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Research article

Cross-view learning with scatters and manifold exploitation in geodesic space


  • Cross-view data correlation analysis is a typical learning paradigm in machine learning and pattern recognition. To associate data from different views, many approaches to correlation learning have been proposed, among which canonical correlation analysis (CCA) is a representative. When data is associated with label information, CCA can be extended to a supervised version by embedding the supervision information. Although most variants of CCA have achieved good performance, nearly all of their objective functions are nonconvex, implying that their optimal solutions are difficult to obtain. More seriously, the discriminative scatters and manifold structures are not exploited simultaneously. To overcome these shortcomings, in this paper we construct a Discriminative Correlation Learning with Manifold Preservation, DCLMP for short, in which, in addition to the within-view supervision information, discriminative knowledge as well as spatial structural information are exploited to benefit subsequent decision making. To pursue a closed-form solution, we remodel the objective of DCLMP from the Euclidean space to a geodesic space and obtain a convex formulation of DCLMP (C-DCLMP). Finally, we have comprehensively evaluated the proposed methods and demonstrated their superiority on both toy and real datasets.

    Citation: Qing Tian, Heng Zhang, Shiyu Xia, Heng Xu, Chuang Ma. Cross-view learning with scatters and manifold exploitation in geodesic space[J]. Electronic Research Archive, 2023, 31(9): 5425-5441. doi: 10.3934/era.2023275

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  • Cross-view data correlation analysis is a typical learning paradigm in machine learning and pattern recognition. To associate data from different views, many approaches to correlation learning have been proposed, among which canonical correlation analysis (CCA) is a representative. When data is associated with label information, CCA can be extended to a supervised version by embedding the supervision information. Although most variants of CCA have achieved good performance, nearly all of their objective functions are nonconvex, implying that their optimal solutions are difficult to obtain. More seriously, the discriminative scatters and manifold structures are not exploited simultaneously. To overcome these shortcomings, in this paper we construct a Discriminative Correlation Learning with Manifold Preservation, DCLMP for short, in which, in addition to the within-view supervision information, discriminative knowledge as well as spatial structural information are exploited to benefit subsequent decision making. To pursue a closed-form solution, we remodel the objective of DCLMP from the Euclidean space to a geodesic space and obtain a convex formulation of DCLMP (C-DCLMP). Finally, we have comprehensively evaluated the proposed methods and demonstrated their superiority on both toy and real datasets.



    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 [1,2,3,4,5].

    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 [6,7,8,9,10,11,12,13].

    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 [14,15,16,17]. Also, this topic has been densely studied and several significant results have been recorded in [18,19,20,21].

    The concepts of mixed monotone property (MMP) and a coupled fixed point (CFP) for a contractive mapping Ξ:χ×χχ, where χ is 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 [23,24,25]. Many authors worked in this direction and obtained some nice results concerned with CFPs in various spaces [26,27,28].

    Definition 1.1. [22] Consider a set χ. A pair (a,b)χ×χ is called a CFP of the mapping Ξ:χ×χχ if a=Ξ(a,b) and b=Ξ(b,a).

    Definition 1.2. [22] Assume that (χ,) is a partially ordered set and Ξ:χ×χχ is a given mapping. We say that Ξ has a MMP if for any a,bχ,

    a1,a2χ, a1a2Ξ(a1,b)Ξ(a2,b),

    and

    b1,b2χ, b1b2Ξ(a,b1)Ξ(a,b2).

    Theorem 1.1. [22] Let (χ,,d) be a complete POMS and Ξ:χ×χχ be 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))τ2(d(a,k)+d(b,l)),

    for all ak and bl. If there are a0,b0χ so that a0Ξ(a0,b0) and b0Ξ(b0,a0), then Ξ has a CFP, that is, there exist a0,b0χ such 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 χ has the property below:

    () if a non-decreasing sequence {am}a, then ama for all m;

    () if a non-increasing sequence {bm}b, then bbm for all m.

    The following auxiliary results are taken from [29,30], which are used efficiently in the next section.

