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A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type

  • In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.

    Citation: Godwin Amechi Okeke, Akanimo Victor Udo, Rubayyi T. Alqahtani, Nadiyah Hussain Alharthi. A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type[J]. AIMS Mathematics, 2023, 8(12): 28488-28516. doi: 10.3934/math.20231458

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  • In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.



    The Atangana-Baleanu fractional differential operator (ABFDO) [1] has recently been used to define fractional derivatives (see [2,3,4]). Additionally, fractional derivatives with nonsingular kernels are crucial because some models of dissipation processes cannot be properly represented by the conventional fractional operators (see[5,6,7]). Applications are presented using ABFDO together with different types of polynomials, such as the Chebyshev polynomial, B-spline polynomials and Alexander polynomials [8,9,10,11,12].

    For decades, classical fractional calculus based on the Riemann-Liouville fractional differential and integral operators has been used to define many classes of fractional analytic functions in an open unit disk. The recent work has demonstrated the ability to modify these classes and it has offered a combination between the most important special functions, called the generalized Mittag-Leffler function (the queen of special functions) and the formula for the fractional integral operator. This work can be suggested to develop linear operators (convolution operators), as well as the integral formula for the Bulboaca integral operator, Breaz integral operator and their generalizations. In addition, since the ABFDO involves the Mittag-Leffler function, it can be extended to k-calculus and q-calculus.

    In a recent effort, we have extended the ABFDO to a complex domain (an open unit disk) to obtain the ABFDOs of a complex variable. To explore the geometric properties of the main operators, we have acted the operators on a special type of class of analytic functions with the m-fold symmetry characteristic in a bounded symmetric domain. This class of analytic functions is a natural generalization of the normalized analytic functions, when m = 1. The most discoveries in this direction involve demonstrating that the operators are convex and have starlike shapes in the open unit disk under some conditions. Moreover, the boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. Duality relations are presented for these spaces. Our method is based on Young's convolution inequality.

    The paper is divided into the following sections. Section 2 deals with the definition of the m-fold symmetric class of analytic functions and the formula that will be studied. Section 3 involves the preliminaries that will be utilized in the proof of our results. Section 4 includes the extended ABFDO and it contains the study of its geometrical characteristics. Sections 5 and 6 discuss the Bergman spaces for a bounded symmetric domain with applications. Section 7 presents the conclusion of the results and the future work.

    In this section, we deduce the meaning of the m-fold symmetric class of analytic functions in the open unit disk D:={ζC:|ζ|<1} (see [13]). In this investigation, we consider the class of m-fold symmetric functions Ωm (see [14,15,16]), as follows:

    φm(ζ)=ζ+n=1anm+1ζnm+1,ζD.

    Akgul [14] modified the class of m-fold symmetric functions in [13] to determine some coefficient results and complex inequalities. Seker and Taymur [15] described two new subclasses of bivalent functions, which are both m-fold symmetric analytic functions. In their study, they determined the upper bounds for the coefficients. Hamzat [16] has analyzed various features of fractional analytic functions belonging to two novel subclasses of m-fold symmetric starlike and convex functions in an open unit disk. Furthermore, features of a new subclass of m-fold symmetric bi-Bazilevic functions associated with modified sigmoid functions are addressed, as are numerous related minor repercussions.

    Note that for m=1, we have the normalized function (the class is denoted by Ω)

    φ(ζ)=ζ+n=2anζn,ζD.

    Corresponding to φm(ζ), we have the following class of function denoted by TΩm

    φm(ζ)=ζn=1|anm+1|ζnm+1,ζD,

    where TΩ is a special class of TΩm,m=1 with

    φ(ζ)=ζn=2|an|ζn,ζD.

    Definition 2.1. Functions φmΩm are considered to belong to the class of (κ,m)-Janowski starlike functions symbolized by (κ,m)ST(u,v),1v<u1,κ0, whenever the following inequality is true

    ((v1)(ζφm(ζ)φm(ζ))(u1)(v+1)(ζφm(ζ)φm(ζ))(u+1))>κ|(v1)(ζφm(ζ)φm(ζ))(u1)(v+1)(ζφm(ζ)φm(ζ))(u+1)1|,

    where indicates the symbol of the real part.

    Also, we have the following class of convex functions:

    Definition 2.2. Functions φmΩm are supposed to belong to the class of (κ,m)-Janowski convex functions symbolized by (κ,m)CT(u,v),1v<u1,κ0, whenever the following inequality is true

    ((v1)((ζφm(ζ))φm(ζ))(u1)(v+1)((ζφm(ζ))φm(ζ))(u+1))>κ|(v1)((ζφm(ζ))φm(ζ))(u1)(v+1)((ζφm(ζ))φm(ζ))(u+1)1|.

    When m=1, we have the Noor-Malik class described in [17].

    This section deals with the supplement results.

    Lemma 3.1. ([18], Theorem 2.4] or [19]HY__HY, Theorem 11.2]) If σ,ς,τC with (σ)>0,(ς)>0,(τ)>0, then

    ζ0χς1Eσς,τ(wχσ)dχ=ζςEσς+1,τ(wζσ).

    Lemma 3.2. [20] For the function φ(ζ)=ζ+n=2anζn, ζD, if n=2(nq)|an|ζn1q then φ is starlike of order q. Moreover, if n=2n(nq)|an|ζn1q then φ is convex of order q.

    Lemma 3.3. [21] Let ρ10, ρ2>0, and ω>1/2. If f is starlike and g is convex then the integral

    (ζω1ζ0(f(τ)τ)ρ1(g(τ)τ)ρ2dτ)1/ω

    is starlike of order (2ω1)/2ω.

    Lemma 3.4. [22] For some integer m1, let

    ρ(ζ)=1+ρmζm+ρm+1ζm+1+...

    be analytic in D with its nonpositive real part in D. Then, there exists a point ζ0D with ρ(ζ0)=iξ and ζ0ρ(ζ0)=ϑ, where ϑm(1+ξ2)/2.

    Numerous mathematical, physical and engineering fields make use of the Mittag-Leffler function, particularly in relation to fractional calculus, fractional differential equations and fractional order systems. It appears in issues with anomalous diffusion, viscoelasticity and memory effects. Recursive relations, integral representations and linkages to other special functions are only a few of the Mittag-Leffler function's intriguing characteristics. It is essential to fractional calculus and serves as a potent tool for comprehending and resolving issues involving fractional derivatives and integrals.

