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Novel generalized inequalities involving a general Hardy operator with multiple variables and general kernels on time scales

  • This paper introduced novel multidimensional Hardy-type inequalities with general kernels on time scales, extending existing results in the literature. We established generalized inequalities involving a general Hardy operator with multiple variables and kernels on arbitrary time scales. Our findings not only encompassed known results in the realm of real numbers (T=R), but also provided refinements and generalizations thereof. The proposed inequalities offered versatile applications in mathematical analysis and beyond, contributing to the ongoing exploration of inequalities on diverse time scales.

    Citation: M. Zakarya, Ghada AlNemer, A. I. Saied, H. M. Rezk. Novel generalized inequalities involving a general Hardy operator with multiple variables and general kernels on time scales[J]. AIMS Mathematics, 2024, 9(8): 21414-21432. doi: 10.3934/math.20241040

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  • This paper introduced novel multidimensional Hardy-type inequalities with general kernels on time scales, extending existing results in the literature. We established generalized inequalities involving a general Hardy operator with multiple variables and kernels on arbitrary time scales. Our findings not only encompassed known results in the realm of real numbers (T=R), but also provided refinements and generalizations thereof. The proposed inequalities offered versatile applications in mathematical analysis and beyond, contributing to the ongoing exploration of inequalities on diverse time scales.


    Fractional differential inclusions (FDIs), as an extension of fractional differential equations (FDEs), have gained popularity among mathematical researchers due to their importance and value in optimization and stochastic processes in economics [1,2] and finance [3]. In addition to their applications in understanding engineering [4] and dynamic systems [5,6] in biological [7,8], medical [9], physics [10] and chemical sciences [11], FDIs are also relevant in various other scientific fields [12].

    Sousa and Oliveira [13] introduced the new fractional derivative ς-Hilfer to unify different types of fractional derivatives into a single operator, expanding fractional derivatives to the types of operators with potentially applicable value. After that, Asawasamrit et al. [14] investigated the following Hilfer-FDE under nonlocal integral boundary conditions (BCs):

    {HDr1,r2ϰ(υ)=(υ,ϰ(υ)),1<r1<2,0r21,υ:=[˜s,˜b],ϰ(˜s)=0,ϰ(˜b)=mi=1δiRLIςiϰ(˜ξi),φi>0,δiR,˜ξi. (1.1)

    In [15], an existence outcome was shown by employing the FPTs (fixed-point theorems) for a sequential FDE of the type,

    {(HDr1,r2,ς(CDr3,ςϰ))(s)=(s,ϰ(s),RLIr5,ςϰ(υ),˜b0ϰ(v)dv),υB:=[0,˜b],ϰ(0)+η1ϰ(˜b)=0,CDr4+r31,ςϰ(0)+ηC2Dδ+r31,ςϰ(˜b)=0,

    where ri(0,1), i=1,2,3, r4=r1+r2(1r1), r4+r3>1, η1,η2R, r5>0, and C(B×R3) is a nonlinear function. In 2024, Ahmed et al. [16] investigated a class of separated boundary value problems of the form

    {(HDr1,r2q(CDr3qϰ))(υ)=(υ,ϰ(υ)),q(0,1),υB,ϰ(0)+λC1Dr3+r41qϰ(0)=ϰ(˜b)+λC2Dr3+r41qϰ(˜b)=0,λ1,λ2R,

    where 0<r1,r3<1, r2[0,1] with r4=r1+r2(1r1), r3+r4>1 and C(B×R). Lachouri et al., in [17], established the existence of solutions to the nonlinear neutral FDI involving ς-Caputo fractional derivative with ς-Riemann–Liouville (RL) fractional integral boundary conditions:

    {CDr1,ς(CDr2,ςϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),υ[0,˜b),ϰ(a)=RLIr3,ςϰ(˜b)=0,a(0,˜b),

    where :B×RP(R) is a set-valued map. Surang et al., in [18], studied the ς-Hilfer type sequential FDEs and FDIs subject to integral multi-point BCs of the form

    {(HDr1,r2,ς+kHDr11,r2,ς)ϰ(υ)=(υ,ϰ(υ)),υ,ϰ(˜s)=0,ϰ(˜b)=ni=1μiηi˜sψ(s)ϰ(s)ds+mj=1θjϰ(ξj), (1.2)

    where r1(1,2), r2[0,1], C(×R), k,μi,θjR, and ηi,ξj(˜s,˜b], i=¯1,n, j=¯1,m. Etemad et al. [19] introduced and studied a novel existence technique based on some special set-valued maps (SVMs) to guarantee the existence of a solution for the following fractional jerk inclusion problem involving the derivative operator in the sense of Caputo–Hadamard

    {(CHDr11+(CHDr21+(CHDr31+ϰ)))(υ)(υ,ϰ(υ),CHDr31+ϰ(υ),(CHDr21+(CHDr31+ϰ))(υ)),ϰ(1)+ϰ(exp(1))=CHDr31+ϰ(η)=(CHDr21+(CHDr31+ϰ))(exp(1))=0,

    for υ[1,exp(1)], where ri(0,1], i=1,2,3, η(1,exp(1)), and the operator :[1,exp(1)]×R3P(R) is an SVM, where P(R) denotes all nonempty subsets of R.

    The boundary conditions (BCs) used in (1.1) and (1.2), share a common feature: the requirement of a zero initial condition, which is essential for the solution to be well-defined. Consequently, certain classes of Hilfer FDEs cannot be addressed, including cases with BCs such as,

    ϰ(0)=ϰ(˜b), ϰ(0)=ϰ(˜b) (anti-periodic),

    ϰ(0)+η1ϰ(0)=0, ϰ(˜b)+η2ϰ(˜b)=0 (separated),

    ϰ(0)+η1ϰ(˜b)=0, ϰ(0)+η2ϰ(˜b)=0 (non-separated), etc.