    Let Θ represent a family of non-decreasing functions θ:[0,)[0,) so that m=1θm(τ)< for all τ>0, where θn is the n-th iterate of θ justifying:

    (i) θ(τ)=0τ=0;

    (ii) for all τ>0, θ(τ)<τ;

    (iii) for all τ>0, limsτ+θ(s)<τ.

    Lemma 1.1. [30] If θ:[0,)[0,) is right continuous and non-decreasing, then limmθm(τ)=0 for all τ0 iff θ(τ)<τ for all τ>0.

    Let ˜L be the set of all functions ˜:[0,)[0,1) which verify the condition:

    limm˜(τm)=1 implies limmτm=0.

    Recently, Samet et al. [29] reported exciting FP results by presenting the concept of α-θ-contractive mappings.

    Definition 1.3. [29] Let χ be a non empty-set, Ξ:χχ be a map and α:χ×χR be a given function. Then, Ξ is called α-admissible if

    α(a,b)1α(Ξa,Ξb))1, a,bχ.

    Definition 1.4. [29] Let (χ,d) be a metric space. Ξ:χχ is called an α-θ-contractive mapping, if there exist two functions α:χ×χ[0,+) and θΘ such that

    α(a,b)d(Ξ(a,b))θ(d(a,b)),

    for all a,bχ.

    Theorem 1.2. [29] Let (χ,d) be a metric space, Ξ:χχ be an α-ψ-contractive mapping justifying the hypotheses below:

    (i) Ξ is α-admissible;

    (ii) there is a0χ so that α(a0,Ξa0)1;

    (iii) Ξ is continuous.

    Then Ξ has 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 χ so that α(am,am+1)1 for all m and limm+am=aχ, then for all m, α(am,a)1.

    The idea of an α-admissible mapping has spread widely, and the FPs obtained under this idea are not small, for example, see [31,32,33,34].

    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 [35,36,37]. In respect of completeness, we state Geraghty's theorem.

    Theorem 1.3. [30] Let Ξ:χχ be an operator on a complete metric space (χ,d). Then Ξ has a unique FP if Ξ satisfies the following inequality:

    d(Ξa,Ξb)˜(d(a,b))d(a,b), for any a,bχ,

    where ˜˜L.

    We need the following results in the last part.

    Definition 1.5. [5] Let σH1(s,t), s<t, and ν[0,1). The Atangana–Baleanu fractional derivative in the Caputo sense of σ of order ν is described by

    (ABCsDνσ)(ζ)=Q(ν)1νζsσ(ϑ)Mν(ν(ζϑ)ν1ν)dϑ,

    where Mν is the Mittag-Leffler function given by Mν(r)=m=0rmΓ(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

    (ABsIνσ)(ζ)=1νQ(ν)σ(ζ)+νQ(ν)(sIνσ)(ζ), (1.1)

    where sIν is the left Riemann-Liouville fractional integral defined by

    (sIνσ)(ζ)=1Γ(ν)ζs(ζϑ)ν1σ(ϑ)dϑ. (1.2)

    Lemma 1.2. [38] For ν(0,1), we have

    (ABsIνABCDνσ)(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 ηθ-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.

    Let L be the set of all functions :[0,)[0,1) satisfying the following condition:

    limm(τn)=1 implies limmτn=1.

    We begin this part with the following definitions:

    Definition 2.1. Suppose that Ξ:χ×χχ and η:χ2×χ2[0,) are two mappings. The mapping Ξ is called η-admissible if

    η((a,b),(k,l))1η((Ξ(a,b),Ξ(b,a)),(Ξ(k,l),Ξ(l,k)))1, a,b,k,lχ.

    Definition 2.2. Let (χ,ϖ) be a POMS and Ξ:χ×χχ be a given mapping. Ξ is termed as an ηθ-coupled contraction mapping if there are two functions η:χ2×χ2[0,) and θΘ so that

    η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l))(θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2), (2.1)

    for all a,b,k,lχ with ak and bl, where L.