    A branch of fractional calculus, which is an extension of ordinary calculus, is the fractional operator based on the Mittag-Leffler function. It has been applied to simulate a variety of physical events and is especially helpful when representing non-local or memory effects-based systems. There are numerous scientific and engineering domains for which the Mittag-Leffler function and the fractional operator it defines are applied, including physics, biology, economics and signal processing. These methods offer a more thorough framework for comprehending and examining intricate systems involving fractional order dynamics. The advantages of using the Mittag-Leffler function include, but are not limited to the following observations. It is simpler to deal with the Mittag-Leffler function in theoretical analysis and modeling since it has good features. It makes the exploration of fractional operators more approachable by allowing mathematicians and scientists to find closed-form solutions to fractional differential equations. In order to ensure that numerical simulations and approximations are well-behaved and accurately converge to the true solution, it gives stable solutions to fractional differential equations. Regarding its relationship to practical applications, in numerous real-world applications, such as the modeling of biological systems, financial mathematics, control systems, and diffusion operations in porous media, fractional calculus with the Mittag-Leffler function has shown great potential.

    In this section, we proceed to extend the ABFDO in D.

    Definition 4.1. The generalized Mittag-Leffler function is defined by

    Eαβ,γ(ζ)=n=0(α)nΓ(βn+γ)ζnn!,(ζ,α,β,γC,Reβ>0),

    where (α)n represents the Pochhammer symbol.

    We will employ double Mittage-Leffler functions in definition to display the modified ABFDOs of a complex variable.

    Definition 4.2. For φmΩm, the extended fractional operators are given, as follows:

    ABCΔνζφm(ζ)=w(ν)1νζ0φm(η)Eν,ω(μνην)Eν(μν(ζη)ν)dη (4.1)

    and

    ABRΔνζφm(ζ)=w(ν)1νddζζ0φm(ζ)Eν,ω(μνην)Eν(μν(ζη)ν)dη, (4.2)

    where ω indicates the power of ζ in the power series of φm(ζ).

    Example 4.3. Suppose that φm(ζ)=ζ. By Lemma 3.1, we have

    ABCΔνζ(ζ)=w(ν)1νζ0Eν(μνην)Eν(μν(ζη)ν)dη=w(ν)1ν(ζE2ν,2(μν(ζ)ν))=w(ν)1ν(ζk=0(2)kζkk!Γ(kν+2)),

    where (y)n=y(y+1)...(y+n1). And,

    ABRΔνζ(ζ)=w(ν)1νddζζ0Eν(μνην)Eν(μν(ζη)ν)ηdη=w(ν)1ν(ζ2E2ν,3(μν(ζ)ν))=w(ν)1ν(ζE2ν,2(μν(ζ)ν)).

    As a result, we obtain the relation ABCΔνζ(ζ)=ABRΔνζ(ζ). In general, we get

    ABCΔνζ(ζmn)=(w(ν)1ν)nζmn(E2ν,1+mn(μν(ζ)ν)),n1,
    ABRΔνζ(ζmn)=(w(ν)1ν)ζmn(E2ν,1+mn(μν(ζ)ν)).

    We have the following result:

    Proposition 4.4. Let φmΩm and b(ν):=w(ν)(1ν). Then,

    ABCΔνζφm(ζ):=ABCΔνζφm(ζ)b(ν)E2ν,2(μν(ζ)ν)Ωm

    and

    ABRΔνζφm(ζ):=ABRΔνζφm(ζ)b(ν)E2ν,2(μν(ζ)ν)Ωm.

    Proof. Let φmΩm. A calculation implies that

    ABCΔνζφm(ζ)=ABCΔνζφm(ζ)b(ν)E2ν,2(μν(ζ)ν)=ζνnABCΔνζφm(ζ)b(ν)ζνnE2ν,2(μν(ζ)ν)=ζνn(b(ν)E2ν,2(μν(ζ)ν)ζ+n=1anm+1b(ν)(nm+1)(E2ν,2+nm(μν(ζ)ν))ζnm+1)b(ν)ζνnE2ν,2(μν(ζ)ν)=ζ+ζνn(n=1anm+1b(ν)(nm+1)([n=0(2)nΓ(νn+(mn+2))(μν)nζνnn!])ζnm+1)b(ν)ζνn[n=0(2)nΓ(νn+2)(μν)nζνnn!]=ζ+n=1anm+1(nm+1)([n=0(2)nΓ(νn+(nm+2))(μν)nn!])ζnm+1[n=0(2)nΓ(νn+(nm+2))(μν)nn!]=ζ+n=1anm+1(mn+1)(E2ν,2+mn(μν)E2ν,2(μν))ζnm+1:=ζ+n=1Δnm+1ζnm+1=(ζ+n=1anm+1ζnm+1)(ζ+n=1σν,μmn+1ζnm+1)=:φm(ζ)σm(ζ),

    where σν,μmn:=(nm+1)(E2ν,2+mn(μν)E2ν,2(μν)), and the notation "" indicates the convolution product. Thus, we conclude that ABCΔνζφm(ζ)Ωm. Similarly, we have that ABRΔνζφm(ζ)Ωm. Note that the integral corresponding to ABCΔνζφm(ζ) is given by the following series:

    ABCIνζφm(ζ)=ζ+n=1anm+1(E2ν,2(μν)(mn+1)E2ν,2+mn(μν))ζnm+1

    satisfying

    ABCIνζABCΔνζφm(ζ)=ABCΔνζABCIνζφm(ζ)=φm(ζ).

    A modification of the ABFDO is given for the normalized univalent functions and quantum analytic functions described in [23].

    In this part, we shall investigate the most important geometric properties of the operator ABCΔνζφm(ζ)Ωm.

    Theorem 4.5. The operator ABCΔνζφm(ζ)Ωm can be included in the class (κ,m)ST(u,v),1v<u1,κ0, if it satisfies the following condition:

    n=1(2(mn)(κ+1)+|(mn+1)(v+1)(1+u)|)|anm+1σν,μnm+1|<|vu|. (4.3)

    Proof. We aim to show that

    κ|(v1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u1)(v+1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u+1)1|((v1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u1)(v+1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u+1)1)<1.

    A computation yields

    κ|(v1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u1)(v+1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u+1)1|((v1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u1)(v+1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u+1)1)(κ+1)|(v1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u1)(v+1)(ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])(u+1)1|=2(κ+1)|[ABCΔνζφm(ζ)]ζ[ABCΔνζφm(ζ)](v+1)ζ[ABCΔνζφm(ζ)](1+u)[ABCΔνζφm(ζ)]|2(κ+1)(n=1(mn)|anm+1σν,μnm+1||vu|n=1|(nm+1)(1+v)(1+u)||anm+1σν,μnm+1|).

