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Research article

Global co-dynamics of viral infections with saturated incidence

  • Several mathematical models of two competing viruses (or viral strains) that have been published in the literature assume that the infection rate is determined by bilinear incidence. These models do not show co-existence equilibrium; moreover, they might not be applicable in situations where the virus concentration is high. In this paper, we developed a mathematical model for the co-dynamics of two competing viruses with saturated incidence. The model included the latently infected cells and three types of time delays: discrete (or distributed): (ⅰ) The formation time of latently infected cells; (ⅱ) The activation time of latently infected cells; (ⅲ) The maturation time of newly released virions. We established the mathematical well-posedness and biological acceptability of the model by examining the boundedness and nonnegativity of the solutions. Four equilibrium points were identified, and their stability was examined. Through the application of Lyapunov's approach and LaSalle's invariance principle, we demonstrated the global stability of equilibria. The impact of saturation incidence, latently infected cells, and time delay on the viral co-dynamics was examined. We demonstrated that the saturation could result in persistent viral coinfections. We established conditions under which these types of viruses could coexist. The coexistence conditions were formulated in terms of saturation constants. These findings offered new perspectives on the circumstances under which coexisting viruses (or strains) could live in stable viral populations. It was shown that adding the class of latently infected cells and time delay to the coinfection model reduced the basic reproduction number for each virus type. Therefore, fewer treatment efficacies would be needed to keep the system at the infection-free equilibrium and remove the viral coinfection from the body when utilizing a model with latently infected cells and time delay. To demonstrate the associated mathematical outcomes, numerical simulations were conducted for the model with discrete delays.

    Citation: Ahmed M. Elaiw, Ghadeer S. Alsaadi, Aatef D. Hobiny. Global co-dynamics of viral infections with saturated incidence[J]. AIMS Mathematics, 2024, 9(6): 13770-13818. doi: 10.3934/math.2024671

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  • Several mathematical models of two competing viruses (or viral strains) that have been published in the literature assume that the infection rate is determined by bilinear incidence. These models do not show co-existence equilibrium; moreover, they might not be applicable in situations where the virus concentration is high. In this paper, we developed a mathematical model for the co-dynamics of two competing viruses with saturated incidence. The model included the latently infected cells and three types of time delays: discrete (or distributed): (ⅰ) The formation time of latently infected cells; (ⅱ) The activation time of latently infected cells; (ⅲ) The maturation time of newly released virions. We established the mathematical well-posedness and biological acceptability of the model by examining the boundedness and nonnegativity of the solutions. Four equilibrium points were identified, and their stability was examined. Through the application of Lyapunov's approach and LaSalle's invariance principle, we demonstrated the global stability of equilibria. The impact of saturation incidence, latently infected cells, and time delay on the viral co-dynamics was examined. We demonstrated that the saturation could result in persistent viral coinfections. We established conditions under which these types of viruses could coexist. The coexistence conditions were formulated in terms of saturation constants. These findings offered new perspectives on the circumstances under which coexisting viruses (or strains) could live in stable viral populations. It was shown that adding the class of latently infected cells and time delay to the coinfection model reduced the basic reproduction number for each virus type. Therefore, fewer treatment efficacies would be needed to keep the system at the infection-free equilibrium and remove the viral coinfection from the body when utilizing a model with latently infected cells and time delay. To demonstrate the associated mathematical outcomes, numerical simulations were conducted for the model with discrete delays.



    Fractional calculus has attempted to be accessed as a promising technique in fluid mechanics [1], nano-material [2], thermal energy [3], epidemics [4] and other scientific disciplines over recent decades. For example, by provoking interest in both cutting-edge and conventional pure and applied analytical techniques, it has reinforced creative collaboration between different disciplines, existence, and relevant applications in real-world manifestations, see [5,6,7]. In 1965, the possibility of fractional calculus depended on the conversation between L'hospital and Leibnitz as letters in [8]. After that, many researchers started experimenting in this field, and a large portion of them concentrated on describing novel fractional formulations [9,10,11]. Various classifications were raised in this process and were expressed by the advancement of research.

    Recently, extensive investigation has been proposed for the qualitative characterization of verification for various fractional differential equations (FDEs) with initial and boundary value problems. Several significant approaches regarding the existence, uniqueness, multiplicity and stability have been reported by proposing certain fixed point theorems. Although many of the important problems have been tackled by the classical fractional derivatives (RL and Caputo) [12,13,14], it has a few limitations when used to design physical issues as a result of the necessary assumptions are themselves fractional and may be unsuitable for physical problems. The Caputo derivative has the opportunity of being appropriate for physical problems because it only necessitates classical initial conditions [15,16].

    Khalil et al. [17] invented an interesting definition of fractional derivative, which is known to be a conformable derivative. In actuality, this so-called derivative is not a fractional derivative, however, it is it is essentially a first derivative duplicated by an extra straightforward factor. Consequently, this novel concept appears to be a regular extension of the classical derivative. More characterizations and the extended form of this derivative have been expounded in [18]. Then authors [19] explored an extension of the conformable derivative by considering proportional derivative. This fact leads to the modified conformable (proportional) differential operator of order λ. The researchers investigated numerous integral inequalities using classical [20], conformable and generalized conformable fractional integrals [21,22]. Qurashi et al. [23] proposed new fractional derivatives and integrals that have nonsingular kernels. By using the generalized proportional Hadamard fractional integral operator, Zhou et al. [24] investigated some general inequalities and their variant forms.

    This recently characterized local derivative approaches to the original function as λ0. In this manner, they had the option to improve the conformable derivative. Jarad et al. [25] presented another kind of fractional operators created from the extended conformable derivatives. The exponential function appeared as a kernel in their examination with outstanding performance [26,27,28]. For the interest of readers, we draw in their thoughtfulness regarding some new papers [29,30].

    In parallel with the concentrated exploration of the fractional derivative, the existence-uniqueness of verification fits to the intense prominent qualitative characterizations of FDEs, see [31,32,33,34].

    Inspired by the work, we utilize a novel fractional derivative which is known as Hilfer-GPF derivative for finding the existence-uniqueness of solutions for a new class of nonlinear FDEs having non-local boundary conditions. For this we consider the subsequent BVP for a class of Hilfer-FDEs:

    {Dλ,ζ;ϑϖ+1y()=G(,y()),λ(0,1),ζ[0,1],(ϖ1,ϖ2],I1δ,;ϑϖ+1[m1y(ϖ+1)+m2y(ϖ2)]=eȷ,δ=λ+ζ(1λ),eȷR, (1.1)

    where G:(ϖ1,ϖ2]×RR be a continuous mapping, Dλ,ζ;ϑϖ+1(.) is the Hilfer-GPF derivative of order λ(0,1) and I1δ;ϑϖ+1(.) is the GPF integral of order 1δ>0. We find the existence consequences by the fixed point techniques of Schauder, Schaefer and Kransnoselskii. Additionally, the investigation of nonlinear FDEs as far as their information sources (fractional orders, related boundaries, and suitable function) has fascinated the interest of mathematicians because of its importance in Orlicz space (see [35]). Rely upon this, the subject of coherence of verification of the Hilfer-FDEs regarding inputs is significant and worth assuming.