    To address this limitation and study Hilfer FDEs with such BCs, regardless of whether they are anti-periodic, separated, or non-separated, we propose a novel approach in this research. Specifically, we combine the Hilfer and Caputo fractional derivatives, enabling the study of boundary value problems under these conditions. More specifically, we aimed to analyze a class of FDEs for FDI, subject to non-separated BCs of the form,

    {HDr1,r2,ς(CDr3,ςϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),υB,ϰ(0)+η1ϰ(˜b)=0,CDδ+r31,ςϰ(0)+ηC2Dδ+r31,ςϰ(˜b)=0, (1.3)

    where ri(0,1), i=1,2,3, δ=r1+r2(1r1), δ+r3>1, η1,η2R, yC(B×R) and :B×RP(R) denotes a SVM, with power set P(R) of R.

    The paper is structured as follows. Section 2 is devoted to discussing the fundamental concepts fractional calculus and set-valued analysis, while Section 3 presents important findings on the qualitative properties of solutions to the ς-Hilfer inclusion FDI (1.3) utilizing FPTs. Finally, Section 4, includes three illustrative examples.

    We outline the background material that is pertinent to our study. We consider the Banach spaces E=C(B) and L1(B) of the Lebesgue integrable functions equipped with the norms ϰ=sup{|ϰ(υ)|:υB} and

    ϰL1=B|ϰ(υ)|dυ,

    respectively. Let ςCn(B) be an increasing function such that ς(υ)0, for any υB.

    Definition 2.1 ([20]). The ς-RL fractional integral and derivative of order r1 for a given function ϰ are expressed by

    RLIr1,ςϰ(υ)=υ0ς(u)Γ(r1)(ςu(υ))r11ϰ(u)du,ςu(υ):=ς(υ)ς(u),

    and

    RLDr1,ςϰ(υ)=D[n]ςRLInr1,ςϰ(υ),D[n]ς:=(1ς(υ)ddυ)n,

    where n=[r1]+1, nN, respectively.

    Definition 2.2 ([21,22]). The Caputo sense of ς-fractional derivative of the ϰCn(B) of order r1 is given as,

    CDr1,ςϰ(s)=RLI(nr1),ςϰ[n](s),ϰ[n](s)=(1ς(s)dds)nϰ(s).

    Lemma 2.3 ([20,22]). Let r1,r2>0. Then

    i)RLIr1,ς(ς0(υ))r21=Γ(r2)Γ(r1+r2)(ς0(υ))r1+r21,ii)CDr1,ς(ς0(υ))r21=Γ(r2)Γ(r2r1)(ς0(υ))r1+r21.

    Lemma 2.4 ([20]). For ϰCn(B), we have

    RLIr1,ςCDr1,ςϰ(s)=ϰ(s)n1k=0ϰ[n](0+)k!(ς0(s))k,n1<r1<n,

    and 0<r2<1. Furthermore, if r1(0,1), then RLIr1,ςCDr1,ς ϰ(υ)=ς0(υ).

    Definition 2.5 ([13]). The ς-Hilfer fractional derivative for ϰCn(B), of order n1<r1<n and type 0r21, is defined by

    HDr1,r2,ςϰ(υ)=(RLIr2(nr1),ς(D[n]ς(RLI(1r2)(nr1),ςϰ)))(υ).

    Lemma 2.6 ([13,23]). Let r1,r2,μ>0. Then

    i)RLIr1,ςRLIr2,ςϰ(υ)=RLIr1+r2,ςϰ(υ),ii)RLIr1,ς(ς0(υ))μ1=Γ(μ)Γ(r1+μ)(ς0(υ))r1+μ1.

    Lemma 2.7 ([13]). For μ>0, r1(n1,n), and 0r21,

    HDr1,r2,ς(ς0(υ))μ1=Γ(μ)Γ(μr1)(ς0(υ))μr11,μ>n.

    In particular, if r1(1,2) and 1<μ2, then HDr1,r2,ς(ς0(υ))μ1=0.

    Lemma 2.8 ([13]). If ϰCn(B), n1<r1<n and type 0<r2<1, then

    i)RLIr1,ςHDr1,r2,ςϰ(υ)=ϰ(υ)nk=1(ς0(υ))δkΓ(δk+1)D[nk]ςRLI(1r2)(nr1),ςϰ(0),ii)HDr1,r2,ςRLIr1,ςϰ(υ)=ϰ(υ).

    Consider the Banach space (E,) and SVM Θ:EP(E). Θ is a) closed (convex), b) bounded and c) measurable, whenever Θ(ϰ) is closed (convex) for every ϰE, Θ(B)=ϰBΘ(ϰ) is bounded for any bounded set BE, that is

    supϰB{sup|ρ|:ρΘ(ϰ)}<,

    and ρR, the function

    υd(ρ,Θ(υ))=inf{|ρλ|:λΘ(υ)},

    is measurable, respectively. One can find the definitions of completely continuous and upper semi-continuous in [24]. Additionally, the set of selections of is described as

    R,ρ={σL1(B):σ(υ)(υ,ρ),υB}.

    Next, we take

    Pβ(E)={ΩP(E):Ω with has a property β},

    where Pcl, Pc, Pb, and Pcp represent the classes of every compact, bounded, closed, and convex subset of E, respectively.

    Definition 2.9 ([25]). An SVM :B×RP(R) is called Carathéodory if the mapping υ(υ,ϰ) is measurable for all ϰR, and ϰ(υ,ϰ) is upper semicontinuous for almost every υB. Additionally, we say is L1-Carathéodory whenever for all m>0, exists zL1(B,R+) such that for a.e. υB,

    (υ,ϰ)=sup{|σ|:σ(υ,ϰ)}z(υ),zm.

