Research article

Synchronization of Clifford-valued neural networks with leakage, time-varying, and infinite distributed delays on time scales

  • Received: 08 February 2024 Revised: 22 April 2024 Accepted: 06 May 2024 Published: 04 June 2024
  • MSC : 93C10, 93C43, 93D23

  • Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.

    Citation: Călin-Adrian Popa. Synchronization of Clifford-valued neural networks with leakage, time-varying, and infinite distributed delays on time scales[J]. AIMS Mathematics, 2024, 9(7): 18796-18823. doi: 10.3934/math.2024915

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  • Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.



    Lotfi A. Zadeh published a paper in 1965 [1] that developed the theory of fuzzy sets. Fuzzy sets theory was included in 2011 [2], in the study of complex-valued functions related to subordination. The connection of this theory with the field of complex analysis was motivated by the many successful attempts of researchers to connect fuzzy sets with established fields of mathematical study. The differential subordination concept was first presented by the writers of [3,4]. Fuzzy differential subordination was first proposed in 2012 [5]. A publication released in 2017 [6] provides a good overview of the evolution of the fuzzy set concept and its connections to many scientific and technical domains. It also includes references to the research done up until that moment in the context of fuzzy differential subordination theory. The research revealed in 2012 [7] showed how to adapt the well-established theory of differential subordination to the specific details that characterizes fuzzy differential subordination and provided techniques for analyzing the dominants as well as for providing the best dominants of fuzzy differential subordinations. Also, some researchers applied fuzzy differential subordination to different function classes; see [8,9]. After that, the specific Briot–Bouquet fuzzy differential subordinations were taken into consideration for the studies [10].

    Let Ap (p a positive integer) denote the class of functions of the form:

    f(η)=ηp+n=p+1anηn ,(pN={1,2,3,..}, ηU) (1.1)

    which are analytic and multivalent (or p-valent) in the open unit disk U given by

    U={η:|η|<1}.

    For p=1, the class Ap=A represents the class of normalized analytic and univalent functions in U.

    Jackson [11,12] was the first to employ the q-difference operator in the context of geometric function theory. Carmichael [13], Mason [14], Trijitzinsky [15], and Ismail et al. [16] presented for the first time some features connected to the q -difference operator. Moreover, many writers have studied different q-calculus applications for generalized subclasses of analytic functions; see [17,18,19,20,21,22,23,24,25,26].

    The Jackson's q-difference operator dq:ApAp defined by

    dq,pf(η):={f(η)f(qη)(1q)η(η0;0<q<1),f(0)(η=0), (1.2)

    provided that f(0) exists. From (1.1) and (1.2), we deduce that

    dq,pf(η):=[p]qηp1+κ=p+1[κ]qaκηκ1, (1.3)

    where

    [κ]q=1qκ1q=1+κ1n=1qn        [0]q=0,[κ]q!={[κ]q[κ1]q.........[2]q[1]q       κ=1,2,3,...   1                                                   κ=0.           (1.4)

    We observe that

    limq1dq,pf(η):=limq1f(η)f(qη)(1q)η=f(η).

    The q-difference operator is subject to the following basic laws:

    dq(cf(η)±d(η))=cdqf(η)±ddq(η) (1.5)
    dq(f(η)(η))=f(qη)dq((η))+(η)dq(f(η))  (1.6)
    dq(f(η)(η))=dq(f(η))(η)f(η)dq((η))(qη)(η) hbar(qη)(η)0 (1.7)
    dq(logf(η))=lnqq1dq(f(η))f(η) (1.8)

    where f,A, and c and d are real or complex constants.

    Jackson in [12] introduced the q-integral of f as:

    η0f(t)dqt =η(1q)κ=0qκf(ηqκ),

    and

    limq1η0f(t)dqt =η0f(t)dt, 

    where η0f(t)dt, is the ordinary integral.

    The study of linear operators is an important topic for research in the field of geometric function theory. Several prominent scholars have recently expressed interest in the introduction and analysis of such linear operators with regard to q-analogues. The Ruscheweyh derivative operator's q-analogue was examined by the writers of [27], who also examined some of its properties. It was Noor et al. [28] who originally introduced the q-Bernardi integral operator.

    In [29], Aouf and Madian investigate the q-p-valent Că tas operator Isq,p(λ,):ApAp (sN0=N{0},,λ0, 0<q<1,pN) as follows:

    Isq,p(λ,)f(η)=ηp+κ=p+1([p+]q+λ([κ+]q[p+]q)[p+]q)saκηκ(sN0,,λ0,0<q<1,pN).