    The organization of the paper is as follows. In Section 2, we proceed with some basics concepts and detailed consequences as a review literature. In Section 3, we establish an equivalence criterion of integral equation of BVP (1.1) and then proposed the existence consequences for GPF-derivative by well-noted fixed point theorem. Also, in Section 4, illustrative examples are presented to check the applicability of the findings developed in Section 3. The conclusion with some open problems is presented in Section 5.

    In what follows, we demonstrate some preliminaries, initial results and spaces which are essential for proving further consequences. Throughout this investigation, let Lp(ϖ1,ϖ2),p1, is the space of Lebesgue integrable mappings on (ϖ1,ϖ2).

    Assume that ϖ1,ϖ2(,+) be a finite and infinite intervals on R.

    Furthermore, we elaborate the subsequent weighted spaces with induced norms defined by (see[8]). Suppose that C[ϖ1,ϖ2] is said to be the space of continuous functions defined on [ϖ1,ϖ2] and the norm is defined as follows:

    GC[ϖ1,ϖ2]=max[ϖ1,ϖ2]|G|,

    and ACn[ϖ1,ϖ2] represents the space of n-times absolutely continuous differentiable mappings defined as follows:

    ACn[ϖ1,ϖ2]={G:(ϖ1,ϖ2]R:Gn1AC[ϖ1,ϖ2]},

    Cδ[ϖ1,ϖ2] denotes the weighted space of G on (ϖ1,ϖ2] is defined as

    Cδ[ϖ1,ϖ2]={G:(ϖ1,ϖ2]R:(ϖ1)δG()C[ϖ1,ϖ2]},δ[0,1)

    with the norm

    GCδ[ϖ1,ϖ2]=(ϖ1)δG()C[ϖ1,ϖ2]=max[ϖ1,ϖ2]|(ϖ1)δG|.

    Also, the weighted space of a function G on (ϖ1,ϖ2] is denoted by Cnδ(ϖ1,ϖ2] defined as

    Cnδ[ϖ1,ϖ2]={G:(ϖ1,ϖ2]R:G()Cn1[ϖ1,ϖ2];Gn()Cδ[ϖ1,ϖ2]},δ[0,1)

    with the norm

    GCnδ[ϖ1,ϖ2]=n1κ=0GκC[ϖ1,ϖ2]+GnCδ[ϖ1,ϖ2],nN.

    For n=0, Cnδ[ϖ1,ϖ2] coincides with Cδ[ϖ1,ϖ2].

    Definition 2.1. ([8]) Assume that GL1([ϖ1,ϖ2],R), then the RL fractional integral operator of G of order λ>0 is stated as

    Jλϖ+1G()=1Γ(λ)ϖ1()λ1G()d,>ϖ1, (2.1)

    where Γ(.) represents the classical Gamma function.

    Definition 2.2. ([8]) Assume that GC([ϖ1,ϖ2]), then the RL fractional derivative operator of G of order λ>0 is stated as

    Dλϖ+1G()=1Γ(nλ)dndnϖ1()nλ1G()d,>ϖ1,n1<λ<n,nN, (2.2)

    where Γ(.) represents the Gamma function.

    Definition 2.3. ([8]) Assume that GCn([ϖ1,ϖ2]), then the Caputo fractional derivative operator of G of order λ>0 is stated as

    cDλϖ+1G()=1Γ(nλ)ϖ1()nλ1Gn()d,>ϖ1,n1<λ<n,nN, (2.3)

    where Γ(.) represents the Gamma function.

    Definition 2.4. ([25]) For ϑ(0,1],λC,(λ)>0, then the left-sided generalized proportional integral of G of order λ>0 is stated as

    Jλ;ϑϖ+1G()=1ϑλΓ(λ)ϖ1eϑ1ϑ()()λ1G()d,>ϖ1. (2.4)

    Definition 2.5. ([25]) For ϑ(0,1],λC,(λ)>0, then the left-sided generalized proportional derivative of G of order λ>0 is stated as

    Dλ;ϑϖ+1G()=Dn,ϑϑnλΓ(nλ)ϖ1eϑ1ϑ()()nλ1G()d,>ϖ1, (2.5)

    where n=[λ]+1.

    Definition 2.6. ([25]) For ϑ(0,1],λC,(λ)>0, then the left-sided generalized proportional integral of G of order λ>0 is stated as

    cDλ;ϑϖ+1G()=1ϑnλΓ(nλ)ϖ1eϑ1ϑ()()nλ1(Dn,ϑG)()d,>ϖ1, (2.6)

    where n=[λ]+1.

    Remark 2.1. Specifically, if ϑ=1 Definitions 2.4–2.6 reduces to Definitions 2.1–2.3, respectively.

    Definition 2.7. ([25]) For nN,λ(n1,n),ϑ(0,1],ζ[0,1], then the left/right-sided Hilfer-GPF derivative having order λ, type ζ of G is stated as follows:

    (Dϖ+1G)(y)=Iζ(nλ),ϑϖ+1[Dϑ(I(1ζ)(nλ),ϑϖ+1G)](y), (2.7)

    where DϑG(y)=(1ϑ)G(y)+ϑG(y) and I assumed to be GPF-integral stated in 2.4.

    Specifically, if n=1, then Definition 2.7 reduces to

    (Dϖ+1G)(y)=Iζ(1λ),ϑϖ+1[Dϑ(I(1ζ)(1λ),ϑϖ+1G)](y). (2.8)

    In the present investigation, we discuss the case where n=1,λ(0,1),ζ[0,1] and δ=λ+ζλζ.

    Remark 2.2. It is remarkable to mention that:

    (a)The Hilfer fractional derivative can be considered as an interpolator between the GPF-derivative and Caputo GPF-derivative, respectively, as

    Dλ,ζ;ϑϖ+1G={DϑI(1λ);ϑϖ+1G,ζ=0(seeDefinition2.5),I1λ;ϑϖ+1DϑG,ζ=1(seeDefinition2.6), (2.9)

    (b) The following assumptions holds true:

    0<δ1δλ,δ>ζ,1δ<1ζ(1λ).

    (c) Particularly, if λ(0,1),ζ[0,1] and δ=λ+ζλζ, then

    (Dλ,ζ;ϑϖ+1G)()=(Iζ(1λ)ϖ+1[Dϑ(I(1ζ)(1λ)ϖ+1G)])(y),

    therefore, we have

    (Dλ,ζ;ϑϖ+1G)()=(Iζ(1λ)ϖ+1(Dδ;ϑϖ+1G))(),

    where (Dδ;ϑϖ+1g1)()=dd(I(1ζ)(1λ);ϑϖ+1G)().