    To achieve the intended outcomes in this search, the following lemmas are necessary.

    Lemma 2.10 ([25], Proposition 1.2). Consider SVM Θ:EPcl(Z) with the graph, Gr(Θ)={(ϰ,ρ)E×Z:ρΘ(ϰ)}. Gr(Θ) is a closed subset of E×Z whenever Θ is upper semi-continuous. Conversely, Θ is upper semi-continuous, when it has a closed graph and is completely continuous.

    Lemma 2.11 ([26]). Consider a separable Banach space E along with a L1-Carathéodory SVM :B×RPcp,c(E) and a linear continuous map Υ:L1(B,E)C(B,E). Then, the composition

    {ΥR:C(B,E)Pcp,c(C(B,E)),ϰ(ΥR)(ϰ)=Υ(R,ϰ),

    is a closed graph map in C(B,E)×C(B,E).

    In relation to the FDI (1.3), the auxiliary Lemma 3.1 is required.

    Lemma 3.1. For y,C(B), the solution of linear-type problem

    {HDr1,r2,ς(CDr3,ςϰ(υ)y(υ))=(υ),υB{˜b},ϰ(0)+η1ϰ(˜b)=0,CDδ+r31,ςϰ(0)+ηC2Dδ+r31,ςϰ(˜b)=0, (3.1)

    is obtained as follows:

    ϰ(υ)=RLIr3,ςy(υ)+RLIr1+r3,ς(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b)+RLIr1δ+1,ς(˜b))Λ2(RLIr3,ςy(˜b)+RLIr1+r3,ς(˜b)), (3.2)

    where for η1,η21,

    Λ1=η2(η2+1)Γ(r3+δ),Λ2=η1η1+1,Λ3=η1η2(η2+1)Γ(r3+δ). (3.3)

    Proof. Applying the ς-fractional integral RLIr1,ς to the first equation of (1.3), and using Lemma 2.4, we get

    CDr3,ςϰ(υ)=y(υ)+RLIr1,ς(υ)+c1(ς0(υ))δ1,υB,c1R, (3.4)

    where δ=r1+r2(1r1). Now, by taking RLIr3,ς in (3.4) from Lemma 2.3, we get

    ϰ(υ)=RLIr3,ςy(υ)+RLIr1+r3,ς(υ)+c1Γ(δ)Γ(r3+δ)(ς0(υ))r3+δ1+c2,c2R. (3.5)

    According to Lemma 2.3, we can obtain

    CDδ+r31,ςϰ(υ)=RLI1δ,ςy(υ)+RLIr1δ+1,ς(υ)+c1Γ(δ). (3.6)

    Next, by combining the BCs ϰ(0)+η1ϰ(˜b)=0 and

    CDδ+r31,ςϰ(0)+ηC2Dδ+r31,ςϰ(˜b)=0

    with (3.6), we get

    c2(1+η1)+ηRL1Ir3,ςy(˜b)+ηRL1Ir1+r3,ς(υ)+c1η1Γ(δ)Γ(r3+δ)(ς0(˜b))r3+δ1=0, (3.7)
    c1(1+η2)Γ(δ)+ηRL2I1δ,ςy(˜b)+ηRL2Ir1δ+1,ς(˜b)=0. (3.8)

    From (3.7) and (3.8), we find

    c1=η2(1+η2)Γ(δ)(RLI1δ,ςy(˜b)+RLIr1δ+1,ς(˜b)),c2=η1η2(1+η2)Γ(r3+δ)(ς0(υ))r3+δ1(RLI1δ,ςy(ς0(υ))+RLIr1δ+1,ς(˜b))η1(1+η1)(RLIr3,ςy(˜b)+RLIr1+r3,ς(υ)).

    By substituting the values of c1 and c2 into (3.5), we arrive at the fractional integral equation (3.2).

    Definition 3.2. An element ϰC1(B) can be a solution of (1.3), if there is σL1(B) with σ(υ)(υ,ϰ) for every υB fulfilling the non-separated BC's, ϰ(0)+η1ϰ(˜b)=0,

    CDδ+r31,ζϰ(0)+ηC2Dδ+r31,ζϰ(˜b)=0,

    and

    ϰ(υ)=RLIr3,ςy(υ,ϰ(υ))+RLIr1+r3,ςσ(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ(˜b))+RLIr1δ+1,ςσ(˜b))Λ2(RLIr3,ςy(˜b,ϰ(˜b))+RLIr1+r3,ςσ(˜b)). (3.9)

    The first consequence addresses the convex-valued using the nonlinear alternative for contractive maps [27].

    Theorem 3.3. Suppose that

    P1) :B×RPcp,c(R) is a L1-Carathéodory SVM;

    P2) There is a exist ˜ϖ1C(B,R+) and a nondecreasing ˜ϖ2C(R+,R+) with,

    (υ,ϰ)P=sup{|ρ|:ρ(υ,ϰ)}˜ϖ1(υ)˜ϖ2(ϰ),(υ,ϰ)B×R;

    P3) There is a constant ly<λ12 such that |y(υ,ϰ1)y(υ,ϰ2)|ly|ϰ1ϰ2|;

    P4) There is a exist ϑyC(B,R+) such that |y(υ,ϰ)|ϑy(υ), for each (υ,ϰ)B×R;

    P5) There is an N>0 satisfying

    Nλ1˜ϖ1˜ϖ2(N)+λ2ϑy>1, (3.10)

    where

    λ1=(ς0(˜b))r3+r1[|Λ3|+|Λ1|Γ(r1δ+2)+1+|Λ2|Γ(r1+r3+1)],λ2=(ς0(˜b))r3[|Λ3|+|Λ1|Γ(2δ)+1+|Λ2|Γ(r3+1)]. (3.11)

    Then, (1.3) admits a solution of B.