    If the condition (4.3) is true, the last assertion is bounded by 1, which completes the proof.

    For a comparison with other works, we have the following observations:

    m=1 and σν,μnm+1=1 [17];

    m=1,u=1,v=1 and σν,μnm+1=1 [24];

    m=1,u=12a,a[0,1),v=1 and σν,μnm+1=1 [25];

    m=1,u=12a,a[0,1),v=1,κ=0 and σν,μnm+1=1 [20].

    In a similar proof of Theorem 4.5, we have the following result:

    Theorem 4.6. The operator ABCΔνζφm(ζ)Ωm is included in the class (κ,m)CT(u,v),1v<u1,κ0, if it satisfies the following condition:

    n=1(mn+1)(2(nm)(κ+1)+|(mn+1)(v+1)(1+u)|)|anm+1σν,μnm+1|<|vu|. (4.4)

    Theorem 4.7. Let φmΩm be starlike of order ,[0,1) with non-positive coefficients (anm+10). Moreover, let

    n=1(1+nm1)anm+1σν,μnm+11.

    Then,

    (1) [ABCΔνζφm(ζ)] achieves the starlikeness under the order .

    (2) It satisfies the boundedness inequality

    |ζ|11+m|ζ|1+m|[ABCΔνζφm(ζ)]||ζ|+11+m|ζ|1+m.

    (3) Its derivative reaches the following maximum bound and minimum bound:

    1(1+m)(1)(1+m)|ζ|m|[ABCΔνζφm(ζ)]|1+(1+m)(1)(1+m)|ζ|m.

    (4) The maximal function is given by the formula

    [ABCΔνζφm(ζ)]=ζ(11+m)ζ1+m.

    (5) If σm(ζ) and φm(ξ) are starlike of order , then [ABCΔνζφm(ζ)] is starlike of order q, where

    q:=1+m2m+22.

    Proof. By the positivity of the connections, φm can be represented by the power series

    φm(ζ)=ζn=1anm+1ζmn+1,ζD,mN.

    In addition, since φm is starlike of order , where [0,1) and the following inequality is satisfied:

    n=1(1+nm1)amn+1σν,μnm+11,

    then, according to Lemma 3.2, we get the starlikeness of [ABCΔνζφm(ζ)] under the order .

    Using the leader component, we can obtain

    (1+m1)n=1amn+1σν,μnm+1n=1(1+nm1)amn+1σν,μnm+11,

    which yields

    n=1amn+1σν,μnm+111+m.

    Consequently, we get

    |[ABCΔνζφm(ζ)]||ζ||ζ|1+mn=1amn+1σν,μnm+1|ζ||ζ|1+m(11+m)

    and

    |[ABCΔνζφm(ζ)]||ζ|+|ζ|1+mn=1amn+1σν,μnm+1|ζ|+|ζ|1+m(11+m).

    We get the second portion by combining the two inequalities above.

    Using the information below,

    n=1(nm+1)amn+1σν,μnm+11+(1)1+m=(1+m)(1)1+m,

    we have

    |[ABCΔνζφm(ζ)]|1|ζ|mn=1(1+nm)amn+1σν,μnm+11(1)(1+m)(1+m)|ζ|m

    and

    |[ABCΔνζφm(ζ)]|1+|ζ|mn=1(1+nm)amn+1σν,μnm+11+(1)(1+m)(1+m)|ζ|m.

    We get the third item when we combine the above inequalities. The maximal function obtained from a direct calculation is as follows:

    [ABCΔνζφm(ζ)]=ζ(11+m)ζ1+m,

    which completes part four.

    Using the definition of the convolution product, we get

    ABCΔνζφm(ζ)=(σmφm)(ζ),

    where σm and φm are starlike of order . To prove the starlikeness of ABCΔνζφm(ζ), it is sufficient to show that

    n=1(1+nmq1q)amn+1σν,μnm+11.

    Since

    n=1(1+nm1)amn+11

    and

    n=1(1+nm1)σν,μnm+11,

    the Cauchy-Schwarz inequality implies that

    n=1(1+nm1)amn+1σν,μnm+11,

    where

    amn+1σν,μnm+111+nm.

    But,

    11+nm(1+nm)(1q)(1)(1+nmq);

    or, equivalently,

    q(1+nm)2(1+nm)(1)2(1+nm)2(1)2.

    But the above fraction is an increasing function; thus, by letting n=1, the inequality of the above conclusion yields

    q:=1+m22+m2.

    Hence, according to Lemma 3.2, we have that [ABCΔνζφm(ζ)] is starlike of order q.

    Theorem 4.8. Assume the convexity of φmΩm with order ,[0,1) and non-positive coefficients (anm0). Moreover, suppose that

    n=1((mn+1)(1+nm)1)amn+1σν,μmn+11.

    Then,

    (1) [ABCΔνζφm(ζ)] achieves convexity under order .

    (2) It satisfies the boundedness inequality

    |ζ|1(1+m)(1+m)|ζ|1+m|[ABCΔνζφm(ζ)]||ζ|+1(1+m)(1+m)|ζ|1+m.

    (3) Its derivative admits the following boundedness inequality:

    11(1+m)|ζ|m|[ABCΔνζφm(ζ)]|1+1(1+m)|ζ|m.

    (4) The maximal function is given by the formula

    [ABCΔνζφm(ζ)]=ζ((1)(1+m)(1+m))ζ1+m.

    (5) If σm and φm are convex of order , then [ABCΔνζφm(ζ)] is convex of order , where

    q:=(1+m)22(1+m)(1)2(1+m)22(1)2.

    Proof. Assume that

    φm(ζ)=ζn=1amn+1ζmn+1,ζD,

    satisfies the inequality

    n=1((1+mn)(1+mn)1)amn+1σν,μnm+11.

    And in view of Lemma 3.2 (the second part), we have that [ABCΔνζφm(ζ)] admits a convexity under order .

    As a consequence of the above conclusion, we get

    ((1+m)(1+m)1)n=1amn+1σν,μnm+1n=1(1+mn)(1+mn1)amn+1σν,μnm+1,1

    which yields

    n=1amn+1σν,μnm+11(1+m)(1+m).

    Moreover, we have

    n=1nmamn+1σν,μnm+11(1+m).

    Thus, we are left with the second and third sections, respectively. Clearly, the formula gives the greatest sharp function, as follows:

    [ABCΔνζφm(ζ)]=ζ((1)(1+m)(1+m))ζ1+m.

    A convolution property implies that

    [ABCΔνζφm(ζ)]=σm(ζ)φm(ζ),

    where σm and φm are convex of order . To obtain that [ABCΔνζφm(ζ)] is convex of order q, we obtain that

    n=1(1+mn)(1+mnq1q)amn+1σν,μnm+11.