    Now we define the weighted spaces of continuous mappings on (ϖ1,ϖ2]:

    Cλ,ζ;ϑ1δ[ϖ1,ϖ2]={GC1δ[ϖ1,ϖ2],Dλ,ζ;ϑϖ+1GC1δ[ϖ1,ϖ2]},δ=λ+ζ(1λ) (2.10)

    and

    Cδ1δ[ϖ1,ϖ2]={GC1δ[ϖ1,ϖ2],Dδ;ϑϖ+1GC1δ[ϖ1,ϖ2]}. (2.11)

    Since Dλ,ζ;ϑϖ+1=Iζ(1λ),ϑϖ+1Dδ;ϑϖ+1, therefore, we have Cδ1δ[ϖ1,ϖ2]Cλ,ζ1δ[ϖ1,ϖ2].

    Theorem 2.1. ([25]) For ϖ1,ϑ(0,1],(λ),(ζ)>0. If GC([ϖ1,ϖ2],R), then

    Iλ;ϑϖ+1(Iζ;ϑϖ+1G)()=Iζ;ϑϖ+1(Iλ;ϑϖ+1G)()=(Iλ+ζϖ+1G)().

    Theorem 2.2. ([25]) For ϖ1,ϑ(0,1] and (λ)>0 and let GL1([ϖ1,ϖ2]), then

    Dλ;ϑϖ+1Iλ;ϑϖ+1G()=G(),n=[(λ)]+1.

    Lemma 2.1. ([25]) For λ,ςC such that (λ)0 and (ς)>0. Then for any ϑ(0,1] we have

    (a)(Iλ;ϑϖ+1eϑ1ϑ(ϖ1)ς1)()=Γ(ς)ϑλΓ(ς+λ)eϑ1ϑ(ϖ1)ς+λ1,
    (b)(Dλ;ϑϖ+1eϑ1ϑ(ϖ1)ς1)()=ϑλΓ(ς)Γ(ςλ)eϑ1ϑ(ϖ1)ςλ1.

    Lemma 2.2. ([30]) For λ(0,1),ϑ(0,1],ζ(0,1) and δ=λ+ζλζ. If GCδ1δ[ϖ1,ϖ2], then

    Iδ;ϑϖ+1Dδ;ϑϖ+1G=Iδ;ϑϖ+1Dλ,ζ;ϑϖ+1G

    and

    Dδ;ϑϖ+1Iλ;ϑϖ+1G=Dζ(1λ);ϑϖ+1G.

    Lemma 2.3. ([30])For y(ϖ1,ϖ2],λ(0,1),ϑ(0,1),ζ[0,1] and δ(0,1). If GC1δ[ϖ1,ϖ2] and I1δ,ϑϖ+1G, then

    Iλ;ϑϖ+1Dλ,ζ;ϑϖ+1G(y)=G(y)eϑ1ϑ(yϖ1)(yϖ1)δ1ϑδ1Γ(δ)(I1δ;ϑϖ1)(ϖ+1).

    Lemma 2.4. ([30]) For δ[0,1),ϑ(0,1] and g1Cδ. If GCδ[ϖ1,ϖ2], then

    Iλ;ϑϖ+1G(ϖ1)=limyϖ+1Iλ;ϑϖ+1G(y)=0,δ[0,λ).

    Lemma 2.5. ([30]) For λ(0,1),ζ[0,1] and δ=λ+ζλζ and let G:(ϖ1,ϖ2]×RR be a mapping such that GC1δ[ϖ1,ϖ2] for any yCδ1δ[ϖ1,ϖ2], then y satisfies problem (1.1) if and only if y satisfies the Volterra integral equation

    y()=(ϖ1)δ1eϑ1ϑ(ϖ1)I1δ;ϑϖ+1y(ϖ+1)ϑδ1Γ(δ)+1ϑλΓ(λ)ϖ1eϑ1ϑ(s1)()λ1G(,y())d. (2.12)

    This section consists of the existence of solution to BVP (1.1) in Cλ,ζ;ϑ1δ[ϖ1,ϖ2].

    Lemma 3.1. For λ(0,1),ζ[0,1], where δ=λ+ζλζ and suppose there be a function G:(ϖ1,ϖ2]×RR such that GC1δ[ϖ1,ϖ2] for any yC1δ[ϖ1,ϖ2]. If yCδ1δ[ϖ1,ϖ2], then y fulfills BVP (1.1) if and only if y holds the following identity

    y()=eȷm1+m2eϑ1ϑ(ϖ1)(ϖ1)δ1ϑδΓ(δ)m2m1+m2eϑ1ϑ(ϖ1)(ϖ1)δ1ϑδΓ(δ)×1ϑ1δ+λΓ(1δ+λ)ϖ2ϖ1eϑ1ϑ(ϖ2)(ϖ2)λδG(,y())d+1ϑλΓ(λ)ϖ1eϑ1ϑ()()λ1G(,y())d. (3.1)

    Proof. By means of Lemma 2.5 and utilizing the solution of (1.1) can be expressed as

    y()=J1δ;ϑϖ+1y(ϖ+1)ϑδ1Γ(δ)eϑ1ϑ(ϖ1)(ϖ1)δ1+1ϑδΓ(δ)ϖ1eϑ1ϑ()()λ1G(,y())d,>ϖ1. (3.2)

    Employing J1δ;ϑϖ+1 on (3.2) and applying the limit ϖ12, we find

    J1δ;ϑϖ+1y(ϖ2)=J1δ;ϑϖ+1y(ϖ+1)+1ϑ1δ+λΓ(1δ+λ)ϖ2ϖ1eϑ1ϑ(ϖ2)(ϖ2)λδG(,y())d. (3.3)

    Again, employing J1δ;ϑϖ+1 on (3.3), we have

    J1δ;ϑϖ+1y(ϖ2)=J1δ;ϑϖ+1y(ϖ+1)+1ϑ1δ+λΓ(1δ+λ)ϖ1eϑ1ϑ(ϖ2)()λδG(,y())d=J1δ;ϑϖ+1y(ϖ+1)+J1ζ(1λ);ϑϖ+1G(,y()). (3.4)

    Applying limit ϖ+1 and utilizing Lemma 2.4 having 1δ<1ζ(1λ), yields

    J1δ;ϑϖ+1y(ϖ+1)=J1δ;ϑϖ+1y(ϖ+1), (3.5)

    thus

    J1δ;ϑϖ+1y(ϖ2)=J1δ;ϑϖ+1y(ϖ+1)+1ϑ1δ+λΓ(1δ+λ)ϖ2ϖ1eϑ1ϑ(ϖ2)(ϖ2)λδG(,y())d. (3.6)

    From boundary condition (1.1), we have

    J1δ;ϑϖ+1y(ϖ2)=eȷm2m1m2J1δ;ϑϖ+1y(ϖ+1). (3.7)

    From (3.5) and (3.6), and utilizing (3.4), we have

    J1δ;ϑϖ+1y(ϖ+1)=m2m1+m2(eȷm21ϑ1δ+λΓ(1δ+λ)ϖ2ϖ1eϑ1ϑ(ϖ2)(ϖ2)λδG(,y())d). (3.8)