    Proof. At first, to convert the sequential-type FDI (1.3) into a problem of the FP type, we write Θ:EP(E) as follows:

    Θ(ϰ)={zC(B):z(υ)={RLIr3,ςy(υ,ϰ(υ))+RLIr1+r3,ςσ(υ)+(Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1)×(RLI1δ,ςy(˜b,ϰ(˜b))+RLIr1δ+1,ςσ(˜b))Λ2(RLIr3,ςy(˜b,ϰ(˜b))+RLIr1+r3,ςσ(˜b))}, (3.12)

    for σR,ϰ. Consider two operators Ψ1:EE and Ψ2:EP(E) as follows:

    Ψ1ϰ(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLI1δ,ςy(˜b,ϰ(˜b))+RLIr3,ςy(υ,ϰ(υ))ΛRL2Ir3,ςy(˜b,ϰ(˜b)),

    and

    Ψ2(ϰ)={zE:z(υ)={[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b)}.

    Obviously, Θ=Ψ1+Ψ2. In the following, we demonstrate that Ψ1 and Ψ2 fulfill the conditions of the nonlinear alternative for contractive maps [27, Corollary 3.8]. Initially, we consider the set,

    Ωγ={ϰE:ϰγ},γ>0, (3.13)

    which is bounded, and show that Ψ˚ȷ define the SVMs Ψ˚ȷ:ΩγPcp,c(E), ˚ȷ=1,2. To achieve this, we need to prove that Ψ1 and Ψ2 are compact and convex-valued. The proof will proceed in five steps.

    Step 1. Ψ2 is bounded on bounded sets of E. Let Ωγ be a bounded set in E. Then for every zΨ2(ϰ) and ϰΩγ, σR,ϰ exists such that,

    z(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b).

    Let (P1) holds. For any υB, we obtain,

    |z(υ)|[|Λ3|(ς0(˜b))r3+δ1+|Λ1|(ς0(υ))r3+δ1]RLIr1δ+1,ς|σ(˜b)|+RLIr1+r3,ς|σ(υ)|+|Λ2|RLIr1+r3,ς|σ(˜b)|˜ϖ1˜ϖ2(γ)(ς0(˜b))r3+r1[|Λ3|+|Λ1|Γ(r1δ+2)+1+|Λ2|Γ(r1+r3+1)].

    Indeed, zλ1˜ϖ1˜ϖ2(γ).

    Step 2. Ψ2 maps bounded sets of E into equicontinuous sets. Let ϰΩγ and zΨ2(ϰ). In this case, an element σR,ϰ exists such that

    z(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b),υB.

    Let υ1,υ2B, υ1<υ2. Then

    |z(υ2)z(υ1)|˜ϖ1˜ϖ2(γ)(ς0(˜b))r1δ+1Γ(r1δ+2)(|Λ1|(ς0(υ2))r3+δ1(ς0(υ1))r3+δ1)+˜ϖ1˜ϖ2(γ)Γ(r1+r3+1)[(ς0(υ2))r1+r3(ς0(υ1))r1+r3].

    As υ1υ2, we obtain, |z(υ2)z(υ1)|0. Therefore, Ψ2(Ωγ) is equicontinuous. Combining the results from Steps 1 and 2, and employing the theorem of Arzelà-Ascoli, we can confirm the completely continuity of Ψ2.

    Step 3. Ψ2(ϰ) is convex for all ϰE. Let z1,z2Ψ2(ϰ). Then σ1,σ2R,ϰ exist such that for each υB

    zj(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσj(˜b)+RLIr1+r3,ςσj(υ)ΛRL2Ir1+r3,ςσj(˜b),j=1,2.

    Let μ[0,1]. Then for any υB,

    (μz1(υ)+(1μ)z2(υ))=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ς(μσ1(˜b)+(1μ)σ2(˜b))+RLIr1+r3,ς(μσ1(υ)+(1μ)σ2(υ))ΛRL2Ir1+r3,ς(μσ1(˜b)+(1μ)σ2(˜b)).

    Since has convex values, R,ϰ is convex, and for μ[0,1], (μσ1(υ)+(1μ)σ2(υ))R,ϰ. Therefore, μz1(υ)+(1μ)z2(υ)Ψ2(ϰ), which shows that Ψ2 is convex-valued. Moreover, Ψ1 is compact and convex-valued.

    Step 4. We prove that Gr(Ψ2) is closed. Let ϰnϰ, znΨ2(ϰn) and znz. We show that zΨ2(ϰ). Since znΨ2(ϰn), there is a σnR,ϰn such that,

    zn(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσn(˜b)+RLIr1+r3,ςσn(υ)ΛRL2Ir1+r3,ςσn(˜b).

    Therefore, we need to prove the existence of σR,ϰ such that for each υB,

    z(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b),υB.

    Let Υ:L1(B,R)C(B,R) be a continuous linear operator defined as follows:

    σΥ(σ)(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]Ia1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b),υB.

    Notice that

    znz=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ς(σn(˜b)σ(˜b))+RLIr1+r3,ς(σn(υ)σ(υ))ΛRL2Ir1+r3,ς(σn(˜b)σ(˜b))0,

    when n. Therefore, by Lemma 2.11, ΥR,ϰ is a closed graph operator. Additionally, znΥ(R,ϰn). Since ϰnϰ, Lemma 2.11 gives

    z(υ)=[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1]RLIr1δ+1,ςσ(˜b)+RLIr1+r3,ςσ(υ)ΛRL2Ir1+r3,ςσ(˜b),

    for some σR,ϰ. Thus, the graph of Ψ2 is closed. As a result, Ψ2 is compact and upper semi-continuous.