    Also, Arif et al. [30] introduced the extended q -derivative operator μ+p1q:ApAp for p-valent analytic functions is defined as follows:

    μ+p1qf(η)=Qp(q,μ+p;η)f(η)     (μ>p),=ηp+κ=p+1[μ+p,q]κp[κp,q]!aκηκ.

    Setting

    Gs,pq,λ,(η)=ηp+κ=p+1([p+]q+λ([κ+]q[p+]q)[p+]q)sηκ.

    Now, we define a new function Gs,μq,p,λ,(η) in terms of the Hadamard product (or convolution) by:

    Gs,pq,λ,(η)Gs,μq,p,λ,(η)=ηp+κ=p+1[μ+p,q]κp[κp,q]!ηκ   (pN).

    Then, motivated essentially by the q-analogue of the Ruscheweyh operator and the q-analogue Cătas operator, we now introduce the operator Is,pq,μ(λ,):ApAp defined by

    Is,pq,μ(λ,)f(η)=Gs,μq,p,λ,(η)f(η), (1.9)

    where sN0,,λ0,μ>p,0<q<1,pN. For fAp and (1.9), it is clear that

    Is,pq,μ(λ,)f(η)=ηp+κ=p+1([p+]q[p+]q+λ([κ+]q[p+]q))s[μ+p]κp,q[κp]q!aκηκ. (1.10)

    We use (1.10) to deduce the following:

    η dq(Is+1,pq,μ(λ,)f(η))=[+p]qλqIs,pq,μ(λ,)f(η)([+p]qλq1)Is+1,pq,μ(λ,)f(η),(λ>0), (1.11)
    qμη dq(Is,pq,μ(λ,)f(η))=[μ+p]qIs,pq,μ+1(λ,)f(η)[μ]qIs,pq,μ(λ,)f(η). (1.12)

    We note that :

    (i) If s=0 and q1 the operator defined in (1.10) reduces to the differential operator investigated by Goel and Sohi [31], and further, by making p=1, we get the familiar Ruscheweyh operator [32] (see also [33]). Also, for more details on the q-analogue of different differential operators, see the works [34,35];

    (ii) If we set q1,p=1, we obtain Isλ,,μf(η) that was defined by Aouf and El-Ashwah [36];

    (iii) If we set μ=0, and q1, we obtain Jsp(λ,)f(η) that was introduced by El-Ashwah and Aouf [37];

    (iv) If μ=0, =λ=1,p=1, and q1, we obtain Isf(η) that was investigated by Jung et al. [38];

    (v) If μ=0, λ=1,=0,p=1, and q1, we obtain Isf(η) that was defined by S ălăgean [39];

    (vi) If we set μ=0, λ=1, and p=1, we obtain Iq,sf(η) that was presented by Shah and Noor [40];

    (vii) If we set μ=0,λ=1,p=1, and q1, we obtain Jsq,, the Srivastava–Attiya operator; see [41,42];

    (vii) I1,1q,0(1,0)=η0f(t)tdqt (q-Alexander operator [40]);

    (viii) I1,1q,0(1,)=[1+ϱ]qηϱη0tϱ1f(t)dqt (q-Bernardi operator [28]);

    (ix) I1,1q,0(1,1)=[2]qηη0f(t)dqt (q-Libera operator [28]).

    We also observe that:

    (i)Is,pq,μ(1,0)f(η)=Is,pq,μf(η)

    f(η)A:Is,pq,μf(η)=ηp+κ=p+1([p]q[κ]q)s[μ+p]κp,q[κp]q!aκηκ, (sN0,μ0,0<q<1,pN,ηU).

    (ii) Is,pq,μ(1,)f(η)=Is,p,q,μf(η)

    f(η)A:Is,p,q,μf(η)=ηp+κ=p+1([p+]q[κ+]q)s[μ+p]κp,q[κp]q!aκηκ, (sN0,>0,μ0,0<q<1,pN,ηU).

    (iii) Is,pq,μ(λ,0)f(η)=Is,p,λq,μf(η)

    f(η)A:Is,p,λq,μf(η)=ηp+κ=p+1([p]q[p]q+λ([κ]q[p]q))s[μ+p]κp,q[κp]q!aκηκ, (sN0,λ>0,μ0,0<q<1,pN,ηU).