    Setting (3.2) in (3.8), one can find

    y()=eȷm1+m2eϑ1ϑ(ϖ1)(ϖ1)δ1ϑδ1Γ(δ)m2m1+m2eϑ1ϑ(ϖ1)(ϖ1)δ1ϑδ1Γ(δ)×1ϑ1δ+λΓ(1δ+λ)ϖ2ϖ1eϑ1ϑ(ϖ2)(ϖ2)λδG(,y())d+1ϑλΓ(λ)ϖ1eϑ1ϑ()()λ1G(,y())d. (3.9)

    Conversely, employing \mathcal{J}_{\varpi_{1}^{+}}^{1-\delta; \vartheta} on (3.1), utilizing Lemmas 2.1 and 2.2, and simple computations yields

    \begin{eqnarray} &&\mathcal{J}_{\varpi_{1}^{+}}^{1-\delta;\vartheta}m_{1}y(\varpi_{1}^{+})+ \mathcal{J}_{\varpi_{1}^{+}}^{1-\delta;\vartheta}m_{2}y(\varpi_{2}^{-})\\&& = \frac{m_{1}m_{2}}{m_{1}+m_{2}}\bigg(\frac{e_{\jmath}}{m_{2}}-\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{G}(\ell, y(\ell))d\ell\bigg)\\&&\quad+\frac{m_{2}^{2}}{m_{1}+m_{2}}\bigg(\frac{e_{\jmath}}{m_{2}}-\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{G}(\ell, y(\ell))d\ell\bigg)\\&&\quad+\frac{m_{2}}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{G}(\ell, y(\ell))d\ell\\&& = e_{\jmath}, \end{eqnarray} (3.10)

    This shows that y(\wp) satisfies boundary condition (1.1).

    Furthermore, employing \mathcal{D}_{\varpi_{1}^{+}}^{\delta; \vartheta} on (3.1) and applying Lemmas 2.1 and 2.2, we have

    \begin{eqnarray} \mathcal{D}_{\varpi_{1}^{+}}^{\delta;\vartheta} y(\wp) = \mathcal{D}_{\varpi_{1}^{+}}^{\zeta(1-\lambda);\vartheta}\mathcal{G}(\wp, y(\wp)). \end{eqnarray} (3.11)

    Since y\in\mathbb{C}_{1-\delta}^{\delta; \vartheta}[\varpi_{1}, \varpi_{2}] and in view of definition of \mathbb{C}_{1-\delta}^{\delta; \vartheta}[\varpi_{1}, \varpi_{2}], we have \mathcal{D}_{\varpi_{1}^{+}}^{\delta}y\in\mathbb{C}_{n-\delta}^{\vartheta}[\varpi_{1}, \varpi_{2}], thus, \mathcal{D}_{\varpi_{1}^{+}}^{\zeta(1-\lambda); \vartheta}\mathcal{G} = \mathcal{D}\mathcal{I}_{\varpi_{1}^{+}}^{1-\zeta(1-\lambda)}\mathcal{G}\in\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}].

    For \mathcal{G}\in\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}], it is noting that \mathcal{I}_{\varpi_{1}^{+}}^{1-\zeta(1-\lambda)}\mathcal{G}\in \mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}]. So, \mathcal{G} and \mathcal{I}_{\varpi_{1}^{+}}^{1-\zeta(1-\lambda); \vartheta}\mathcal{G} holds the assumptions of Lemma 2.3. Now, employing \mathcal{I}_{\varpi_{1}^{+}}^{\zeta(1-\lambda); \vartheta} on (3.11), we have

    \begin{eqnarray} \mathcal{I}_{\varpi_{1}^{+}}^{\zeta(1-\lambda);\vartheta}\mathcal{D}_{\varpi_{1}^{+}}^{\delta;\vartheta} y(\wp) = \mathcal{I}_{\varpi_{1}^{+}}^{\zeta(1-\lambda);\vartheta}\mathcal{D}_{\varpi_{1}^{+}}^{\zeta(1-\lambda);\vartheta}\mathcal{G}(\wp, y(\wp)). \end{eqnarray} (3.12)

    Considering (2.8), (3.11) and Lemma 2.3, we have

    \mathcal{I}_{\varpi_{1}^{+}}^{\delta;\vartheta}\mathcal{D}_{\varpi_{1}^{+}}^{\delta;\vartheta} y(\wp) = \mathcal{G}(\wp, y(\wp))-\frac{\mathcal{I}_{\varpi_{1}^{+}}^{1-\zeta(1-\lambda);\vartheta}\mathcal{G}(\varpi_{1}, y(\varpi_{1}))}{\vartheta^{\zeta(1-\lambda)}\Gamma(\zeta(1-\lambda))}e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi_{1})}(\wp-\varpi_{1})^{\zeta(1-\lambda)-1}, \\ \forall\wp\in(\varpi_{1}, \varpi_{2}]. (3.13)

    By Lemma 2.4, we have \mathcal{I}_{\varpi_{1}^{+}}^{1-\zeta(n-\lambda)}\mathcal{G}(\varpi_{1}, y(\varpi_{1})) = 0. Thus, we have \mathcal{D}_{\varpi_{1}^{+}}^{\lambda, \zeta; \vartheta}y(\wp) = \mathcal{G}(\wp, y(\wp)). Hence, this completes the proof.

    Let us evoke some essential assumptions which are required to prove the existence of solutions for the problem mentioned.

    (A_{1}) Let a function \mathcal{G}:(\varpi_{1}, \varpi_{2}]\times\mathbb{R}\mapsto\mathbb{R} with \mathcal{G}(., y(.))\in\mathbb{C}_{1-\delta}^{\zeta(1-\lambda); \vartheta}[\varpi_{1}, \varpi_{2}]. For any y\in\mathbb{C}_{1-\delta}^{\vartheta}[\varpi_{1}, \varpi_{2}] and there exist two constants \mathcal{M}_{1}, \mathfrak{m} such that

    \begin{eqnarray} \big\vert \mathcal{G}(\wp_{1}, y)\big\vert\leq \mathcal{M}_{1}\Big(1+\mathfrak{m}\|y\|_{\mathbb{C}_{1-\delta}^{\vartheta}}\Big). \end{eqnarray} (3.14)

    (A_{2}) The inequality

    \begin{eqnarray} \mathcal{G}: = \frac{\mathfrak{m}\mathcal{M}_{1}\Gamma(\delta)}{\vartheta^{\lambda}\Gamma(\lambda+1)}\Big[(\varpi_{2}-\varpi_{1})^{\lambda}+(\varpi_{2}-\varpi_{1})^{\lambda+1-\delta}\Big] < 1 \end{eqnarray} (3.15)

    holds.

    Now we are in a position to show the existence results for the BVP (1.1) by employing Schauder's fixed point theorem (see [36]).