    Since

    n=1(1+mn)(1+mn1)amn+11

    and

    n=1(1+mn)(1+mn1)σν,μnm1,

    the Cauchy-Schwarz inequality yields

    n=1(1+nm)(1+nm1)anm+1σν,μnm+11,

    where

    anm+1σν,μnm+11(1+nm)(1+mn).

    But,

    1(1+nm)(1+mn)(1+nm)(1q)(1+mn)(1)1+(nmq),

    or, equivalently, we have the increasing inequality

    q(1+mn)22(1+mn)(1)2(1+mn)22(1)2.

    By assuming that n=1, computation yields

    q=(1+m)22(1+m)(1)2(1+m)22(1)2.

    Hence, [ABCΔνζφm(ζ)] addresses the convexity under order q.

    Theorem 4.9. Consider the operator [ABCΔνζφm(ζ)]. Then,

    [ABCΔνζφm(ζ)]S(ζω1ζ0([ABCΔνζφm(τ)]τ)ρ1(gm(τ)τ)ρ2dτ)1/ωS(2ω12ω),

    where gm is a convex univalent function, ω>1/2, ρ10 and ρ2>0.

    Moreover, if

    gm(ζ)=ζ1ζm,ρ2=1,

    then

    [ABCΔνζφm(ζ)]S(ζω1ζ0([ABCΔνζφm(τ)]τ(1τm)1/ρ1)ρ1dτ)1/ωS(2ω12ω),

    where g is a convex univalent function and ρ10 and ρ2>0.

    Proof. Let

    gm(ζ)=ζ+n=1gmn+1ζnm+1.

    First, we must show that

    (ζω1ζ0([ABCΔνζφm(τ)]τ)ρ1(gm(τ)τ)ρ2dτ)1/ωΩm. (4.5)

    By the definition of [ABCΔνζφm(ζ)], we have

    I[F,G]m(ζ):=(ζω1ζ0([ABCΔνζφm(τ)]τ)ρ1(gm(τ)τ)ρ2dτ)1/ω=(ζω1ζ0(τ+n=1Δnm+1τmn+1τ)ρ1(τ+n=1gmn+1τnm+1τ)ρ2dτ)1/ω=(ζω1ω0(1+n=1Δnm+1τmn)ρ1(1+n=1gmn+1τnm)ρ2dτ)1/ω=(ζω1ζ0(1+ρ1n=1Δnm+1τmn+...)(1+ρ2n=1gmn+1τmn+...)dτ)1/ω=(ζω1ζ0((1+ρ1n=1Δnm+1τnm)+...)dτ)1/ω.

    As a consequence, we obtain (4.5). By the convexity of gm, we attain that it is in the class S(1/2). Since the multiplication of starlike functions implies starlikeness, I[F,G]m(ζ) admits starlikeness of order (2ω1)/2ω (see Lemma 3.3). The second part of the theorem is valid when gm(ζ)=ζ/(1ζm) and ρ2=1.

    We have the following outcome for some geometric inequalities:

    Theorem 4.10. Consider the operator [ABCΔνζφm(ζ)]. If

    |[ABCΔνζφm(ζ)]"[ABCΔνζφm(ζ)]|m2+1,

    then

    (1) (ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])<12;

    (2) or, equivalently, |ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)]1|<1.

    Proof. Note that (ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)])<12 is equivalent to ([ABCΔνζφm(ζ)]ζ[ABCΔνζφm(ζ)])>12. Based on the above inequality, we define the following function:

    ρ(ζ):=2[ABCΔνζφm(ζ)]ζ[ABCΔνζφm(ζ)]1.

    Then one can write the formula series ρ(ζ)=1+ρmζm+ρm+1ζm+1+... by using

    ζ[ABCΔνζφm(ζ)]"[ABCΔνζφm(ζ)]=1ρ(ζ)ζρ(ζ)1+ρ(ζ).

    Assume that ρ(ζ) does not have a positive real component. Lemma 3.4 states that a point ζ0 belongs to D, where ρ(ζ0)=iξ and ζ0ρ(ζ0)=ϑ, where ϑm(1+ξ2)/2. Direct computation yields

    |ζ[ABCΔνζφm(ζ)]"[ABCΔνζφm(ζ)]|=(1ϑ)2+ξ21+ξ2(1+m(1+ξ2)/2)2+ξ21+ξ2(1+m2)2.

    This leads to the assertion made by this theorem.

    Not that, when m=1 and σν,μnm+1=1, we obtain the result presented in [26], and when σν,μnm=1, we have the result presented in [13]. Moreover, Theorem 4.10 can be considered for the integral operators in Theorem 4.9 and the fractional operator corresponds to [ABCΔνζφm(ζ)].

    Theorem 4.11. Consider the operator [ABCΔνζφm(ζ)]. If

    |[ABCΔνζφm(ζ)]"[ABCΔνζφm(ζ)]k|m214m,

    for kZ+ and

    [ABCΔνζφm(ζ)]k=12kk1n=0([ABCΔνζφm(ϖnζ)]ϖn)+ϖn¯[ABCΔνζφm(ϖnˉζ)],ϖ=exp(2πi/k),

    then

    (ζ[ABCΔνζφm(ϖnζ)][ABCΔνζφm(ϖnζ)]k)>0.

    Proof. By the assumption of the theorem, we have

    [ABCΔνζφm(ζ)]k=12kk1n=0([ABCΔνζφm(ϖnζ)])+¯[ABCΔνζφm(ϖnˉζ)]

    and

    [ABCΔνζφm(ζ)]k"=12kk1n=0([ABCΔνζφm(ϖnζ)])"+¯[ABCΔνζφm(ϖnˉζ)]".

    Direct computation yields

    |ϖk[ABCΔνζφm(ϖkζ)]"[ABCΔνζφm(ζ)]k|m214m

    and

    |ϖk¯[ABCΔνζφm(ϖkζ)]"[ABCΔνζφm(ζ)]k|m214m.

    Combining the above inequalities, we obtain

    |[ABCΔνζφm(ζ)]"k[ABCΔνζφm(ζ)]k|m214m.

    We proceed to show that (ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)]k)>0. Let ρ(ζ)=ζ[ABCΔνζφm(ζ)][ABCΔνζφm(ζ)]k. Then, by the formula series ρ(ζ)=1+ρmζm+ρm+1ζm+1+... can be applied

    ζ[ABCΔνζφm(ζ)]"[ABCΔνζφm(ζ)]k=([ABCΔνζφm(ζ)]kζ[ABCΔνζφm(ζ)]k)(ζρ(ζ)ρ(ζ)(1ζ[ABCΔνζφm(ζ)]k[ABCΔνζφm(ζ)]k)).