    We provide an overview of a number of fundamental ideas that are important to our research.

    Definition 2.1. [43] A mapping F is said to be a fuzzy subset on Yϕ, if it maps from Y to [0,1].

    Alternatively, it is defined as:

    Definition 2.2. [43] A pair (U,FU) is said to be a fuzzy subset on Y, where FU:Y[0,1] is the membership function of the fuzzy set (U,FU) and U ={xY:0<FU(x)1}=sup(U,FU) is the support of fuzzy set (U,FU).

    Definition 2.3. [43] Let (U1,FU1) and (U2,FU2) be two subsets of Y. Then, (U1,FU1)(U2,FU2) if and only if FU1(t)FU2(t), tY, whereas (U1,FU1) and (U2,FU2) of Y are equal if and only if U1=U2.

    The subordination method for two analytic functions f and h was established by Miller and Mocanu [44]. Specifically, if f(η)=h(κ(η)), where κ(η) is a Schwartz function in U, then, f is subordinate to h, symbolized by fh.

    According to Oros [5], the subordination technique of analytic functions can be generalized to fuzzy notions as follows:

    Definition 2.4. If f(η0)=h(η0) and F(f(η))F(h(η)),(ηUC), where η0U be a fixed point, then f is fuzzy subordinate to h and is denoted by fFh.

    Definition 2.5. [5] Let ψ:C3×UC, and let h be univalent in U If ω is analytic in U and satisfies the (second-order) fuzzy differential subordination:

    Fψ(C3×U)(ψ(ω(η),ηω(η),η2ω(η);η))Fh(U)(h(η)), (2.1)

    i.e.,

    ψ(ω(η),ηω(η),η2ω(η);η))F(h(ζ)),  ηU,

    then, ω is called a fuzzy solution of the fuzzy differential subordination. The univalent function ω is called a fuzzy dominant if ω(η)Fχ(η), for all ω satisfying (2.1). A fuzzy dominant ˜χ that satisfies ˜χ(η)Fχ(η) for all fuzzy dominant χ of (2.1) is said to be the fuzzy best dominant of (2.1).

    Using the concept of fuzzy subordination, certain special classes are next defined.

    The class of analytic functions h(η) that are univalent convex functions in U with h(0)=1 and Re(h(η))>0 is denoted by Ω. We define the following for h(η)Ω, F:C[0,1], sN0,,λ0,μ>p,0<q<1, and pN:

    Definition 2.6. When fAp, we say that fFMpγ(h) if and only if

    (1γ)[p]qηdqf(η)f(η)+γ[p]qdq(ηdqf(η))dqf(η)Fh(η).

    Furthermore,

    FMp0(h)=FSTp(h)={fAp:ηdqf(η)[p]qf(η)Fh(η)},

    and

    FMp1(h)=FCVp(h)={fAp:dq(ηdqf(η))[p]qdqf(η)Fh(η)}.

    It is noted that

    fFCVp(h)ηdqf(η)[p]qFSTp(h). (2.2)

    Particularly, for h(η)=1+η1η, the classes FCVp(h) and FSTp(h) reduce to the classes FCVp, and FSTp, of the fuzzy p-valent convex and fuzzy p-valent starlike functions, respectively.

    With the operator Is,pq,μ(λ,), specified by (1.10), certain new classes of fuzzy p-valent functions are defined as follows:

    Definition 2.7. Let fAp,,λ0,μ>p,0<q<1, pN and s be a real. Then,

    FMs,pq,μ(γ,λ,;h)={fAp:Is,pq,μ(λ,)f(η)FMpγ(h)}
    FSTs,pq,μ(γ,λ,;h)={fAp:Is,pq,μ(λ,)f(η)FSTp(h)}

    and

    FCVs,pq,μ(γ,λ,;h)={fAp:Is,pq,μ(λ,)f(η)FCVp(h)}.

    It is clear that

    fFCVs,pq,μ(γ,λ,;h)ηdqf(η)[p]qFSTs,pq,μ(γ,λ,;h). (2.3)

    Particularly, if s=0,μ=1, then FMs,pq,μ(γ,λ,;h)=FMpγ(h), FSTs,pq,μ(γ,λ,;h)=FSTp(h), and FCVs,pq,μ(γ,λ,;h)= FCVp(h). Moreover, if p=1, then the classes FMpγ(h), FSTp(h), and FCVp(h) reduce to the classes FMγ(h), FST(h), and FCV(h) studied in [45].