    Theorem 3.1. Suppose that the assumptions (A_{1}) and (A_{2}) fulfills. Then by Hilfer- BVP (1.1) has at least one solution in \mathbb{C}_{1-\delta}^{\delta; \vartheta}[\varpi_{1}, \varpi_{2}]\subset\mathbb{C}_{1-\delta}^{\lambda, \zeta; \vartheta}[\varpi_{1}, \varpi_{2}].

    Proof. Defining an operator {\bf T}:\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}]\mapsto\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}] by

    \begin{eqnarray} ({\bf T}y)(\wp)&& = \frac{e_{\jmath}}{m_{1}+m_{2}}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi_{1})}(\wp-\varpi_{1})^{\delta-1}}{\vartheta^{\delta-1}\Gamma(\delta)}-\frac{m_{2}}{m_{1}+m_{2}}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi_{1})}(\wp-\varpi_{1})^{\delta-1}}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{G}(\ell, y(\ell))d\ell\\&&\quad+\frac{1}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}(\wp-\ell)^{\lambda-1}\mathcal{G}(\ell, y(\ell))d\ell. \end{eqnarray} (3.16)

    Assume that \mathbb{B}_{\varrho} = \Big\{y\in\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}]:\|y\|_{\mathbb{C}_{1-\delta}}\leq\varrho\Big\} having \varrho\geq\frac{\Omega}{1-\mathcal{G}}, for \mathcal{G} < 1, we have

    \begin{eqnarray} \Omega&&: = \frac{e_{\jmath}}{(m_{1}+m_{2})\vartheta^{\delta-1}\Gamma(\delta)}+\bigg\vert\frac{m_{2}}{m_{1}+m_{2}}\bigg\vert\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{\mathcal{M}_{1}}{\vartheta^{1-\delta+\lambda}}\bigg[\frac{(\varpi_{2}-\varpi_{1})^{\lambda+1-\delta}}{\Gamma(\lambda-\delta+2)}+\frac{(\varpi_{2}-\varpi_{1})^{2\lambda-\delta+1}}{\Gamma(\lambda+1)}\bigg]. \end{eqnarray} (3.17)

    The proof will be demonstrated by the accompanying three steps:

    Case 1. We will prove that {\bf T}(\mathbb{B}_{\varrho})\subset\mathbb{B}_{\varrho}. Utilizing assumption (A_{2}), we have

    \begin{eqnarray} &&\Big\vert({\bf T}y)(\wp)(\wp-\varpi_{1})^{1-\delta}\Big\vert\\&&\leq\bigg\vert\frac{e_{\jmath}}{(m_{1}+m_{2})\vartheta^{\delta-1}\Gamma(\delta)}\bigg\vert+\bigg\vert\frac{m_{2}}{m_{1}+m_{2}}\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\bigg\vert\\&&\quad\times\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}} \Big\vert e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}\Big\vert(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{M}_{1}(1+\mathfrak{m}\vert y\vert)d\ell\\&&\quad+\frac{(\wp-\varpi_{1})^{1-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp} \Big\vert e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}\Big\vert(\wp-\ell)^{\lambda-1}\mathcal{M}_{1}(1+\mathfrak{m}\vert y\vert)d\ell\\&&\leq\frac{e_{\jmath}}{(m_{1}+m_{2})\vartheta^{\delta-1}\Gamma(\delta)}+\bigg\vert\frac{m_{2}}{m_{1}+m_{2}}\bigg\vert\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}\Big\vert(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{M}_{1}(1+\mathfrak{m}\|y\|_{\mathbb{C}_{1-\delta}})d\ell\\&&\quad+\frac{(\wp-\varpi_{1})^{1-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}\Big\vert(\wp-\ell)^{\lambda-1}\mathcal{M}_{1}(1+\mathfrak{m}\|y\|_{\mathbb{C}_{1-\delta}})d\ell. \end{eqnarray} (3.18)

    Since \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1. Observe that, for any y\in\mathbb{B}_{\varrho}, and for every \wp\in(\varpi_{1}, \varpi_{2}], we have

    \begin{eqnarray} &&\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}\Big\vert(\varpi_{2}-\ell)^{\lambda-\delta}\mathcal{M}_{1}(1+\mathfrak{m}\|y\|_{\mathbb{C}_{1-\delta}})d\ell\\&&\leq\frac{\mathcal{M}_{1}(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{1-\delta+\lambda}}\bigg[\frac{(\varpi_{2}-\varpi_{1})^{1-\delta}}{\Gamma(\lambda-\delta+2)}+\frac{\mathfrak{m}\varrho\Gamma(\delta)}{\Gamma(\lambda+1)}\bigg], \; since\, \, \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1 \end{eqnarray} (3.19)

    and

    \begin{eqnarray} &&\frac{\big\vert(\wp-\varpi_{1})^{1-\delta}\big\vert}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}\Big\vert(\wp-\ell)^{\lambda-1}\mathcal{M}_{1}(1+\mathfrak{m}\|y\|_{\mathbb{C}_{1-\delta}})d\ell\\&&\leq\frac{\mathcal{M}_{1}(\wp-\varpi_{1})^{\lambda-\delta+1}}{\vartheta^{\lambda}}\bigg[\frac{(\wp-\varpi_{1})^{\lambda}}{\Gamma(\lambda+1)}+\frac{\mathfrak{m}\varrho\Gamma(\delta)}{\Gamma(\lambda+1)}\bigg]. \end{eqnarray} (3.20)

    Therefore, we get

    \begin{eqnarray} &&\Big\vert({\bf T}y)(\wp)(\wp-\varpi_{1})^{1-\delta}\Big\vert\\&&\leq\frac{e_{\jmath}}{(m_{1}+m_{2})\vartheta^{\delta-1}\Gamma(\delta)}+\bigg\vert\frac{m_{2}}{m_{1}+m_{2}}\bigg\vert\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{\mathcal{M}_{1}(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{1-\delta+\lambda}}\bigg[\frac{(\varpi_{2}-\varpi_{1})^{1-\delta}}{\Gamma(\lambda-\delta+2)}+\frac{\mathfrak{m}\varrho\Gamma(\delta)}{\Gamma(\lambda+1)}\bigg]\\&&\quad+\frac{\mathcal{M}_{1}(\wp-\varpi_{1})^{\lambda-\delta+1}}{\vartheta^{\lambda}}\bigg[\frac{(\wp-\varpi_{1})^{\lambda}}{\Gamma(\lambda+1)}+\frac{\mathfrak{m}\varrho\Gamma(\delta)}{\Gamma(\lambda+1)}\bigg], \end{eqnarray} (3.21)

    which leads to

    \begin{eqnarray} \|{\bf T}y\|_{\mathbb{C}_{1-\delta}}&&\leq\frac{e_{\jmath}}{(m_{1}+m_{2})\vartheta^{\delta-1}\Gamma(\delta)}+\bigg\vert\frac{m_{2}}{m_{1}+m_{2}}\bigg\vert\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{\mathcal{M}_{1}}{\vartheta^{1-\delta+\lambda}}\bigg[\frac{(\varpi_{2}-\varpi_{1})^{\lambda+1-\delta}}{\Gamma(\lambda-\delta+2)}+\frac{(\varpi_{2}-\varpi_{1})^{2\lambda-\delta+1}}{\Gamma(\lambda+1)}\bigg]\\&&\quad+\frac{\mathcal{M}_{1}\mathfrak{m}\varrho\Gamma(\delta)}{\vartheta^{\lambda}\Gamma(\lambda+1)}\bigg[{(\wp-\varpi_{1})^{\lambda}}+(\varpi_{2}-\varpi_{1})^{\lambda+1-\delta}\bigg]. \end{eqnarray} (3.22)

    of assumption (A_{2}), we conclude that \|{\bf T}y\|_{\mathbb{C}_{1-\delta}}\leq\mathcal{G}\varrho+(1-\mathcal{G})\varrho = \varrho, Therefore, {\bf T}(\mathbb{B}_{\varrho})\subset\mathbb{B}_{\varrho}.