    Remark 2.1. Notice that since :[0,)[0,1), we have

    η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l))(θ(ϖ(a,k)+ϖ(b,l)2))×θ(ϖ(a,k)+ϖ(b,l)2)<θ(ϖ(a,k)+ϖ(b,l)2), for any a,b,k,lχ with abkl.

    Theorem 2.1. Let (χ,,ϖ) be a complete POMS and Ξ be an ηθ-coupled contraction which has the mixed monotone property so that

    (i) Ξ is η-admissible;

    (ii) there are a0,b0χ so that

    η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0)))1 and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0)))1;

    (iii) Ξ is continuous.

    If there are a0,b0χ so that a0Ξ(a0,b0) and b0Ξ(b0,a0), then Ξ has a CFP.

    Proof. Let a0,b0χ be such that η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0)))1, η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0)))1, a0Ξ(a0,b0)=a1 (say) and b0Ξ(b0,a0)=b1 (say). Consider a2,b2χ so that Ξ(a1,b1)=a2 and Ξ(b1,a1)=b2. Similar to this approach, we extract two sequences {am} and {bm} in χ so that

    am+1=Ξ(am,bm) and bm+1=Ξ(bm,am), for all m0.

    Now, we shall show that

    amam+1 and bmbm+1, for all m0. (2.2)

    By a mathematical induction, we have

    (1) At m=0, because a0Ξ(a0,b0) and b0Ξ(b0,a0) and since Ξ(a0,b0)=a1 and Ξ(b0,a0)=b1, we obtain a0a1 and b0b1, thus (2.2) holds for m=0.

    (2) Suppose that (2.2) holds for some fixed m0.

    (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+2am+1 and bm+2bm+1.

    Thus, we conclude that (2.2) is valid for all n0.

    Next, if for some m0, (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 m0. As Ξ is 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 m0. (2.3)

    Using (2.1) and (2.3) and the definition of , we have

    ϖ(am,am+1)=ϖ(Ξ(am1,bm1),Ξ(am,bm))η((am1,bm1),(am,bm))ϖ(Ξ(am1,bm1),Ξ(am,bm))(θ(ϖ(am1,am)+ϖ(bm1,bm)2))θ(ϖ(am1,am)+ϖ(bm1,bm)2)θ(ϖ(am1,am)+ϖ(bm1,bm)2). (2.4)

    Analogously, we get

    ϖ(bm,bm+1)=ϖ(Ξ(bm1,am1),Ξ(bm,am))η((bm1,am1),(bm,am))ϖ(Ξ(bm1,am1),Ξ(bm,am))θ(ϖ(bm1,bm)+ϖ(am1,am)2). (2.5)

    Adding (2.4) and (2.5) we have

    ϖ(am,am+1)+ϖ(bm,bm+1)2θ(ϖ(am1,am)+ϖ(bm1,bm)2).

    Continuing in the same way, we get

    ϖ(am,am+1)+ϖ(bm,bm+1)2θm(ϖ(a0,a1)+ϖ(b0,b1)2), for all mN.

    For ϵ>0, there exists m(ϵ)N so that

    mm(ϵ)θm(ϖ(a0,a1)+ϖ(b0,b1)2)<ϵ2,

    for some θΘ. Let m,jN be so thatj>m>m(ϵ). Then based on the triangle inequality, we obtain

    ϖ(am,aj)+ϖ(bm,bj)2j1i=mϖ(ai,ai+1)+ϖ(bi,bi+1)2j1i=mθi(ϖ(a0,a1)+ϖ(b0,b1)2)mm(ϵ)θm(ϖ(a0,a1)+ϖ(b0,b1)2)<ϵ2,

    this leads to ϖ(am,aj)+ϖ(bm,bj)<ϵ. Because

    ϖ(am,aj)ϖ(am,aj)+ϖ(bm,bj)<ϵ,

    and

    ϖ(bm,bj)ϖ(am,aj)+ϖ(bm,bj)<ϵ,

    hence {am} and {bm} are Cauchy sequences in χ. The completeness of χ implies that the sequences {am} and {bm} are convergent in χ, that is, there are a,bχ so that

    limmam=a and limmbm=b.