    Assume that ρ(ζ) does not have a positive real component. Lemma 3.4 states that ζ0 belongs to D with ρ(ζ0)=iξ and ζ0ρ(ζ0)=ϑ, where ϑm(1+ξ2)/2. Direct computation yields

    |ζ0[ABCΔνζφm(ζ0)]"[ABCΔνζφm(ζ0)]k|=|ϑ+iξ(ζ0[ABCΔνζφm(ζ0)]k[ABCΔνζφm(ζ0)]k1)|2|ϑ||ξ|2m(1+ξ2)/2|ξ|2m214m.

    This contradicts the assertion made by this theorem. Hence, (ζ[ABCΔνζφm(ϖnζ)][ABCΔνζφm(ϖnζ)]k)>0.

    The above theorem is valid for the integrals in Theorem 4.9 and the integral corresponds to [ABCΔνζφm(ϖnζ)].

    Example 4.12. Consider the fractional differential equation

    (ζ[ABCΔνζφm(ϖnζ)][ABCΔνζφm(ϖnζ)])=1. (4.6)

    Equation (4.6) has the following expression:

    [ABCΔνζφm(ϖnζ)]=ζ,

    which satisfies

    (ζ[ABCΔνζφm(ϖnζ)][ABCΔνζφm(ϖnζ)])=1>0;

    thus, it is starlike. Let

    gm(ζ)=ζ1ζm,ρ2=1;

    then, according to Theorem 4.9,

    (ζω1ζ0(1(1τm)1/ρ1)ρ1dτ)1/ωS(2ω12ω),

    where gm is a convex univalent function, ρ10 and ρ2>0.

    In this part, we study the boundedness of the operator ABCΔνζφm(ζ) in some well-known spaces. We shall use the Bergman space of analytic functions in D (bounded symmetric domain). The weighted Bergman space is a modification of the Bergman space in which functions are not only square integrable, but they also have integrability that can be quantified in terms of a specific weight function. The space of functions that are square-integrable with respect to this weight is affected by this weight function, which imposes varying weights on various places in the bounded symmetric domain. Weighted Bergman spaces have been utilized to investigate diverse analytic function qualities for certain domains and particular actions selected by the weight function. Alongside other mathematical disciplines these spaces have uses in complex analysis, potential theory, and harmonic analysis. The characteristics of the related weighted Bergman space might be very different for a given domain and weight function. Understanding how analytic functions behave in relation to the domain's underlying geometry and weight distribution can be accomplished by looking at the properties of functions in these spaces. These realizations may then contribute to a deeper comprehension of sophisticated analysis and associated mathematical ideas.

    The weighted Bergman space is a set of all analytic functions in D (bounded symmetric domain) with the norm [27]

    ϕBβp=((1+β)D(1|ζ|2)β|ϕ(ζ)|pdΛ(ζ))1/p<,(β>1,p(0,)).

    The convex structure is formulated when γ(0,1/2], as follows:

    ϕBγp=((1γ)γD(1|ζ|2)12γγ|ϕ(ζ)|pdΛ(ζ))1/p<,(γ(0,1/2],p(0,)),

    where dΛ=dζ/π is the area measure. Note that, when β=12γγ, we obtain the weighted space. In addition, the non-normal weighted logarithmic Bergman space is defined as follows [28]:

    ϕBβp,log=(D(loge1|ζ|2)β|ϕ(ζ)|pdΛ(ζ)1|ζ|2)1/p<,(β>1,p(0,)).

    The two parameter normal weighted logarithmic Bergman space is defined as follows [29]:

    ϕBβ,γp,log=(D(log11|ζ|)β(1|ζ|)γ|ϕ(ζ)|pdΛ(ζ))1/p<,(β0,γ>1,p(0,)).

    Finally, the general weighted Bergman spaces have the following structure:

    ϕBpω=(D|ϕ(ζ)|pω(ζ)dΛ(ζ))1/p<,ωL1(D).

    Alternatively, they have the following parametric structure [30]:

    ϕBpω=(D|ϕ(ζ)|pω(ζ)dΛ(ζ))1/p<,ω(ζ)=ω(ζ)ϖ(ζ)αL1(D),αR.

    We have the following result of this section:

    Theorem 5.1. Consider the operator ABCΔνζφm(ζ), where φmΩm. Then,

    (1) φmBβpABCΔνζφmBβp,β>1;

    (2) φmBγpABCΔνζφmBγp,γ(0,1/2];

    (3) φmBβp,logABCΔνζφmBβp,log,β>1;

    (4) φmBβ,γp,logABCΔνζφmBβ,γp,log,β0,γ>1;

    (5) φmBpωABCΔνζφmBpω,ωL1(D);

    (6) φmBpωABCΔνζφmBpω,ω(ζ)=ω(ζ)ϖα(ζ)L1(D),αR.

    Proof. Let φmBβp. Assume that

    Σm(β,p):=sup(β,p),|ζ|=r((1+β)D(1|ζ|2)β|σm(ζ)|pdΛ(ζ))1/p<.

    Then, for p1, Young's inequality of the convoluted functions implies that

    ABCΔνζφmBβp=((1+β)D(1|ζ|2)β|ABCΔνζφm(ζ)|pdΛ(ζ))1/p=((1+β)D(1|ζ|2)β|φm(ζ)σm(ζ)|pdΛ(ζ))1/p((1+β)D(1|ζ|2)β|φm(ζ)|pdΛ(ζ))1/p((1+β)D(1|ζ|2)β|σm(ζ)|pdΛ(ζ))1/pΣm(β,p)φmBβp<.

    Thus, ABCΔνζφmBβp. Conversely, let

    λ(β,p):=sup(β,p),|ζ|=r((1+β)D(1|ζ|2)β|ςm(ζ)|pdΛ(ζ))1/p<

    and assume that ABCΔνζφmBβp<. Analogous to σm(ς), define the function ςm(ζ) as follows (see Figures 1 and 2):

    σm(ζ)ςm(ζ)=ζ+n=1ζnm+1=ζ1ζm,ζD.
    Figure 1.  3D plots of the m-fold symmetric function ζ/(1ζm) when m=1,2,3,4 respectively (the graph was plotted by using Mathematica 13.3).
    Figure 2.  3D plot of the m-fold symmetric function ζ/(1ζm) when m=10,20,50,100 respectively (the graph was plotted by using Mathematica 13.3).