    Next we will prove that {\bf T} is completely continuous.

    Case 2. We prove that the operator {\bf T} is completely continuous.

    Assume that \{\bar{z}_{n}\} is a sequence such that \bar{z}_{n}\mapsto \bar{z} in \mathbb{B}_{\varrho} as n\mapsto\infty. Then for every \wp\in(\varpi_{1}, \varpi_{2}], we have

    \begin{eqnarray} &&\Big\vert\Big(({\bf T}\bar{z}_{n})(\wp)-({\bf T}\bar{z})(\wp)\Big)(\wp-\varpi_{1})^{1-\delta}\Big\vert\\&& = \Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\frac{1}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\quad\times\frac{1}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}\Big\vert(\varpi_{2}-\ell)^{\lambda-\delta}\Big\vert \mathcal{G}(\ell, \bar{z}_{n}(\ell))-\mathcal{G}(\ell, \bar{z}(\ell))\Big\vert d\ell\\&&\quad+\frac{(\wp-\varpi_{1})^{1-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}\Big\vert e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}\Big\vert (\wp-\ell)^{\lambda-1}\Big\vert \mathcal{G}(\ell, \bar{z}_{n}(\ell))-\mathcal{G}(\ell, \bar{z}(\ell))\Big\vert d\ell\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\frac{1}{\vartheta^{\lambda}\Gamma(\lambda+1)}(\varpi_{2}-\varpi_{1})^{\lambda}\Big\| \mathcal{G}(., \bar{z}_{n}(.))-\mathcal{G}(., \bar{z}(.))\Big\|_{\mathbb{C}_{1-\delta}}\\&&\quad+\frac{\Gamma(\delta)(\wp-\varpi_{1})^{1-\delta+\lambda}}{\vartheta^{\lambda-\delta}\Gamma(\lambda-\delta)} (\wp-\ell)^{1-\delta+\lambda}\Big\| \mathcal{G}(., \bar{z}_{n}(.))-\mathcal{G}(., \bar{z}(.))\Big\|_{\mathbb{C}_{1-\delta}}. \end{eqnarray} (3.23)

    Since \Big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\Big\vert < 1 and \mathcal{G} is continuous on (\varpi_{1}, \varpi_{2}] and \bar{z}_{n}\mapsto\bar{z}, then

    \begin{eqnarray} \big\|({\bf T}\bar{z}_{n}-{\bf T}\bar{z}\big\|_{\mathbb{C}_{1-\delta}}\rightarrow0\; as\; n\rightarrow \infty, \end{eqnarray} (3.24)

    which shows that operator {\bf T} is continuous on \mathbb{B}_{\varrho}.

    Case 3. We show that {\bf T}(\mathbb{B}_{\varrho}) is relatively compact. In case 1, we have {\bf T}(\mathbb{B}_{\varrho})\subset\mathbb{B}_{\varrho}. It is observed that {\bf T}(\mathbb{B}_{\varrho}) is uniformly bounded. To show operator {\bf T} is equi-continuous on \mathbb{B}_{\varrho}. In fact, for any \varpi_{1} < \wp_{1} < \wp_{2} < \varpi_{2} and \bar{z}\in\mathbb{B}_{\varrho}, we have

    \begin{eqnarray} &&\Big\vert(\wp_{2}-\varpi_{1})^{1-\delta}({\bf T}y)(\wp_{2})-(\wp_{1}-\varpi_{1})^{1-\delta}({\bf T}y)(\wp_{})\Big\vert\\&&\leq\frac{\big\vert(\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\kappa}\big\vert}{\vartheta^{\delta-1}\Gamma(\delta)}\frac{e_{\jmath}}{m_{1}+m_{2}}+\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\frac{\big\vert(\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\kappa}\big\vert}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\times\frac{1}{\vartheta^{\lambda-\delta+1}\Gamma(\lambda-\delta+1)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{\lambda-\delta}\big\vert \mathcal{G}(\ell, y(\ell))\big\vert d\ell\\&&\quad+\frac{1}{\vartheta^{\lambda}\Gamma(\lambda)}\Big\vert(\wp_{2}-\varpi_{1})^{1-\delta}\int\limits_{\varpi_{1}}^{\wp_{2}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{2}-\ell)}(\wp_{2}-\ell)^{\lambda-1} \mathcal{G}(\ell, y(\ell)) d\ell\\&&\quad-(\wp_{1}-\varpi_{1})^{1-\delta}\int\limits_{\varpi_{1}}^{\wp_{1}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{1}-\ell)}(\wp_{1}-\ell)^{\lambda-1} \mathcal{G}(\ell, y(\ell)) d\ell\Big\vert\\&&\leq\frac{\big\vert(\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\kappa}\big\vert}{\vartheta^{\delta-1}\Gamma(\delta)}\\&&\bigg[\frac{e_{\jmath}}{m_{1}+m_{2}}+\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\frac{\|\mathcal{G}\|_{\mathbb{C}_{1-\delta}}}{\vartheta^{1-\delta+\lambda}\Gamma(1-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}(\varpi_{2}-\ell)^{\lambda-\delta}(\ell-\varpi_{1})^{\delta-1}d\ell\bigg]\\&&\quad+\frac{\|\mathcal{G}\|_{\mathbb{C}_{1-\delta}}}{\vartheta^{\lambda}\Gamma(\lambda)}\Big\vert(\wp_{2}-\varpi_{1})^{1-\delta}\int\limits_{\varpi_{1}}^{\wp_{2}}(\wp_{2}-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-1} d\ell\\&&\quad-(\wp_{1}-\varpi_{1})^{1-\delta}\int\limits_{\varpi_{1}}^{\wp_{1}}(\wp_{1}-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-1}d\ell\Big\vert\; \; \; \; \; \; \; \; \; \; \; \; \; \Big(since\; \; \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1\Big)\\&&\leq\frac{\big\vert(\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\kappa}\big\vert}{\vartheta^{\delta-1}\Gamma(\delta)}\bigg[\frac{e_{\jmath}}{m_{1}+m_{2}}+\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\frac{\Gamma(\delta)}{\vartheta^{\lambda}\Gamma(\lambda+1)}(\varpi_{2}-\varpi_{1})^{\lambda}\|\mathcal{G}\|_{\mathbb{C}_{1-\delta}}\bigg]\\&&\quad+\frac{\|\mathcal{G}\|_{\mathbb{C}_{1-\delta}}}{\vartheta^{\lambda}\Gamma(\lambda)}\mathfrak{B}(\delta-n+1, \lambda)\big\vert(\wp_{2}-\varpi_{1})^{\lambda}-(\wp_{1}-\varpi_{1})^{\lambda}\big\vert, \end{eqnarray} (3.25)

    which approaches to zero as \wp_{2}\rightarrow \wp_{1}, independent of y\in\mathbb{\varrho}, where \mathfrak{B}(., .) denotes the Euler Beta function.