    Step 5. We prove that Ψ1 is a contraction in E. Let ϰ1,ϰ2E. By using the assumption (P3), we get,

    |Ψ1ϰ1(υ)Ψ1ϰ2(υ)|ly(ς0(˜b))r3(|Λ3|+|Λ1|Γ(2δ)+1+|Λ2|Γ(r3+1))ϰ1ϰ2.

    Thus, Ψ1ϰ1Ψ1ϰ2lyλ2φ¯φ. As lyλ2<1, we conclude that Ψ1 is a contraction. Thus, the operators Ψ1 and Ψ2 meet the theorem [27] hypotheses. As a result, we conclude that either of the two following conditions holds, (a) Θ has an FP in ¯E, (b) we have ϰE and ξ(0,1) with ϰξF(ϰ). We show that conclusion (b) is not possible. If ϰξΨ1(ϰ)+ξΨ2(ϰ) for ξ(0,1). Then, σR,ϰ exists such that

    |ϰ(υ)|=|ξRLIr3,ςy(υ,ϰ(υ))+ξRLIr1+r3,ςσ(υ)+ξ[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ(˜b))+RLIr1δ+1,ςσ(˜b))ξΛ2(RLIr3,ςy(˜b,ϰ(˜b))+RLIr1+r3,ςσ(˜b))|λ1˜ϖ1˜ϖ2(ϰ)+λ2ϑy,

    which implies that |ϰ(υ)|λ1˜ϖ1˜ϖ2(ϰ)+λ2ϑy, for each υB. If criterion of [27, Theorem-(b)] is true, then ξ(0,1) and ϰE with ϰ=ξΘ(ϰ) exist. Therefore, ϰ is a solution of (1.3) with ϰ=N. Now, thanks to |ϰ(υ)|λ1˜ϖ1˜ϖ2(ϰ)+λ2ϑy, we get

    Nλ1˜ϖ1˜ϖ2(N)+λ2ϑy1,

    which contradicts (3.10). Thus, it follows from the theorem [27] that Θ admits an FP, and it is a solution of (1.3).

    We try to establish a more general existence criterion for the FDI (1.3) under new hypotheses. Specifically, we demonstrate the desired existence result for a nonconvex-valued right-hand side using the theorem of Covitz and Nadler [28]. For a metric space (E,ϱ), we define

    {Hϱ:P(E)×P(E)R+{},Hϱ(˜R1,˜R2)=max{sup˜r1˜R1ϱ(˜r1,˜R2),sup˜r2˜R2ϱ(˜R1,˜r2)},

    where ϱ(˜R1,˜r2)=inf˜r1˜R1ϱ(˜r1,ϱ2) and ϱ(˜r1,˜R2)=inf˜r2˜R2ϱ(˜r1,˜r2). Then (Pb,cl(E),Hϱ) forms a metric space [29].

    Definition 3.4. An SVM Ω:EPcl(E) is a ˜η-Lipschitz if and only if ˜η>0 exists such that

    Hϱ(Ω(ϰ1),Ω(ϰ2))˜ηϱ(ϰ1,ϰ2),ϰ1,ϰ2E.

    In particular, Ω is a contraction whenever ˜η<1.

    Theorem 3.5. Assume that (P3) and the following conditions hold:

    P6) The map :B×RPcp(R) is such that (,φ):BPcp(R) is measurable for any ϰR;

    P7) The condition Hϱ((υ,ϰ1),(υ,ϰ2))n(υ)|ϰ1ϰ2| holds for a.e. υB and ϰ1,ϰ2R with nC(B,R+) and ϱ(0,(υ,0))n(υ) for a.e. υB.

    Then FDI (1.3) has at least one solution for B whenever nλ1+lyλ2<1, where λ1,λ2 are given in (3.11).

    Proof. By assumption (P6) and [30, Theorem III.6], has a measurable selection σ:BR, with σL1(B), which implies that is integrability bounded. Therefore, R,ϰ. We demonstrate that the operator Ω:EP(E) described in (3.12) meets the conditions required by Nadler and Covitz's FPT. Specifically, we prove that Ω(ϰ) is closed for each ϰE. Assume a sequence such that {un}n0Ω(ϰ) and unu(n) in E. Then uE and σnRG,ϰn exists such that

    un(υ)=RLIr3,ςy(υ,ϰ(υ))+RLIr1+r3,ςσn(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ(˜b))+RLIr1δ+1,ςσn(˜b))Λ2(RLIr3,ςy(˜b,ϰ(˜b))+RLIr1+r3,ςσn(˜b)).

    So there is a subsequence σn that converges to σ in L1(B), because has compact values. As a result, σR,ϰ, and we get

    un(υ)u(υ)=RLIr3,ςy(υ,ϰ(υ))+RLIr1+r3,ςσ(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ(˜b))+RLIr1δ+1,ςσ(˜b))Λ2(RLIr3,ςy(˜b,ϰ(˜b))+RLIr1+r3,ςσ(˜b)).

    Hence uΩ(ϰ). Next, we show that a Δ(0,1), (Δ=nλ1+lyλ2) exists such that

    Hϱ(Ω(ϰ1),Ω(ϰ2))Δϰ1ϰ2,ϰ1,ϰ2E.

    Let ϰ1,ϰ2E and v1Ω(ϰ1). Then σ1(υ)(υ,ϰ1(υ)) exists such that for all υB and

    v1(υ)=RLIr3,ςy(υ,ϰ1(υ))+RLIr1+r3,ςσ1(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ1(˜b))+RLIr1δ+1,ςσ1(˜b))Λ2(RLIr3,ςy(˜b,ϰ1(˜b))+RLIr1+r3,ςσ1(˜b)).

    By (P7), we have

    Hϱ((υ,ϰ1(υ)),(υ,ϰ2(υ)))n(υ)|ϰ1(υ)ϰ2(υ)|.