    In the first part of this investigation, the goal is to establish certain inclusion relations between the classes seen in Definitions 5 and 7 using the properties of fuzzy differential subordination and then to obtain connections between the newly introduced subclasses by applying a new generalized q-calculus operator, which will be defined in the second part of this study. This research follows the line established by recent publications like [46,47,48].

    The proofs of the main results require the following lemma:

    Lemma 3.1. [47] Let r1, r2C, r10, and a convex function h satisfies

    Re(r1h(t)+r2)>0tU.

    If g is analytic in U with g(0)=h(0), and Ω(g(t),tdqg(t);t)=g(t)+tdqg(t)r1g(t)+r2 is analytic in U with Ω(h(0),0;0)=h(0), then,

    FΩ(C2×U)[g(t)+tdqg(t)r1g(t)+r2]Fh(U)(h(t))

    implies

    Fg(U)(g(t))Fh(U)(h(t)), tU.

    In this section, we are going to discuss some inclusion properties for the classes defined above.

    Theorem 3.1. Let hΩ, 0γ1,,λ0,μ>p,0<q<1,pN, and s be a real. Then,

    FMs,pq,μ(γ,λ,;h)FSTs,pq,μ(λ,;h).

    Proof. Let fFMs,pq,μ(γ,λ,;h), and let

    ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η)=χ(η), (3.1)

    with χ(η) being analytic in U and χ(0)=1.

    We take logarithmic differentiation of (2.1) to get

    dq(ηdq(Is,pq,μ(λ,)f(η)))ηdqIs,pq,μ(λ,)f(η)dq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)=dq(χ(η))χ(η).

    Equivalently,

    dq(ηdq(Is,pq,μ(λ,)f(η)))[p]qdqIs,pq,μ(λ,)f(η)=χ(η)+1[p]qηdq(χ(η))χ(η). (3.2)

    Since fFMs,pq,μ(γ,λ,;h), from (2.1) and (3.2), we get

    (1γ)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)=χ(η)+γ[p]qηdq(χ(η))χ(η)Fh(η). (3.3)

    We obtain χ(η)Fh(η) by applying (3.3) and Lemma 3.1. Hence, fFSTs,pq,μ(λ,;h).

    Corollary 3.1. FMsq,μ(λ,;h)FSTsq,μ(λ,;h),if p=1. Furthermore, if s=0, μ=1, we obtain FMγ(h)FST(h), and if γ=1, then FCV(h)FST(h). Additionally, for h(η)=1+η1η, we obtain FCVFST.

    Theorem 3.2. Let hΩ, γ>1,,λ0,μ>p,0<q<1, pN, and s be a real. Then,

    FMs,pq,μ(γ,λ,;h)FCVs,pq,μ(γ,λ,;h).

    Proof. Let fFMs,pq,μ(γ,λ,;h). Then, by definition, we write

    (1γ)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)=p1(η)Fh(η).

    Now,

    γ[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)=(1γ)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)+(γ1)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)=(γ1)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+p1(η).

    This implies

    dq(ηdq(Is,pq,μ(λ,)f(η)))[p]qdqIs,pq,μ(λ,)f(η)=1γp1(η)+(11γ)ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η)=1γp1(η)+(11γ)p2(η).

    Since p1,p2Fh(η), dq(ηdq(Is,pq,μ(λ,)f(η)))[p]qηdqIs,pq,μ(λ,)f(η)Fh(η). This is the expected outcome.

    In particular, if p=1, we get FMsq,μ(γ,λ,;h)FCVsq,μ(γ,λ,;h). Additionally, when s=0, μ=1, we have FMγ(h)FCV(h), and considering h(η)=1+η1η, we obtain FMpγFCV.

    Theorem 3.3. Let hΩ, 0γ1<γ2<1,,λ0,μ>p,0<q<1, pN, and s be a real number. Then,

    FMs,pq,μ(γ2,λ,;h)FMs,pq,μ(γ1,λ,;h).

    Proof. For γ1=0, it is obviously true, based on the preceding theorem.