    Since Ξ is continuous, am+1=Ξ(am,bm) and bm+1=Ξ(bm,am), we obtain after taking the limit as m that

    a=limmam=limmΞ(am1,bm1)=Ξ(a,b),

    and

    b=limmbm=limmΞ(bm1,am1)=Ξ(b,a).

    Therefore, Ξ has 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,,ϖ) be a complete POMS and Ξ be an ηθ-coupled contraction and having the mixed monotone property so that

    (a) Ξ is η-admissible;

    (b) there are a0,b0χ so that

    η((a0,b0),(Ξ(a0,b0),Ξ(b0,a0)))1 and η((b0,a0),(Ξ(b0,a0),Ξ(a0,b0)))1;

    (c) if {am} and {bm} are sequences in χ such that

    η((am,bm),(am+1,bm+1))1, η((bm,am),(bm+1,am+1))1

    for all m0, limmam=aχ and limmbm=bχ, then

    η((am,bm),(a,b))1 and η((bm,am),(b,a))1.

    If a0,b0χ are that a0Ξ(a0,b0) and b0Ξ(b0,a0), then Ξ has 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 χ implies that there are a,bχ so that

    limmam=a and limmbm=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 mN. (2.6)

    It follows by (2.3), the definition of and the property of θ(τ)<τ for all τ>0, that

    ϖ(Ξ(a,b),a)ϖ(Ξ(a,b),Ξ(am,bm))+ϖ(Ξ(am,bm),a)η((am,bm),(a,b))ϖ(Ξ(am,bm),Ξ(a,b))+ϖ(am+1,a)(θ(ϖ(am,a)+ϖ(bm,b)2))θ(ϖ(am,a)+ϖ(bm,b)2)+ϖ(am+1,a)θ(ϖ(am,a)+ϖ(bm,b)2)+ϖ(am+1,a)<ϖ(am,a)+ϖ(bm,b)2+ϖ(am+1,a). (2.7)

    Similarly, we find that

    ϖ(Ξ(b,a),b)ϖ(Ξ(b,a),Ξ(bm,am))+ϖ(Ξ(bm,am),b)η((bm,am),(b,a))ϖ(Ξ(bm,am),Ξ(b,a))+ϖ(bm+1,b)(θ(ϖ(bm,b)+ϖ(am,a)2))θ(ϖ(bm,b)+ϖ(am,a)2)+ϖ(bm+1,b)θ(ϖ(bm,b)+ϖ(am,a)2)+ϖ(bm+1,b)<ϖ(bm,b)+ϖ(am,a)2+ϖ(bm+1,b). (2.8)

    As m in (2.7) and (2.8), we have

    ϖ(Ξ(a,b),a)=0 and ϖ(Ξ(b,a),b)=0.

    Hence, a=Ξ(a,b) and b=Ξ(b,a). Thus, Ξ has a CFP and this completes the proof.

    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×χ as follows:

    (a,b)(k,l)ak and bl, 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×χ so that

    η((a,b),(k,l))1 and η((y,z),(k,l))1.

    Suppose also (k,l) is comparable to (a,b) and (y,z). Then Ξ has 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×χ so that (k,l) is comparable to (a,b) and (y,z). Let (a,b)(k,l), k=k0 and l=l0. Choose k1,l1χ×χ so 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 ak1 and bl1. Hence, for m1, we have akm and blm. Because for every (a,b),(y,z)χ×χ, there is (k,l)χ×χ so that

    η((a,b),(k,l))1 and η((y,z),(k,l))1. (2.9)

    Because Ξ is 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=k0 and 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 mN. 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

    ϖ(a,km+1)=ϖ(Ξ(a,b),Ξ(km,lm))η((a,b),(km,lm))ϖ(Ξ(a,b),Ξ(km,lm))(θ(ϖ(a,km)+ϖ(b,lm)2))θ(ϖ(a,km)+ϖ(b,lm)2)θ(ϖ(a,km)+ϖ(b,lm)2). (2.11)