    Thus, χ(υ)(υ,ϰ2) exists such that |σ1(υ)χ|n(υ)|ϰ1(υ)ϰ2(υ)|, for each υB. We build an SVM, O:BP(R) as follows:

    O(υ)={χR:|σ1(υ)χ|n(υ)|ϰ1(υ)ϰ2(υ)|}.

    Notice that σ1 and ω=n|ϰ1ϰ2| are measurable, so it follows that O(υ)(υ,ϰ2) is measurable. Next, we select the function σ2(υ)(υ,ϰ2) such that,

    |σ1(υ)σ2(υ)|n(υ)|ϰ1(υ)ϰ2(υ)|,υB.

    Define

    v2(υ)=RLIr3,ςy(υ,ϰ2(υ))+IRLIr1+r3,ςσ2(υ)+[Λ3(ς0(˜b))r3+δ1Λ1(ς0(υ))r3+δ1](RLI1δ,ςy(˜b,ϰ2(˜b))+RLIr1δ+1,ςσ2(˜b))Λ2(RLIr3,ςy(˜b,ϰ2(˜b))+RLIr1+r3,ςσ2(˜b)).

    As a results, we arrive at,

    |v1(υ)v2(υ)|(nλ1+lyλ2)ϰ1ϰ2,

    which implies v1v2(nλ1+lyλ2)ϰ1ϰ2. Now, by interchanging the roles of ϰ1 and ϰ2, we obtain,

    Hϱ(Ω(ϰ1),Ω(ϰ2))(nλ1+lyλ2)ϰ1ϰ2.

    Since Ω is a contraction, it follows that the Covitz and Nadler theorem that Ω has an FP, which is a solution of the FDI (1.3).

    In order to validate the theoretical findings, we provide specific cases of FDIs in this section. In fact, we focus on the FDI with the following form:

    {HDr1,r2,ς(CDr3,ςϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),υB,ϰ(0)+η1ϰ(˜b)=0,CDδ+r31,ςϰ(0)+ηC2Dδ+r31,ςϰ(˜b)=0. (4.1)

    The examples below are special cases of FDIs given by (4.1).

    Example 4.1. Using the FDIs defined by (4.1) and taking r1{12,23,56}, r2=13, r3=15, ς(υ)=υ2, η1=14, η2=16, δ=0.666,0.777,0.888, and ˜b=1, the problem (4.1) is reduced to

    {HD1/2,1/3,υ2(CD1/5,υ2ϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),ϰ(0)+14ϰ(1)=0,CD2/15,υ2ϰ(0)+16CD2/15,υ2ϰ(1)=0, (4.2)

    for υB. With these data, it follows from (3.3), that we have

    Λ1=η2(η2+1)Γ(r3+δ){0.1302,r1=1/2,0.1409,r1=2/3,0.1494,r1=5/6,Λ2=η1η1+1{0.2000,r1=1/2,0.2000,r1=2/3,0.2000,r1=5/6,Λ3=η1η2(η2+1)Γ(r3+δ){0.0325,r1=1/2,0.0352,r1=2/3,0.0373,r1=5/6.

    We define the function y and the SVM :B×RP(R) as follows:

    y(υ,ϰ)=cos(υ)υ2+2(|ϰ||ϰ|+1),(υ,ϰ)B×R, (4.3)

    and

    (υ,ϰ)=[1(5υ2+7exp(υ))ϰ5(ϰ+3), 1υ2+16|ϰ||ϰ|+1]. (4.4)

    For ϰ,¯ϰR, we have

    |y(υ,ϰ)y(υ,¯ϰ)|=|cos(υ)υ2+2(|ϰ||ϰ|+1|¯ϰ||¯ϰ|+1)|1υ2+2(|ϰ¯ϰ|(1+|ϰ|)(1+|¯ϰ|))ly|ϰ¯ϰ|, (4.5)

    with ly=12 and also,

    y(υ,ϰ)1exp(υ2)+1=ϑy(υ),(υ,ϰ)B×R.

    Thus, the assumptions (P3) and (P4) hold. It is also clear that the SVM satisfies the assumption (P1) and

    (υ,ϰ)P=sup{|η|:η(υ,ϰ)}1υ2+16=˜ϖ1(υ)˜ϖ2(ϰ),

    where ˜ϖ1=14 and ˜ϖ2(ϰ)=1. Thus, (P2) holds, and by (P5),

    λ1=(ς0(˜b))r3+r1[|Λ3|+|Λ1|Γ(r1δ+2)+1+|Λ2|Γ(r1+r3+1)]{1.494,r1=1/2,1.446,r1=2/3,1.374,r1=5/6,λ2=(ς0(˜b))r3[|Λ3|+|Λ1|Γ(2δ)+1+|Λ2|Γ(r3+1)]{1.489,r1=1/2,1.500,r1=2/3,1.504,r1=5/6,

    for which the curves are shown in Figure 1, Moreover,

    N>λ1˜ϖ1˜ϖ2(N)+λ2ϑy{1.140,r1=1/2,1.117,r1=2/3,1.086,r1=5/6,

    whenever N=1.15, which it is shown in Figure 2. As seen in Table 1, the effect of the order of the derivative r1 is very insignificant. So all assumptions of Theorem 3.3 are valid. Hence the FDI (4.2) has a solution for B.