    Then, Young's inequality yields

    φmBβp=((1+β)D(1|ζ|2)β|φm(ζ)|pdΛ(ζ))1/p=((1+β)D(1|ζ|2)β|φm(ζ)(ζ1ζm)|pdΛ(ζ))1/p=((1+β)D(1|ζ|2)β|φm(ζ)(σm(ζ)ςm(ζ))|pdΛ(ζ))1/p=((1+β)D(1|ζ|2)β|(φm(ζ)σm(ζ))ςm(ζ)|pdΛ(ζ))1/p((1+β)D(1|ζ|2)β|(φm(ζ)σm(ζ))|pdΛ(ζ))1/p((1+β)D(1|ζ|2)β|ςm(ζ)|pdΛ(ζ))1/pλ(β,p)ABCΔνζφmBβp<.

    Thus, φmBβp.

    The process is similar for the other above listed cases.

    Remark 5.2. Figures 1 and 2 present the m-symmetrical behavior of the Koebe function, which is the extreme function of the convexity in an open unit disk. The Koebe function is an extreme function in a number of univalent function problems. The m-symmetric Koebe function is a useful mathematical tool for complex analysis and conformal mapping theory. It helps mathematicians and scientists to understand and work with conformal mappings, which have numerous applications in physics and engineering. While it lacks a direct physical interpretation, the characteristics and theorems it references can be used to solve real-world problems involving complicated shapes and locations.

    In the same manner of Theorem 5.1, we have the following result regarding the operator ABRΔνζφm(ζ).

    Theorem 5.3. Consider the operator ABRΔνζφm(ζ), where φmΩm. Then, consider the following:

    (1) φmBβpABRΔνζφmBβp,β>1;

    (2) φmBγpABRΔνζφmBγp,γ(0,1/2];

    (3) φmBβp,logABRΔνζφmBβp,log,β>1;

    (4) φmBβ,γp,logABRΔνζφmBβ,γp,log,β0,γ>1;

    (5) φmBpωABRΔνζφmBpω,ωL1(D);

    (6) φmBpωABRΔνζφmBpω,ω(ζ)=ω(ζ)ϖα(ζ)L1(D),αR.

    Numerous mathematical disciplines, such as complex analysis, functional analysis, harmonic analysis, operator theory and others, all make use of the Bergman space. It is an invaluable instrument for comprehending and resolving issues in these sectors because of its adaptability and connections to diverse mathematical disciplines. A symmetric function is one that does not change when its variables are permuted. An m-fold symmetric function is a special form of symmetric function in which the variables are permuted by separating them into m distinct subgroups and permuting the variables within each subset. The physical meaning of an m-fold symmetric function is determined by the situation. Here are a handful of samples to demonstrate its significance in many fields:

    ● In physics, m-fold symmetric functions can describe the action of material properties that maintain some kind of symmetry when seen from multiple locations or orientations inside the crystal lattice, particularly in the study of solid-state materials and crystals. When investigating the electronic band structure of crystals, for example, m-fold symmetric functions can help scholars to describe the energy levels and wave functions of electrons in the crystal while taking the crystal's symmetry into account (see [31]).

    ● In chemistry, symmetry is important for the classification of molecular structures and their spectroscopic properties. The symmetry of molecular vibrations, electronic states and other features can be described by using m-fold symmetric functions. For instance, when evaluating a molecule's vibration modes, m-fold symmetric functions can aid in the determination of which modes are Raman-active or infrared-active, as predicated on their symmetry-related features (see [32]).

    ● In engineering, m-fold symmetric functions can be utilized to examine signals or systems that display specified symmetries, particularly in signal processing and control theory. By taking advantage of the underlying symmetries, this may simplify system analysis and implementation (see [33]). Other applications are discussed in [34,35,36,37].

    In this part, we study the estimate of the fractional equation

    ¯(ABCΔνζφm)(ζ)=φm(ζ), (6.1)

    when

    φmBpω=(D|φm(ζ)|pω(ζ)dΛ(ζ))1/p<.

    It is well known that the ¯(.)=(.)¯ζd¯ζ equation has many applications in different fields, including mathematical physics and fluids.

    Theorem 6.1. Consider the Eq (6.1). Then, it admits a solution satisfying the finite inequality

    D|(ABCΔνζφm)(ζ)|p(ω(ζ))p/2dΛC1D|φm(ζ)|p(ω(ζ))p/2ϖp(ζ)dΛ, (6.2)

    where ω is the decreasing weight. Furthermore,

    supζD|(ABCΔνζφm)(ζ)|(ω(ζ))1/2C2supζD|φm(ζ)|(ω(ζ))1/2ϖ(ζ), (6.3)

    where C1,C2 are positive constants.

    Proof. It is enough to prove that

    D|φm(ζ)|p(ω(ζ))p/2ϖp(ζ)dΛ<.

    Define an analytic function on the disk D(r0), where

    r0:=ϖ(ξ0)r<1,

    as follows:

    Ψ(ξ)=ω(ξ)1/2,ξD(r0).

    Let χn be a partition covering the disk D(rn),rnr<1 with |χn(ξ)|<1. Suppose that

    Fn(φm)(ξ):=Ψ(ξn)Dφm(ξ)χn(ξ)(ξζ)Ψ(ξn)dΛ(ξ).

    According to the Cauchy-Pompeiu formula, we get

    ¯Fn(φm)(ζ)=φm(ζ)χn(ζ),n=1,2,....

    Then, it can be extended by the power series

    F(φm)(ζ)=n=1Fn(φm)(ζ).

    Thus, we obtain

    ¯F(φm)(ζ)=n=1¯Fn(φm)(ζ)=n=1φm(ζ)χn(ζ)=φm(ζ)n=1χn(ζ)=φm(ζ).

    Assume that

    D|n=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2|dΛ(ξ)ϖ(ξ)1 (6.4)

    and

    D|Ψ(ζ)||ξζ|[ω(ζ)]1/2dΛ(ζ)[ϖ(ξ)]1+α2,ξD(rn). (6.5)

    We aim to show that

    D|Fφm(ζ)|p[ω]p/2dΛ(ζ)D|φm(ζ)|p[ω]p/2ϖp(ζ)dΛ(ζ).

    According to Hölder's inequality, we obtain

    |Dn=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2(φm(ξ)[ω(ξ)]1/2)|p(D|n=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2||φm(ξ)[ω]1/2(ξ)|p[ϖ(ξ)]p1dΛ(ξ))×(D|n=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2|dΛ(ξ)ϖ(ξ))p1(D|n=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2||φm(ξ)[ω]1/2|p[ϖ(ξ)]p1dΛ(ξ)).