    Therefore, we deduce that {\bf T}(\mathbb{B}_{\varrho}) is equicontinuous on \mathbb{B}_{\varrho}, that leads to the relatively compactness. As a result, we conclude that by Arzela-Ascoli theorem, the defined operator {\bf T}:\mathbb{B}_{\varrho}\mapsto\mathbb{B}_{\varrho} is completely continuous operator.

    By Schauder's fixed point theorem, there exists at least one fixed point y of {\bf T} in \mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}]. This fixed point y is the solution of (1.1) in \mathbb{C}_{1-\delta}^{\delta; \vartheta}, and this completes the proof.

    Now we present another existence result via Schaefer fixed point theorem. For this, we need the following assumption.

    (A_{3}) Suppose a function \mathcal{G}:(\varpi_{1}, \varpi_{2}]\times\mathbb{R}\mapsto\mathbb{R} such that \mathcal{G}(., y(.))\in\mathbb{C}_{1-\delta}^{\zeta(1-\lambda); \vartheta}[\varpi_{1}, \varpi_{2}] for any y\in\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}] and there exist a mapping \eta(\wp)\in\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}] such that

    \begin{eqnarray} \big\vert \mathcal{G}(\wp, y)\big\vert\leq\eta(\wp), \; \forall \wp\in(\varpi_{1}, \varpi_{2}], y\in\mathbb{R}. \end{eqnarray} (3.26)

    Theorem 3.2. Suppose that assumption (A_{3}) satisfies. then Hilfer- BVP (1.1) has at least one solution in \mathbb{C}_{1-\delta}^{\delta}\subset\mathbb{C}_{1-\delta}^{\lambda, \zeta; \vartheta}[\varpi_{1}, \varpi_{2}].

    Proof. For the proof of Theorem 3.2, one can adopt the same technique as we did in Theorem 3.1 and easily prove that the operator {\bf T}:\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}]\mapsto\mathbb{C}_{1-\delta}[\varpi_{1}, \varpi_{2}] stated in (3.16) is completely continuous. Now we show that

    \begin{eqnarray} \Delta = \Big\{y\in\mathbb{C}_{n-\delta}[\varpi_{1}, \varpi_{2}]: y = \sigma{\bf T}y, \; for\; some\; \sigma\in(0, 1)\Big\} \end{eqnarray} (3.27)

    is bounded set. Assume that y\in\Delta and \sigma\in(0, 1) be such that y = \sigma {\bf T}y. By assumption (A_{3}) and (3.16), then for all \wp\in[\varpi_{1}, \varpi_{2}], we have

    \begin{eqnarray} &&\big\vert{\bf T}y(\wp)(\wp-\varpi_{1})^{n-\delta}\big\vert\\&&\leq\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi_{1})}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{\delta-\kappa+1}\Gamma(\delta-\kappa+1)}\frac{e_{\jmath}}{m_{1}+m_{2}}\\&&\quad+\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi_{1})}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)\Gamma(n-\delta+\lambda)}\\&&\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n+\lambda-\delta-1}\eta(\ell)d\ell\\&&\quad+\frac{\vert(\wp-\varpi_{1})^{n-\delta}\vert}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}(\wp-\ell)^{\lambda-1}\eta(\ell)d\ell\\&&\leq\sum\limits_{\kappa = 1}^{n}\frac{(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{\delta-\kappa+1}\Gamma(\delta-\kappa+1)}\frac{e_{\jmath}}{m_{1}+m_{2}}\\&&\quad+\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)\Gamma(n-\delta+\lambda)}\\&&\int\limits_{\varpi_{1}}^{\varpi_{2}}(\varpi_{2}-\ell)^{n+\lambda-\delta-1}(\ell-\varpi_{1})^{\delta-n}\|\eta\|_{\mathbb{C}_{n-\delta}}d\ell\\&&\quad+\frac{\vert(\wp-\varpi_{1})^{n-\delta}\vert}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}(\wp-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-n}\|\eta\|_{\mathbb{C}_{n-\delta}}d\ell. \end{eqnarray} (3.28)

    Since \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1, we have

    \begin{eqnarray} &&\|{\bf T}y\|_{\mathbb{C}_{n-\delta}}\\&&\leq\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{\delta-\kappa+1}\Gamma(\delta-\kappa+1)}\frac{e_{\jmath}}{m_{1}+m_{2}}\\&&\quad+\bigg[\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{-\kappa}}{\vartheta^{\delta-\kappa+1}}\frac{\Gamma(\lambda)}{\mathfrak{B}(\lambda, 1)}+\frac{\mathfrak{B}(\delta-n+1, 1)}{\vartheta^{\lambda}(\varpi_{2}-\varpi_{1})^{\delta}}\bigg] (\varpi_{2}-\varpi_{1})^{n+\lambda}\|\eta\|_{\mathbb{C}_{n-\delta}}\\&&: = \tau. \end{eqnarray} (3.29)

    Since \sigma\in(0, 1), then y < {\bf T}y. The last inequality with (3.29) leads us to the conclusion that

    \begin{eqnarray} \|y\|_{\mathbb{C}_{n-\delta}} < \|{\bf T}y\|_{\mathbb{C}_{n-\delta}}\leq\tau, \end{eqnarray} (3.30)

    which proves that \Delta is bounded. Utilizing Schaefer fixed point postulate, this completes the proof.

    Our last result is the existence result for the problem (1.1) by using the Kransnoselskii's fixed point theorem (see [37]), the following assumption is needed:

    (A_{4}) Suppose that \mathcal{G}:(\varpi_{1}, \varpi_{2}]\times\mathbb{R}\mapsto\mathbb{R} is a function such that \mathcal{G}(. y(.))\in\mathbb{C}_{n-\delta}^{\zeta(n-\lambda); \vartheta}[\varpi_{1}, \varpi_{2}] for any y\in\mathbb{C}_{n-\delta}[\varpi_{1}, \varpi_{2}] and there exists a constant L > 0 such that

    \begin{eqnarray} \big\vert \mathcal{G}(\wp, y)-\mathcal{G}(\wp, \omega)\big\vert\leq L\big\vert y-\omega\big\vert, \; \forall\wp\in(\varpi_{1}, \varpi_{2}], \, y, \omega\in\mathbb{R}. \end{eqnarray} (3.31)

    Also, we note the following assumption as follows: (A_{5}) the inequality

    \begin{eqnarray} \mathcal{Q}&&: = \bigg[\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}+\frac{\mathfrak{B}(\delta-n, \lambda+1)}{\vartheta^{\delta}\Gamma(\delta-n)}\bigg]\\&&\quad\times\frac{\Gamma(\delta-n)(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\mathfrak{B}(\delta-n, 1)\Gamma(\lambda+1)}\|\tilde{\mathcal{G}}\|_{\mathbb{C}_{n-\delta}}+\frac{e_{\jmath}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}L < 1 \end{eqnarray} (3.32)

    is hold.

