(b,l) 2!,(2.1) for all a,b,k,l∈χwith a≥kand b≤l,where `∈L. Remark 2.1. Notice that since `: [0,∞)→[0,1),we have η((a,b),(k,l))$(Ξ(a,b),Ξ(k,l)) ≤` θ $(a,k)+$(b,l) 2!!×θ $(a,k)+$(b,l) 2! < θ $(a,k)+$(b,l) 2!,for any a,b,k,l∈χwith a,b,k,l. Theorem 2.1. Let (χ, ≤, $)be a complete POMS and Ξbe an η` θ-coupled contraction which has the mixed monotone property so that (i) Ξis η-admissible; (ii) there are a0,b0∈χso that η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1; (iii) Ξis continuous. If there are a0,b0∈χso that a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0),then Ξhas a CFP. Proof. Let a0,b0∈χbe such that η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1, η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1,a0≤Ξ(a0,b0)=a1(say) and b0≥Ξ(b0,a0)=b1(say). Consider a2,b2∈χso that Ξ(a1,b1)=a2and Ξ(b1,a1)=b2.Similar to this approach, we extract two sequences {am}and {bm}in χso that am+1= Ξ (am,bm)and bm+1= Ξ (bm,am),for all m≥0. Now, we shall show that am≤am+1and bm≥bm+1,for all m≥0.(2.2) By a mathematical induction, we have AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8638 (1) At m=0,because a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0)and since Ξ(a0,b0)=a1and Ξ(b0,a0)=b1, we obtain a0≤a1and b0≥b1,thus (2.2) holds for m=0. (2) Suppose that (2.2) holds for some fixed m≥0. (3) Attempting to prove the validity of (2.2) for any m,by assumption (2) and the mixed monotone property of Ξ,we get am+2= Ξ (am+1,bm+1)≥Ξ(am,bm+1)≥Ξ(am,bm)=am+1, and bm+2= Ξ (bm+1,am+1)≤Ξ(bm,am+1)≤Ξ(bm,am)=bm+1. This implies that am+2≥am+1and bm+2≤bm+1. Thus, we conclude that (2.2) is valid for all n≥0. Next, if for some m≥0,(am+1,bm+1)=(am,bm),then am= Ξ (am,bm)and bm= Ξ (bm,am),i.e., Ξ has a CFP. So, let (am+1,bm+1),(am,bm)for all m≥0.As Ξis η-admissible, we get η((a0,b0),(a1,b1)) =η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1, implies η((Ξ(a0,b0),Ξ(b0,a0)) ,(Ξ(a1,b1),Ξ(b1,a1))) =η((a1,b1),(a2,b2)) ≥1. Thus, by induction, one can write η((am,bm),(am+1,bm+1)) ≥1 and η((bm,am),(bm+1,am+1)) ≥1 for all m≥0.(2.3) Using (2.1) and (2.3) and the definition of `, we have $(am,am+1)=$(Ξ(am−1,bm−1),Ξ(am,bm)) ≤η((am−1,bm−1),(am,bm))$(Ξ(am−1,bm−1),Ξ(am,bm)) ≤` θ $(am−1,am)+$(bm−1,bm) 2!!θ $(am−1,am)+$(bm−1,bm) 2! ≤θ $(am−1,am)+$(bm−1,bm) 2!.(2.4) Analogously, we get $(bm,bm+1)=$(Ξ(bm−1,am−1),Ξ(bm,am)) ≤η((bm−1,am−1),(bm,am))$(Ξ(bm−1,am−1),Ξ(bm,am)) ≤θ $(bm−1,bm)+$(am−1,am) 2!.(2.5) Adding (2.4) and (2.5) we have $(am,am+1)+$(bm,bm+1) 2≤θ $(am−1,am)+$(bm−1,bm) 2!. AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8639 Continuing in the same way, we get $(am,am+1)+$(bm,bm+1) 2≤θm $(a0,a1)+$(b0,b1) 2!,for all m∈N. For  > 0,there exists m()∈Nso that X m≥m() θm $(a0,a1)+$(b0,b1) 2!< 2, for some θ∈Θ.Let m,j∈Nbe so that j>m>m().Then based on the triangle inequality, we obtain $am,aj+$bm,bj 2≤ j−1 X i=m $(ai,ai+1)+$(bi,bi+1) 2 ≤ j−1 X i=m θi $(a0,a1)+$(b0,b1) 2! ≤X m≥m() θm $(a0,a1)+$(b0,b1) 2!< 2, this leads to $am,aj+$bm,bj< . Because $am,aj≤$am,aj+$bm,bj< , and $bm,bj≤$am,aj+$bm,bj< , hence {am}and {bm}are Cauchy sequences in χ. The completeness of χimplies that the sequences {am} and {bm}are convergent in χ, that is, there are a,b∈χso that lim m→∞ am=aand lim m→∞ bm=b. Since Ξis continuous, am+1= Ξ (am,bm)and bm+1= Ξ (bm,am),we obtain after taking the limit as m→ ∞ that a=lim m→∞ am=lim m→∞ Ξ(am−1,bm−1)= Ξ(a,b), and b=lim m→∞ bm=lim m→∞ Ξ(bm−1,am−1)= Ξ(b,a). Therefore, Ξhas a CFP and this ends the proof. In the above theorem, when omitting the continuity assumption on Ξ,we derive the following