    Similarly, we get

    ϖ(b,lm+1)=ϖ(Ξ(b,a),Ξ(lm,km))η((b,a),(lm,km))ϖ(Ξ(b,a),Ξ(lm,km))(θ(ϖ(b,lm)+ϖ(a,km)2))θ(ϖ(b,lm)+ϖ(a,km)2)θ(ϖ(b,lm)+ϖ(a,km)2). (2.12)

    Adding (2.11) and (2.12), we have

    ϖ(a,km+1)+ϖ(b,lm+1)2θ(ϖ(b,lm)+ϖ(a,km)2).

    Thus,

    ϖ(a,km+1)+ϖ(b,lm+1)2θm(ϖ(b,l1)+ϖ(a,k1)2), (2.13)

    for each n1. As m in (2.13) and by Lemma 1.1, we have

    limm(ϖ(a,km+1)+ϖ(b,lm+1))=0,

    which yields that

    limmϖ(a,km+1)=limmϖ(b,lm+1)=0. (2.14)

    Similarly, one obtains

    limmϖ(y,km+1)=limmϖ(z,lm+1)=0. (2.15)

    It follows from (2.14) and (2.15), we find that a=y and b=z. This proves that the CFP is unique.

    Examples below support the theoretical results.

    Example 2.1. (Linear case) Let ϖ:χ×χR be a usual metric on χ=[0,1]. Define the mappings Ξ:χ×χχ and η:χ2×χ2[0,) by Ξ(a,b)=(ab)32 and

    η((a,b),(k,l))={32,if ab, kl,0otherwise, 

    for all a,b,k,lχ, respectively. Consider

    (θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l))=(θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)32ϖ(Ξ(a,b),Ξ(k,l))=(1|ak|4+|bl|4)(|ak|4+|bl|4)32|Ξ(a,b)Ξ(k,l)|=132|132(ab)132(kl)|=1364|(ab)(kl)|0,

    which implies that

    η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l))(θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2).

    Therefore, (2.1) is fulfilled with (τ)=1τ and θ(τ)=τ2, for all τ>0. Also, all the hypotheses of Theorem 2.1 are satisfied and (0,0) is the unique CFP of Ξ.

    Example 2.2. (Nonlinear case) Let ϖ:χ×χR be the usual metric on χ=[0,1]. Define the mappings Ξ:χ×χχ and η:χ2×χ2[0,) by Ξ(a,b)=132(ln(1+a)ln(1+b)) and

    η((a,b),(k,l))={43,if ab, kl,0otherwise, 

    for all a,b,k,lχ, respectively. Then, we have

    (θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l))=(θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)43ϖ(Ξ(a,b),Ξ(k,l))=(1θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)43|Ξ(a,b)Ξ(k,l)|=143×32|(ln(1+a)ln(1+b))(ln(1+k)ln(1+l))|=1124|(ln(1+a1+k)+ln(1+l1+b))|1124(ln(1+|ak|)+ln(1+|lb|))0.

    Note that we used the property ln(1+a1+k)ln(1+(ak)). Hence,

    (θ(ϖ(a,k)+ϖ(b,l)2))θ(ϖ(a,k)+ϖ(b,l)2)η((a,b),(k,l))ϖ(Ξ(a,b),Ξ(k,l)).

    Therefore, (2.1) holds with (τ)=1τ and θ(τ)=τ2, for all τ>0. Furthermore, all the hypotheses of Theorem 2.1 are fulfilled and (0,0) is the unique CFP of Ξ.

    In this section, we apply Theorem 2.2 to discuss the existence solution for the following Atangana–Baleanu fractional differential equation in the Caputo sense:

    {(ABC0Dνσ)(ζ)=φ(ζ,σ(ζ),ρ(ζ)),ζI=[0,1],(ABC0Dνρ)(ζ)=φ(ζ,ρ(ζ),σ(ζ)),0ν1,σ(0)=σ0 and ρ(0)=ρ0, (3.1)

    where Dν is the Atangana-Baleanu derivative in the Caputo sense of order ν and φ:I×χ×χχ is a continuous function with φ(0,σ(0),ρ(0))=0.