    Let fFMs,pq,μ(γ2,λ,;h). Then, by definition, we have

    (1γ2)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ2[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))ηdqIs,pq,μ(λ,)f(η)=g1(η)Fh(η). (3.4)

    Now, we can easily write

    (1γ1)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ1[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)=γ1γ2g1(η)+(1γ1γ2)g2(η), (3.5)

    where we have used (3.4), and ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η)=g2(η)Fh(η). Since g1,g2Fh(η), (3.5) implies

    (1γ1)[p]qηdq(Is,pq,μ(λ,)f(η))Is,pq,μ(λ,)f(η)+γ1[p]qdq(ηdq(Is,pq,μ(λ,)f(η)))dqIs,pq,μ(λ,)f(η)Fh(η).

    This proves the theorem.

    Remark 3.1. If γ2=1, and fFMs,pq,μ(1,λ,;h)=FCVs,pq,μ(λ,;h), then the previously proved result shows that

    fFMs,pq,μ(γ1,λ,;h), for  0γ1<1.

    Consequently, by using Theorem 3.1, we get FCVs,pq,μ(λ,;h)FSTs,pq,μ(1,λ,;h).

    Now, certain inclusion results are discussed for the subclasses given by Definition 2.1.

    Theorem 3.4. Let hΩ, ,λ0,μ>p,0<q<1, pN, and s be a real with [+p]q>λq. Then,

    FSTs,pq,μ+1(λ,;h)FSTs,pq,μ(λ,;h)FSTs+1,pq,μ(λ,;h).

    Proof. Let fFSTs,pq,μ(λ,;h). Then,

    ηdq(Is+1,pq,μ(λ,)f(η))[p]qIs+1,pq,μ(λ,)f(η)Fh(η).

    Now, we set

    ηdq(Is+1,pq,μ(λ,)f(η))[p]qIs+1,pq,μ(λ,)f(η)=P(η) (3.6)

    with analytic P(η) in U and P(0)=1.

    From (1.11) and (3.6), we get

    ηdq(Is+1,pq,μ(λ,)f(η))[p]qIs+1,pq,μ(λ,)f(η)=[+p]qλq(Is,pq,μ(λ,)f(η))[p]qIs+1,pq,μ(λ,)f(η)1[p]q([+p]qλq1),

    equivalently,

    [+p]qλq(Is,pq,μ(λ,)f(η))[p]qIs+1,pq,μ(λ,)f(η)=P(η)+ξq,p.

    where ξq,p=1[p]q([+p]qλq1).

    On q-logarithmic differentiation yields,

    ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η)=P(η)+ηdq(P(η))[p]q(P(η)+ξq,p). (3.7)

    Since fFSTs,pq,μ(λ,;h), (3.7) implies

    P(η)+ηdq(P(η))[p]q(P(η)+ξq,p)Fh(η). (3.8)

    We conclude that P(η)Fh(η) by applying (3.8) and Lemma 3.1. Hence, fFSTs+1,pq,μ(λ,;h). To prove the first part, let fFSTs,pq,μ+1(λ,;h), and set

    χ(η)=ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η),

    where χ is analytic in U with χ(0)=1. Then, it follows χFh(η) that by applying the same arguments as those described before with (1.12). Theorem 3.4's proof is now complete.

    Theorem 3.5. Let hΩ, ,λ0,μ>p,0<q<1, pN, and s be a real. Then,

    FCVs,pq,μ+1(λ,;h)FCVs,pq,μ(λ,;h)FCVs+1,pq,μ(λ,;h).

    Proof. Let fFCVs,pq,μ(λ,;h). Applying (2.3), we show that

    fFCVs,pq,μ(λ,;h)η(dqf)[p]qFSTs,pq,μ(λ,;h)η[p]qdq(Is,pq,μ(λ,)f(η))FSTp(h)η(dqf)[p]qFSTs+1,pq,μ(λ,;h)η[p]qdq(Is+1,pq,μ(λ,)f(η))FSTp(h)Is+1,pq,μ(λ,)(η(dqf)[p]q)FSTp(h)Is+1,pq,μ(λ,)f(η)FCVp(h)fFCVs+1,pq,μ(λ,;h).

    We can demonstrate the first part using arguments similar to those described above. Theorem 3.5's proof is now complete.

    For f(η)Ap, the generalized (p,q)-Bernardi integral operator for p-valent functions Bpn,qf(η):ApAp is defined by

    Bpn,qf(η)={Bp1,q(Bpn1,qf(η)),          (nN),f(η),                            (n=0),

    where Bp1,qf(η) is given by

    Bp1,qf(η)=[p+ϱ]qηϱη0tϱ1f(t)dqt=ηp+κ=p+1([p+ϱ]q[κ+ϱ]q)aκzκ,     (ϱ>p,ηU). (3.9)

    From Bp1,qf(η), we deduce that

    Bp2,qf(η)=Bp1,q(Bp1,qf(η))=ηp+κ=p+1([p+ϱ]q[κ+ϱ]q)2aκzκ,     (ϱ>p),

    and

    Bpn,qf(η)=ηp+κ=p+1([p+ϱ]q[κ+ϱ]q)naκzκ,     (nN, ϱ>p),

    which are defined in [49].