    Now, in view of Fubini's theorem, and by using the inequalities (6.4) and (6.5), we have

    D|n=1(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))ω(ξ)]1/2[ω(ζ)]1/2(φm(ξ)[ω(ξ)]1/2)ϖp1(ξ)|pdΛ(ζ)D(Dn=1|(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2||(φm(ξ)[ω(ξ)]1/2)|pϖp1(ξ)dΛ(ξ))dΛ(ζ)D|(φm(ξ)[ω(ξ)]1/2)|pϖp1(ξ)(Dn=1|(Ψ(ζn)χn(ξ)(ξζ)Ψ(ξn))[ω(ξ)]1/2[ω(ζ)]1/2|dΛ(ζ))dΛ(ξ)D|(φm(ξ)[ω(ξ)]1/2)|pϖp1(ξ)([ω(ξ)]1/2Ψ(ξn)Dn=1|(Ψ(ζn)(ξζ))|[ω(ζ)]1/2dΛ(ζ))|χn(ξ)|dΛ(ξ)n=1D(rn)|(φm(ξ)[ω(ξ)]1/2)|pϖp1(ξ)([ω(ξ)]1/2Ψ(ξn)D|(Ψ(ζn)(ξζ))|[ω(ζ)]1/2dΛ(ζ))|χn(ξ)|dΛ(ξ)n=1D(rn)|(φm(ξ)[ω(ξ)]1/2)|pϖp1α/2(ξ)(D|Ψ(ζn)(ξζ)|[ω(ζ)]1/2dΛ(ζ))dΛ(ξ)n=1D(rn)|(φm(ξ)[ω(ξ)]1/2)|pϖp(ξ)dΛ(ξ)D|(φm(ξ)[ω(ξ)]1/2)|pϖp(ξ)dΛ(ξ)<,

    where

    maxξ[ω(ξ)]1/2Ψ(ξn)1

    and χn(ξ) is considered for D(rn),rn<1 with |χn(ξ)|<1. Then, we obtain (6.2). Since φmBpω, Theorem 5.1 yields (6.3).

    The proof is completed.

    Working on a specific kind of class of analytic functions with the m-fold symmetry feature in a complex domain, we expanded the fractional differential operator. We have illustrated a set of geometric properties of this operator including the uniform starlike and uniform convex shapes (Theorem 4.5). Sufficient conditions on this operator are presented to be starlike in terms of double (κ,m)-symmetric-conjugate points (Theorems 4.10 and 4.11). Under some conditions, the operator preserves some integral formulas (Theorem 4.9). Sharpness for some geometric properties has been indicated. Applications in the field of fractional differential equations are presented to determine the geometric behavior of the solutions in an open unit disk (Example 4.12). The final aim of this work was to study the symmetry of fractional differential operator in Bergman spaces for a symmetric domain. We suggest that the applications can to find the solution of the ¯-equation whenever φmBpω. To summarize, the use of fractional derivatives of complex variables is a particular mathematical technique that involves applying fractional calculus to complex functions. They are applied in a variety of scientific and technical disciplines whereby complex systems or events must be investigated and simulated. A fractional derivative in the complex plane can be converted to a fractional Laplacian operator in some instances, which is a generalization of the Laplacian operator for real variables. Other properties can be considered in the future by using different classes of analytic functions, including the class of meromorphic functions, multi-valent functions and harmonic functions.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare no conflict of interest.