    Let ϖ:χ×χ[0,) be a function defined by

    ϖ(σ,ρ)=σρ=supζI|σ(ζ)ρ(ζ)|,

    where χ=C(I,R) represents the set of continuous functions. Define a partial order on χ by

    (a,b)(k,l)ak and bl, for all a,b,k,lχ.

    It is clear that (χ,,ϖ) is a complete POMS.

    Now, to discuss the existence solution to the problem (3.1), we describe our hypotheses in the following theorem:

    Theorem 3.1. Assume that:

    (h1) there is a continuous function φ:I×χ×χχ so that

    |φ(,σ(),ρ())φ(,σ(),ρ())|Q(ν)Γ(ν)(1ν)Γ(ν)+1(θ(|σ()σ()|+|ρ()ρ()|2))×θ(|σ()σ()|+|ρ()ρ()|2),

    for I, L, θΘ and σ,ρ,σ,ρχ. Moreover, there exists :C2(I)×C2(I)C(I) such that ((σ(),ρ()),(σ(),ρ()))0 and ((ρ(),σ()),(ρ(),σ()))0, for each σ,ρ,σ,ρC(I) and I;

    (h2) there exist σ1,ρ1C(I) with ((σ1(),ρ1()),(Ξ(σ1(),ρ1()),Ξ(ρ1(),σ1())))0 and ((ρ1(),σ1()),(Ξ(ρ1(),σ1()),Ξ(σ1(),ρ1())))0, for I, where Ξ:C(I)×C(I)C(I) is defined by

    Ξ(ρ,σ)()=σ0+AB0Iνφ(,σ(),ρ());

    (h3) for σ,ρ,σ,ρC(I) and I, ((σ(),ρ()),(σ(), ρ()))0 and ((ρ(),σ()), (ρ(),σ()))0 implies

    ((Ξ(σ(),ρ()),Ξ(ρ(),σ())),(Ξ(σ(),ρ()),Ξ(ρ(),σ())))0

    and

    ((Ξ(ρ(),σ()),Ξ(σ(),ρ())),(Ξ(ρ(),σ()),Ξ(σ(),ρ())))0;

    (h4) if {σm},{ρm}C(I), limmσm=σ, limmρm=ρ in C(I), ((σm,ρm),(σm+1,ρm+1)) 0 and ((ρm,σm), (ρm+1,σm+1))0, then ((σm,ρm) ,(σ,ρ))0 and ((ρm,σm) ,(ρ,σ))0, for all mN.

    Then there is at least one solution for the problem (3.1).

    Proof. Effecting the Atangana-Baleanu integral to both sides of (3.1) and applying Lemma 1.2, we have

    σ()=σ0+AB0Iνφ(,σ(),ρ()),

    and

    ρ()=ρ0+AB0Iνφ(,ρ(),σ()).

    Now, we shall prove that the mapping Ξ:C(I)×C(I)C(I) has a CFP. From (1.1) and (1.2) and (h1), we get