theorem. Theorem 2.2. Let (χ, ≤, $)be a complete POMS and Ξbe an η` θ-coupled contraction and having the mixed monotone property so that (a) Ξis η-admissible; AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8640 (b) there are a0,b0∈χso that η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0))) ≥1and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0))) ≥1; (c) if {am}and {bm}are sequences in χsuch that η((am,bm),(am+1,bm+1)) ≥1, η ((bm,am),(bm+1,am+1)) ≥1 for all m ≥0,limm→∞ am=a∈χand limm→∞ bm=b∈χ, then η((am,bm),(a,b)) ≥1and η((bm,am),(b,a)) ≥1. If a0,b0∈χare that a0≤Ξ(a0,b0)and b0≥Ξ(b0,a0),then Ξhas a CFP. Proof. With the same approach as for the proof of Theorem 2.1, the sequences {am}and {bm}are Cauchy sequences in χ. The completeness of χimplies that there are a,b∈χso that lim m→∞ am=aand lim m→∞ bm=b. According to the assumption (c) and (2.3), one can write η((am,bm),(a,b)) ≥1 and η((bm,am),(b,a)) ≥1,for all m∈N.(2.6) It follows by (2.3), the definition of `and the property of θ(τ)< τ for all τ > 0, that $(Ξ(a,b),a)≤$(Ξ(a,b),Ξ(am,bm))+$(Ξ(am,bm),a) ≤η((am,bm),(a,b)) $(Ξ(am,bm),Ξ(a,b))+$(am+1,a) ≤` θ $(am,a)+$(bm,b) 2!!θ $(am,a)+$(bm,b) 2!+$(am+1,a) ≤θ $(am,a)+$(bm,b) 2!+$(am+1,a) <$(am,a)+$(bm,b) 2+$(am+1,a).(2.7) Similarly, we find that $(Ξ(b,a),b)≤$(Ξ(b,a),Ξ(bm,am))+$(Ξ(bm,am),b) ≤η((bm,am),(b,a)) $(Ξ(bm,am),Ξ(b,a))+$(bm+1,b) ≤` θ $(bm,b)+$(am,a) 2!!θ $(bm,b)+$(am,a) 2!+$(bm+1,b) ≤θ $(bm,b)+$(am,a) 2!+$(bm+1,b) <$(bm,b)+$(am,a) 2+$(bm+1,b).(2.8) As m→ ∞ in (2.7) and (2.8), we have $(Ξ(a,b),a)=0 and $(Ξ(b,a),b)=0. Hence, a= Ξ(a,b) and b= Ξ(b,a).Thus, Ξhas a CFP and this completes the proof. AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8641 In order to show the uniqueness of a CFP, we give the theorem below. If (χ, ≤) is a partially ordered set, we define a partial order relation ≤on the product χ×χas follows: (a,b)≤(k,l)⇔a≤kand b≥l,for all (a,b),(k,l)∈χ×χ. Theorem 2.3. In addition to the assertions of Theorem 2.1, assume that for each (a,b),(y,z)in χ×χ, there is (k,l)∈χ×χso that η((a,b),(k,l)) ≥1and η((y,z),(k,l)) ≥1. Suppose also (k,l)is comparable to (a,b)and (y,z).Then Ξhas a unique CFP. Proof. Theorem 2.1 asserts that the set of CFPs is non-empty. Let (a,b) and (y,z) be CFPs of the mapping Ξ,that is, a= Ξ(a,b),b= Ξ(b,a) and y= Ξ(y,z),z= Ξ(z,y).By hypothesis, there is (k,l)∈χ×χso that (k,l) is comparable to (a,b) and (y,z).Let (a,b)≤(k,l),k=k0and l=l0. Choose k1,l1∈χ×χso that k1= Ξ(k1,l1),l1= Ξ(l1,k1).Thus, we can construct two sequences {km}and {lm}as km+1= Ξ(km,lm) and lm+1= Ξ(lm,km). Since (k,l) is comparable to (a,b),in an easy way we can prove that a≤k1and b≥l1.Hence, for m≥1,we have a≤kmand b≥lm.Because for every (a,b),(y,z)∈χ×χ, there is (k,l)∈χ×χso that η((a,b),(k,l)) ≥1 and η((y,z),(k,l)) ≥1.(2.9) Because Ξis η-admissible, then by (2.9), we get η((a,b),(k,l)) ≥1 implies η((Ξ(a,b),Ξ(b,a)),(Ξ(k,l),Ξ(l,k))) ≥1. Since k=k0and l=l0,we obtain η((a,b),(k,l)) ≥1 implies η((Ξ(a,b),Ξ(b,a)),(Ξ(k0,l0),Ξ(l0,k0))) ≥1. Hence, η((a,b),(k,l)) ≥1 implies η((a,b),(k1,l1)) ≥1. So, by induction, we conclude that η((a,b),(km,lm)) ≥1,(2.10) for all m∈N.Analogously, one can obtain that η((b,a),(lm,km)) ≥1.Therefore, the obtained results hold if (a,b)≤(k,l). Based on (2.9) and (2.10), we can write $(a,km+1)=$(Ξ(a,b),Ξ(km,lm)) ≤η((a,b),(km,lm)) $(Ξ(a,b),Ξ(km,lm)) ≤` θ $(a,km)+$(b,lm) 2!!θ $(a,km)+$(b,lm) 2! ≤θ $(a,km)+$(b,lm) 2!.(2.11) AIMS Mathematics Volume 8, Issue 4, 8633–8649. 8648 13. H. Tajadodi, A. Khan, J. F. G´ omez-Aguilar, H. 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