    Figure 1.  Graphical representation of the λi, i=1,2 of the FDI (4.2) with three different values of r1.
    Figure 2.  Graphical representation of the N of the FDI (4.2) for r1{12,23,56}.
    Table 1.  The data obtained for the FDI (4.2) with three different values of r1.
    υ r1=12r1=12 r1=23r1=23 r1=56r1=56
    λ1 λ2 N>... λ1 λ2 N>... λ1 λ2 N>...
    0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.10 0.059 0.593 0.316 0.027 0.597 0.307 0.012 0.599 0.303
    0.20 0.157 0.782 0.443 0.089 0.788 0.423 0.049 0.790 0.411
    0.30 0.277 0.920 0.552 0.179 0.927 0.523 0.114 0.929 0.502
    0.40 0.414 1.032 0.653 0.295 1.040 0.618 0.207 1.043 0.590
    0.50 0.566 1.129 0.751 0.435 1.137 0.712 0.328 1.140 0.678
    0.60 0.731 1.214 0.849 0.597 1.223 0.809 0.478 1.226 0.771
    0.70 0.907 1.291 0.945 0.779 1.301 0.908 0.657 1.304 0.869
    0.80 1.093 1.362 1.042 0.982 1.372 1.011 0.866 1.376 0.974
    0.90 1.289 1.428 1.140 1.205 1.438 1.117 1.105 1.442 1.086
    1.00 1.494 1.489 1.238 1.446 1.500 1.228 1.374 1.504 1.206

     | Show Table
    DownLoad: CSV

    In the next example, we check the changes in the derivative order r2.

    Example 4.2. Using the FDI defined by (4.1) and taking r1=12, r2{115,17,13}, r3=15, ς(υ)=υ, η1=14, η2=16, δ=0.533,0.571,0.666, and ˜b=1, 4.1 is reduced to

    {HD1/2,1/3,υ(CD1/5,υϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),υB,ϰ(0)+14ϰ(1)=0,CD2/15,υϰ(0)+16CD2/15,υϰ(1)=0. (4.6)

    With these data, we find

    Λ1{0.114,r2=1/15,0.119,r2=1/7,0.130,r2=1/3,Λ2{0.200,r2=1/15,0.200,r2=1/7,0.200,r2=1/3,Λ3{0.028,r2=1/15,0.029,r2=1/7,0.032,r2=1/3.

    Consider the SVM :B×RP(R) is defined by, φ(υ,ϰ)=[0,sin(ϰ)5υ2+4+112], and the function y defined in (4.3). From (4.5), we see that the assumption (P3) is satisfied with ly=12. Next, we have Hϱ((υ,ϰ),(υ,¯ϰ))n(υ)|ϰ¯ϰ|, where n(υ)=15υ2+4 and ϱ(0,(υ,0))=112n(υ) for a.e. υB. Figure 3 shows the curves of λi, i=1,2, whenever r2 varies in the interval B. By comparing the curves and data in Table 2, it can be clearly seen that as r2 approaches zero, λi decreases.

    Figure 3.  Graphical representation of the λi, i=1,2 of the FDI (4.6) with three different values of r2.
    Table 2.  The data obtained for the FDI (4.6) with three different values of r2.
    υ r2=115r2=115 r2=17r2=17 r2=13r2=13
    λ1 λ2 nλ1+lyλ2 λ1 λ2 nλ1+lyλ2 λ1 λ2 nλ1+lyλ2
    0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.10 0.292 0.927 0.493 0.294 0.931 0.495 0.298 0.940 0.500
    0.20 0.475 1.064 0.580 0.478 1.069 0.582 0.484 1.079 0.588
    0.30 0.631 1.154 0.640 0.635 1.159 0.643 0.643 1.171 0.650
    0.40 0.772 1.223 0.688 0.776 1.228 0.692 0.787 1.240 0.699
    0.50 0.902 1.278 0.729 0.907 1.284 0.733 0.919 1.296 0.740
    0.60 1.025 1.326 0.765 1.031 1.332 0.769 1.045 1.345 0.777
    0.70 1.142 1.367 0.798 1.148 1.374 0.802 1.164 1.387 0.810
    0.80 1.254 1.404 0.828 1.261 1.411 0.831 1.278 1.424 0.840
    0.90 1.361 1.438 0.855 1.369 1.444 0.859 1.388 1.458 0.868
    1.00 1.466 1.468 0.881 1.474 1.475 0.885 1.494 1.489 0.894

     | Show Table
    DownLoad: CSV

    Furthermore, we obtain n=110, resulting in

    nλ1+lyλ2{0.881,r2=1/15,0.885,r2=1/7,0.894,r2=1/3.}<1. (4.7)

    These results are shown in Table 2. Furthermore, the curves of Eq (4.7) for three cases of r2 are shown in Figure 4.

    Figure 4.  Graphical representation of nλ1+lyλ2 in Eq (4.7) of the FDI (4.6) for r2{115,17,13}.

    Therefore, all the assumptions of Theorem 3.5 are satisfied, which implies that at least one solution to the problem (4.6) for B.

    In Example 4.3, we examine our proven theorems for changes of function ς(υ).

    Example 4.3. Using the FDIs defined by (4.1) and taking r123, r2=13, r3=15,

    ς1(υ)=υ2,ς2(υ)=υ,ς3(υ)=υ,ς4(υ)=ln(υ+0.01), (4.8)

    η1=14, η2=16, δ=0.777, ˜b=1, the problem (4.1) is reduced to

    {HD2/3,1/3,ςj(υ)(CD1/5,ςj(υ)ϰ(υ)y(υ,ϰ(υ)))(υ,ϰ(υ)),ϰ(0)+14ϰ(1)=0,CD2/15,ςj(υ)ϰ(0)+16CD2/15,ςj(υ)ϰ(1)=0, (4.9)

    for υB. With these data, it follows from (3.3) that

    Λ1=η2(η2+1)Γ(r3+δ)0.1409,Λ2=η1η1+10.2000,Λ3=η1η2(η2+1)Γ(r3+δ)0.0352.