    [1] A. Atangana, D. Baleanu, Application of fixed point theorem for stability analysis of a nonlinear Schrodinger with Caputo-Liouville derivative, Filomat, 31 (2017), 2243–2248. https://doi.org/10.2298/FIL1708243A doi: 10.2298/FIL1708243A
    [2] E. Karapinar, T. Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Adv. Differ. Equ., 2019 (2019), 421. https://doi.org/10.1186/s13662-019-2354-3
    [3] G. A. Okeke, D. Francis, C. A. Nse, A generalized contraction mapping applied in solving modified implicit ϕ-Hilfer pantograph fractional differential equations, J. Anal., 31 (2023), 1143–1173.
    [4] M. Syam, M. Al-Refai, Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications, Chaos Soliton. Fract. X, 2 (2019), 100013. https://doi.org/10.1016/j.csfx.2019.100013 doi: 10.1016/j.csfx.2019.100013
    [5] X. Zhang, P. Chen, Y. Wu, B. Wiwatanapataphee, A necessary and sufficient condition for the existence of entire large solutions to a k-Hessian system, Appl. Math. Lett., 145 (2023), 108745. https://doi.org/10.1016/j.aml.2023.108745 doi: 10.1016/j.aml.2023.108745
    [6] X. Zhang, P. Xu, Y. Wu, B. Wiwatanapataphee, The uniqueness and iterative properties of solutions for a general Hadamard-type singular fractional turbulent flow model, Nonlinear Anal.-Model., 27 (2022), 428–444. https://doi.org/10.15388/namc.2022.27.25473 doi: 10.15388/namc.2022.27.25473
    [7] X. Zhang, J. Jiang, Y. Wu, B. Wiwatanapataphee, Iterative properties of solution for a general singular n-Hessian equation with decreasing nonlinearity, Appl. Math. Lett., 112 (2021), 106826. https://doi.org/10.1016/j.aml.2020.106826 doi: 10.1016/j.aml.2020.106826
    [8] X. Zhang, J. Jiang, L. Liu, Y. Wu, Extremal solutions for a class of tempered fractional turbulent flow equations in a porous medium, Math. Probl. Eng., 2020 (2020), 2492193. https://doi.org/10.1155/2020/2492193
    [9] X. Zhang, L. Liu, Y. Wu, Y. Cui, A sufficient and necessary condition of existenceof blow-up radial solutions for a k-Hessian equationwith a nonlinear operator, Nonlinear Anal.-Model., 25 (2020), 126–143. https://doi.org/10.15388/namc.2020.25.15736 doi: 10.15388/namc.2020.25.15736
    [10] X. Zhang, J. Xu, J. Jiang, Y. Wu, Y. Cui, The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general k-Hessian equations, Appl. Math. Lett., 102 (2020), 106124. https://doi.org/10.1016/j.aml.2019.106124 doi: 10.1016/j.aml.2019.106124
    [11] J. Wu, X. Zhang, L. Liu, Y. Wu, Y. Cui, The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity, Bound. Value Probl. 2018 (2018), 82. https://doi.org/10.1186/s13661-018-1003-1
    [12] J. Ali, M. Jubair, F. Ali, Stability and convergence of F iterative scheme with an application to the fractional differential equation, Eng. Comput., 38 (2022), 693–702. https://doi.org/10.1007/s00366-020-01172-y doi: 10.1007/s00366-020-01172-y
    [13] M. Jubair, J. Ali, S. Kumar, Estimating fixed points via new iterative scheme with an application, J. Funct. Space., 2022 (2022), 3740809. https://doi.org/10.1155/2022/3740809 doi: 10.1155/2022/3740809
    [14] J. Ahmad, K. Ullah, H. A. Hammad, R. George, A solution of a fractional differential equation via novel fixed-point approaches in Banach spaces, AIMS Mathematics, 8 (2023), 12657–12670. https://doi.org/10.3934/math.2023636 doi: 10.3934/math.2023636
    [15] S. Khatoon, I. Uddin, D. Baleanu, Approximation of fixed point and its application to fractional to fractional differential equation, J. Appl. Math. Comput., 66 (2021), 507–525. http://doi.org/10.1007/s12190-020-01445-1 doi: 10.1007/s12190-020-01445-1
    [16] M. Kaur, S. Chandok, Convergence and stability of a novel M-iterative algorithm with application, Math. Probl. Eng., 2022 (2022), 9327527. https://doi.org/10.1155/2022/9327527
    [17] G. A. Okeke, A. E. Ofem, H. Isik, A faster iterative method for solving nonlinear third-order BVPs based on Green's function, Bound. Value Probl., 2022 (2022), 103. https://doi.org/10.1186/s13661-022-01686-y doi: 10.1186/s13661-022-01686-y
    [18] G. A. Okeke, A. E. Ofem, T. Abdeljawad, M. A. Alqudah, A. Khan, A solution of a nonlinear Volterra integral equation with delay via a faster iteration method, AIMS Mathematics, 8 (2023), 102–124. https://doi.org/10.3934/math.2023005
    [19] S. Panja, K. Roy, M. V. Paunovic, M. Saha, V. Parvaneh, Fixed points of weakly K-nonexpansive mappings and a stability result for fixed point iterative process with an application, J. Inequal. Appl., 2022 (2022), 90. https://doi.org/10.1186/s13660-022-02826-9 doi: 10.1186/s13660-022-02826-9
    [20] I. Uddin, C. Garodia, T. Abdelwajad, N. Mlaiki, Convergence analysis of a novel iteration process with application to a fractional differential equation, Adv. Cont. Discr. Mod., 2022 (2022), 16. https://doi.org/10.1186/s13662-022-03690-z
    [21] K. Ullah, S. T. M. Thabet, A. Kamal, J. Ahmad, Convergence analysis of an iteration process for a class of generalized nonexpansive mappings with application to fractional differential equations, Discrete Dyn. Nat. Soc., 2023 (2023), 8432560. https://doi.org/10.1155/2023/8432560 doi: 10.1155/2023/8432560
    [22] F. A. Khan, Approximating fixed points and the solution of a nonlinear fractional difference equation via an iterative method, J. Math., 2022 (2022), 6962430. https://doi.org/10.1155/2022/6962430 doi: 10.1155/2022/6962430
    [23] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023 doi: 10.1016/j.jmaa.2007.09.023
    [24] K. Goebel, W. A. Kirk, Topics in metric fixed theory, Cambridge: Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511526152
    [25] W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear Anal.-Hybri., 5 (2011), 583–590. https://doi.org/10.1016/j.nahs.2010.12.006 doi: 10.1016/j.nahs.2010.12.006
    [26] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597. https://doi.org/10.1090/S0002-9904-1967-11761-0 doi: 10.1090/S0002-9904-1967-11761-0
    [27] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operator, Fixed Point Theory Appl., 2004 (2004), 716359, http://doi.org/10.1155/s1687182004311058 doi: 10.1155/s1687182004311058
    [28] A. M. Harder, T. L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon, 33 (1988), 687–692.
    [29] V. Berinde, On the stability of some fixed procedure, Bul. Ştiinţ. Univ. Baia Mare, Ser. B, 18 (2002), 7–14.
    [30] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, B. Aust. Math. Soc., 43 (1991), 153–159. https://doi.org/10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
    [31] Ş. M. Şultuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory Appl., 2008 (2008), 242916. https://doi.org/10.1155/2008/242916 doi: 10.1155/2008/242916
    [32] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, P. Am. Math. Soc., 113 (1991), 727–731.
    [33] W. R. Mann, Mean value method in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
    [34] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013 (2013), 69. https://doi.org/10.1186/1687-1812-2013-69 doi: 10.1186/1687-1812-2013-69
    [35] B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147–155, http://doi.org/10.1016/j.amc.2015.11.065 doi: 10.1016/j.amc.2015.11.065
    [36] K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki's generalized nonexpansive mapping via new iteration process, Filomat, 32 (2018), 187–196. https://doi.org/10.2298/FIL1801187U doi: 10.2298/FIL1801187U
    [37] M. A. Noor, New approximation scheme for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
    [38] G. A. Okeke, Convergence of the Picard-Ishikawa hybrid iterative process with applications, Afr. Mat., 30 (2019), 817–835. https://doi.org/10.1007/s13370-019-00686-z doi: 10.1007/s13370-019-00686-z
    [39] M. A. Krasnosel'skii, Two observations about the method of successive approximations, Uspeh Mat. Nauk., 10 (1957), 131–140.
    [40] S. Ishikawa, Fixed points by a new iteration method, P. Am. Math. Soc., 44 (1974), 147–150.
    [41] G. A. Okeke, M. Abbas, A solution of delay differential equation via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math., 6 (2017), 21–29. https://doi.org/10.1007/s40065-017-0162-8 doi: 10.1007/s40065-017-0162-8
    [42] E. Ameer, H. Aydi, H. Işik, M. Nazam, V. Parvaneh, M. Arshad, Some existence results for a system of nonlinear fractional differential equations, J. Math., 2020 (2020), 4786053. https://doi.org/10.1155/2020/4786053 doi: 10.1155/2020/4786053
    [43] S. Kuma, A. Kumar, J. J. Nieto, B. Sharma, Atangana–Baleanu derivative with fractional order applied to the gas dynamics equations, In: Fractional derivatives with Mittag-Leffler Kernel, Springer, 2019,235–251. https://doi.org/10.1007/978-3-030-11662-0_14
    [44] N. A. Sheikh, F. Ali, M. Saqib, I. Khan, S. A. A. Jan, A. S. Alshomrani, et al., Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results phys., 7 (2017), 789–800. https://doi.org/10.1016/j.rinp.2017.01.025 doi: 10.1016/j.rinp.2017.01.025
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