    |Ξ(ρ,σ)()Ξ(ρ,σ)()|=|AB0Iν[φ(,σ(),ρ())φ(,σ(),ρ())]|=|1νQ(ν)[φ(,σ(),ρ())φ(,σ(),ρ())]+νQ(ν) 0Iν[φ(,σ(),ρ())φ(,σ(),ρ())]|1νQ(ν)|φ(,σ(),ρ())φ(,σ(),ρ())|+νQ(ν) 0Iν|φ(,σ(),ρ())φ(,σ(),ρ())|1νQ(ν)×Q(ν)Γ(ν)(1ν)Γ(ν)+1(θ(|σ()σ()|+|ρ()ρ()|2))×θ(|σ()σ()|+|ρ()ρ()|2)+νQ(ν)×Q(ν)Γ(ν)(1ν)Γ(ν)+1 0Iν(1)(θ(|σ()σ()|+|ρ()ρ()|2))×θ(|σ()σ()|+|ρ()ρ()|2)={Q(ν)Γ(ν)(1ν)Γ(ν)+1(θ(|σ()σ()|+|ρ()ρ()|2))×θ(|σ()σ()|+|ρ()ρ()|2)}(1νQ(ν)+νQ(ν)νΓ(ν)){Q(ν)Γ(ν)(1ν)Γ(ν)+1(θ(supI|σ()σ()|+supI|ρ()ρ()|2))×θ(supI|σ()σ()|+supI|ρ()ρ()|2)}(1νQ(ν)+νQ(ν)νΓ(ν))=(Q(ν)Γ(ν)(1ν)Γ(ν)+1(θ(ϖ(σ,σ)+ϖ(ρ,ρ)2))θ(ϖ(σ,σ)+ϖ(ρ,ρ)2))×(1νQ(ν)+1Q(ν)Γ(ν))=(θ(ϖ(σ,σ)+ϖ(ρ,ρ)2))θ(ϖ(σ,σ)+ϖ(ρ,ρ)2).

    Hence, for σ,ρC(I), I, with ((σ(),ρ()),(σ(),ρ()))0 and ((ρ(),σ()),(ρ(),σ()))0, we get

    ϖ(Ξ(ρ,σ)(),Ξ(ρ,σ))(θ(ϖ(σ,σ)+ϖ(ρ,ρ)2))θ(ϖ(σ,σ)+ϖ(ρ,ρ)2).

    Define η:C2(I)×C2(I)[0,) by

    η((σ(),ρ()),(σ(),ρ()))={1,if ((σ(),ρ()),(σ(),ρ()))0,0,otherwise.

    So

    η((σ(),ρ()),(σ(),ρ()))ϖ(Ξ(ρ,σ)(),Ξ(ρ,σ))(θ(ϖ(σ,σ)+ϖ(ρ,ρ)2))θ(ϖ(σ,σ)+ϖ(ρ,ρ)2).

    Then, Ξ is an ηθ-coupled contraction mapping. Now, for each ρ,σ,ρ,σC(I) and I, we have

    η((σ(),ρ()),(σ(),ρ()))1,

    due to definition of and η. So, hypothesis (h3) gives

    {η((Ξ(σ(),ρ()),Ξ(ρ(),σ())),(Ξ(σ(),ρ()),Ξ(ρ(),σ())))1,η((Ξ(ρ(),σ()),Ξ(σ(),ρ())),(Ξ(ρ(),σ()),Ξ(σ(),ρ())))1,

    for ρ,σ,ρ,σC(I). Therefore, Ξ is η-admissible. From (h2), there are σ0,ρ0C(I) with η((σ0(),ρ0()),Ξ(σ0(),ρ0()))1 and η((ρ0(),σ0()),Ξ(ρ0(),σ0()))1. Using (h4) and Theorem 2.2, we conclude that there is (ˆσ,ˆρ)C(I) with ˆσ=Ξ(ˆσ,ˆρ) and ˆρ=Ξ(ˆρ,ˆσ), that is, Ξ has a CFP, which is a solution of the system (3.1).

    Many physical phenomena can be described by nonlinear differential equations (both ODEs and PDEs), so the study of numerical and analytical methods used in solving nonlinear differential equations are an interesting topic for analyzing scientific engineering problems. From this perspective, some coupled fixed point results for the class of ηθ-contractions in POMSs are obtained. These results are reinforced by their applications in a study of the existence of a solution for a CFDE with the Mittag-Leffler kernel. In the future, our findings may be applied to differential equations of an arbitrary fractional order, linear and nonlinear fractional integro-differential systems, Hadamard fractional derivatives, Caputo-Fabrizio's kernel, and so on.

    This work was supported in part by the Basque Government under Grant IT1555-22.

    The authors declare that they have no competing interests.



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