    We define the function y and the SVM :B×RP(R) as follows:

    y(υ,ϰ)=cos(υ)υ2+2(|ϰ||ϰ|+1),(υ,ϰ)B×R,

    and

    (υ,ϰ)=[1(5υ2+7exp(υ))ϰ5(ϰ+3), 1υ2+16|ϰ||ϰ|+1].

    For ϰ,¯ϰR, we have

    |y(υ,ϰ)y(υ,¯ϰ)|=|cos(υ)υ2+2(|ϰ||ϰ|+1|¯ϰ||¯ϰ|+1)|1υ2+2(|ϰ¯ϰ|(1+|ϰ|)(1+|¯ϰ|))ly|ϰ¯ϰ|,

    with ly=12, as well as y(υ,ϰ)1exp(υ2)+1=ϑy(υ), for each (υ,ϰ)B×R. Thus, the assumptions (P3) and (P4) hold. It is also clear that the SVM satisfies the assumption (P1) and

    (υ,ϰ)P=sup{|η|:η(υ,ϰ)}1υ2+16=˜ϖ1(υ)˜ϖ2(ϰ),

    where ˜ϖ1=14 and ˜ϖ2(ϰ)=1. Thus, (P2) holds, and by (P5)

    λ1=(ς0(˜b))r3+r1[|Λ3|+|Λ1|Γ(r1δ+2)+1+|Λ2|Γ(r1+r3+1)]{1.494,ς1(υ)=υ2,1.446,ς2(υ)=υ,1.374,ς3(υ)=υ,1.374,ς4(υ)=ln(υ+0.01),λ2=(ς0(˜b))r3[|Λ3|+|Λ1|Γ(2δ)+1+|Λ2|Γ(r3+1)]{1.494,ς1(υ)=υ2,1.446,ς2(υ)=υ,1.374,ς3(υ)=υ,1.374,ς4(υ)=ln(υ+0.01),

    for which the curves are shown in Figure 5. Moreover

    N>λ1˜ϖ1˜ϖ2(N)+λ2ϑy{1.117,ς1(υ)=υ2,1.114,ς2(υ)=υ,1.138,ς3(υ)=υ,1.142,ς4(υ)=ln(υ+0.01),

    whenever N=1.15, which is shown in Figure 6. As seen in Table 3, the effect of ς(υ) is very remarkable.

    Figure 5.  Graphical representation of the λi, i=1,2 of the FDI (4.9) with four cases of ς(υ).
    Figure 6.  Graphical representation of the N of the FDI (4.9) with four cases of ς(υ) as defined in (4.8).
    Table 3.  The data obtained for the FDI (4.2) with four cases of ς(υ).
    υ ς1(υ)=υ2ς1(υ)=υ2 ς2(υ)=υς2(υ)=υ ς3(υ)=υς3(υ)=υ ς4(υ)=ln(υ+0.01)ς4(υ)=ln(υ+0.01)
    λ1 λ2 N>... λ1 λ2 N>... λ1 λ2 N>... λ1 λ2 N>
    0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.10 0.027 0.597 0.307 0.197 0.946 0.530 0.533 1.192 0.739 3.086 1.787 1.008
    0.20 0.089 0.788 0.423 0.358 1.087 0.647 0.720 1.277 0.832 3.795 1.874 1.078
    0.30 0.179 0.927 0.523 0.509 1.179 0.736 0.858 1.330 0.895 4.213 1.920 1.116
    0.40 0.295 1.040 0.618 0.654 1.249 0.812 0.972 1.369 0.945 4.508 1.950 1.142
    0.50 0.435 1.137 0.712 0.793 1.306 0.881 1.071 1.400 0.987 4.737 1.973 1.162
    0.60 0.597 1.223 0.809 0.929 1.354 0.944 1.159 1.425 1.023 4.923 1.990 1.178
    0.70 0.779 1.301 0.908 1.062 1.397 1.004 1.239 1.447 1.056 5.081 2.005 1.191
    0.80 0.982 1.372 1.011 1.192 1.435 1.060 1.313 1.467 1.085 5.216 2.017 1.202
    0.90 1.205 1.438 1.117 1.320 1.469 1.114 1.382 1.484 1.113 5.336 2.027 1.212
    1.00 1.446 1.500 1.228 1.446 1.500 1.166 1.446 1.500 1.138 5.443 2.037 1.220

     | Show Table
    DownLoad: CSV

    So all the assumptions of Theorem 3.3 are valid. Hence the FDI (4.9) has a solution for B.

    In the investigation of FDEs and FDIs that contain Hilfer fractional derivative operators, a zero initial condition is typically required. To address this limitation, we proposed a novel approach that combines Hilfer and Caputo fractional derivatives. In this research, we applied this method to study a class of FDEs for FDIs with non-separated BCs, incorporating both Hilfer and Caputo fractional derivative operators. The existence results are established by examining cases where the set-valued map has either convex or nonconvex values. For convex SVMs, the Leray-Schauder FPT was applied, whereas Nadler's and Covitz's FPTs are used for nonconvex SVMs. The findings are well demonstrated with two relevant illustrative examples. The findings of this study contribute significantly to the emerging field of FDIs. In future work, we aim to apply this method to study other types of FDEs with nonzero initial conditions, as well as coupled systems of FDEs that incorporate both Hilfer and Caputo FDs.

    Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

    Adel Lachouri: Actualization, methodology, formal analysis, validation, investigation, initial draft and a major contribution to writing the manuscript. Naas Adjimi: Actualization, methodology, formal analysis, validation, investigation and review. Mohammad Esmael Samei: Actualization, methodology, formal analysis, validation, investigation, software, simulation, review and a major contribution to writing the manuscript. Manuel De la Sen: Validation, review, funding. All authors read and approved the final manuscript.

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

    This research was funded by the Basque Government, Grant IT1555-22.

    The authors declare that they have no competing interests.



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