    If n=1, we obtain the q-Bernardi integral operator for a p -valent function [50].

    Theorem 3.6. Let fFMs,pq,μ(γ,λ,;h), and define

    Bpq,ϱ(η)=[p+ϱ]qηϱη0tϱ1f(t)dqt    (ϱ>0). (3.10)

    Then, Bpq,ϱFSTs,pq,μ(λ,;h).

    Proof. Let fMs,pq,μ(γ,λ,;h), and Bs,pq,μ,ϱ(λ,)(η)=Is,pq,μ(λ,)(Bpq,ϱ(η)). We assume

    ηdq(Bs,pq,μ,ϱ(λ,)(η))[p]qBs,pq,μ,ϱ(λ,)(η)=N(η), (3.11)

    where N(η) is analytic in U with N(0)=1.

    From (3.10), we obtain

    dq(ηϱBs,pq,μ,ϱ(λ,)(η))[p+ϱ]q=ηϱ1f(η).

    This implies

    ηdq(Bs,pq,μ,ϱ(λ,)(η))=([p]q+[ϱ]qqϱ)f(η)[ϱ]qqϱBs,pq,μ,ϱ(λ,)(η). (3.12)

    We use (3.11), (3.12), and (1.10), to obtain

    N(η)=([p]q+[ϱ]qqϱ)ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)(Bpq,ϱ(η))[ϱ]qqϱ[p]q.

    We use logarithmic differentiation to obtain

    ηdq(Is,pq,μ(λ,)f(η))[p]qIs,pq,μ(λ,)f(η)=N(η)+ηdqN(η)[p]qN(η)+[ϱ]qqϱ. (3.13)

    Since fFMs,pq,μ(γ,λ,;h)FSTs,pq,μ(λ,;h), (3.13) implies

    N(η)+ηdqN(η)[p]qN(η)+[ϱ]qqϱFh(η).

    The intended outcome follows from applying Lemma 3.1.

    When p=1, the following corollary can be stated:

    Corollary 3.2. Let fFMsq,μ(γ,λ,;h), and define

    Bq,ϱf(η)=[1+ϱ]qηϱη0tϱ1f(t)dqt  (ϱ>0).

    Then, Bq,ϱf(η)FSTsq,μ(λ,;h).

    The means of the fuzzy differential subordiantion theory are employed in order to introduce and initiate investigations on certain subclasses of multivalent functions. The q-p-analogue multiplier-Ruscheweyh operator Is,pq,μ(λ,) is developed using the notion of a q-difference operator and the concept of convolution. The q-analogue of the Ruscheweyh operator and the q-p-analogue of the Cătas operator are further used to introduce a new operator applied for defining particular subclasses. In the second section, we obtained some inclusion properties between the classes FMs,pq,μγ,λ,;h), FSTs,pq,μ(λ,;h), and FCVs,pq,μ(λ,;h). The investigations concern the (p,q)-Bernardi integral operator for the p-valent function preservation property and certain inclusion outcomes for the newly defined classes. Another new generalized q-calculus operator is defined in this investigation that helps establish connections between the classes introduced and investigated in this study. For instance, many researchers used fuzzy theory in different branches of mathematics [51,52,53,54].

    This work is intended to motivate future studies that would contribute to this direction of study by developing other generalized subclasses of q-close-to-convex and quasi-convex multivalent functions as well as by presenting other generalized q-calculus operators.

    Ekram E. Ali1, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi: conceptualization, methodology, validation, formal analysis, formal analysis, writing-review and editing; Ekram E. Ali1, Rabha M. El-Ashwah and Abeer M. Albalahi: writing-original draft preparation; Ekram E. Ali1 and Georgia Irina Oros: supervision; Ekram E. Ali1: project administration; Georgia Irina Oros: funding acquisition. All authors have read and approved the final version of the manuscript for publication.

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

    The publication of this paper was supported by University of Oradea, Romania.

    The authors declare that they have no conflicts of interest.



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