    Theorem 3.3. Suppose that the assumptions (A_{4}) and (A_{5}) are satisfied. If

    \begin{eqnarray} \Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa+\lambda}}{{\vartheta^{n-\delta+\lambda}}\Gamma(\delta-\kappa+1)}\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}L < 1. \end{eqnarray} (3.33)

    Then the Hilfer- BVP (1.1) has at least one solution in \mathbb{C}_{n-\delta}^{\delta}[\varpi_{1}, \varpi_{2}]\subset\mathbb{C}_{n-\delta}^{\lambda, \zeta; \vartheta}.

    Proof. Considering the operator {\bf T} stated in Theorem 3.1.

    First, surmise the operator {\bf T} into sum of two operators {\bf T}_{1}+{\bf T}_{2} as follows

    {\bf T}_{1}y(\wp) = \frac{-m_{2}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{\delta-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\\ \frac{1}{\Gamma(n-\delta+\lambda)}\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n-\delta+\lambda-1}\mathcal{G}(\ell, y(\ell))d\ell (3.34)

    and

    \begin{eqnarray} {\bf T}_{2}y(\wp) = \frac{e_{\jmath}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{\delta-\kappa}}{\vartheta^{\delta-\kappa+1}\Gamma(\delta-\kappa+1)}+\frac{1}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}(\wp-\ell)^{\lambda-1}\mathcal{G}(\ell, y(\ell))d\ell. \end{eqnarray} (3.35)

    Setting \tilde{\mathcal{G}} = \mathcal{G}(\ell, 0) and suppose the ball \mathbb{B}_{\epsilon} = \{y\in\mathbb{C}_{n-\delta; \psi([\varpi_{1}, \varpi_{2}])}:\|y\|_{\mathbb{C}_{n-\delta}; \psi}\leq\epsilon\} having \epsilon\geq\frac{\sigma}{1-\mathcal{Q}}, \mathcal{Q} < 1, where

    \begin{eqnarray} \sigma&& = \bigg[\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}+\frac{\mathfrak{B}(\delta-n, \lambda+1)}{\vartheta^{\delta}\Gamma(\delta-n)}\bigg]\\&&\quad\times\frac{\Gamma(\delta-n)(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\mathfrak{B}(\delta-n, 1)\Gamma(\lambda+1)}L < 1 \end{eqnarray} (3.36)

    The proof will be done in three cases.

    Case 1. We show that {\bf T}_{1}y+{\bf T}_{1}\omega\in\mathbb{B}_{\varrho} for every y, \omega\in\mathbb{B}_{\epsilon}.

    Utilizing assumption (A_{4}), then for every y\in\mathbb{B}_{\epsilon} and \wp\in(\varpi_{1}, \varpi_{2}], we have

    \begin{eqnarray} &&\big\vert(\wp-\varpi_{1})^{n-\delta}{\bf T}_{1}(\wp)\big\vert\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{1}{\Gamma(n-\delta+\lambda)}\\&&\quad\times\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n-\delta+\lambda-1}\Big[\big\vert \mathcal{G}(\ell, y(\ell))-\mathcal{G}(\ell, 0)\big\vert+\big\vert \mathcal{G}(\ell, 0)\big\vert\Big]d\ell\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{1}{\Gamma(n-\delta+\lambda)}\\&&\quad\times\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n-\delta+\lambda-1}(\ell-\varpi_{1})^{\delta-n}\Big[L\|y\|_{\mathbb{C}_{n-\delta}}+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]d\ell\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}{\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}}\Big[L\epsilon+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]. \end{eqnarray} (3.37)

    Since \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1. Therefore, we get

    \begin{eqnarray} &&\|{\bf T}_{1}y\|_{\mathbb{C}_{n-\delta}}\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa+\lambda}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}\Big[L\epsilon+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]. \end{eqnarray} (3.38)

    For operator {\bf T}_{2}, we have

    \begin{eqnarray} &&\big\vert(\wp-\varpi_{1})^{n-\delta}{\bf T}_{2}\omega(\wp)\big\vert\\&&\leq\Big\vert\frac{e_{\jmath}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\\&&\quad\times\frac{(\wp-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}(\wp-\ell)^{\lambda-1}\Big[\big\vert \mathcal{G}(\ell, \omega(\ell))-\mathcal{G}(\ell, 0)\big\vert+\big\vert \mathcal{G}(\ell, 0)\big\vert\Big]d\ell\\&&\leq\Big\vert\frac{e_{\jmath}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\\&&\quad\times\frac{(\wp-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp}e^{\frac{\vartheta-1}{\vartheta}(\wp-\ell)}(\wp-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-n} \Big[L\|\omega\|_{\mathbb{C}_{n-\delta}}+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]d\ell. \end{eqnarray} (3.39)

    For every \omega\in\mathbb{B}_{\epsilon} and \wp\in(\varpi_{1}, \varpi_{2}], this shows

    \begin{eqnarray} &&\|{\bf T}_{1}\omega\|_{\mathbb{C}_{n-\delta}}\\&&\leq\Big\vert\frac{e_{\jmath}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\\&&\quad\times\frac{(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\Gamma(\delta-n+\lambda+1)} \Big[L\epsilon+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]. \end{eqnarray} (3.40)

    From (3.38), (3.40) and utilizing assumption (A_{5}) with (3.36), we find

    \begin{eqnarray} &&\|{\bf T}_{1}y+{\bf T}_{2}\omega\| _{\mathbb{C}_{n-\delta}}\\&&\leq\|{\bf T}_{1}y\| _{\mathbb{C}_{n-\delta}}+\|{\bf T}_{1}\omega\| _{\mathbb{C}_{n-\delta}}\\&&\leq\frac{(\varpi_{2}-\varpi_{1})^{\lambda}\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}\Big[L\epsilon+\big\|\tilde{\mathcal{G}}\big\|_{\mathbb{C}_{n-\delta}}\Big]\\&&\bigg[\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}+\frac{\Gamma(\lambda+1)}{\Gamma(\delta-n+\lambda+1)}\bigg]\\&&\quad+\frac{e_{\jmath}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\\&&\leq\mathcal{Q}\epsilon+(1-\mathcal{Q})\epsilon = \epsilon. \end{eqnarray} (3.41)

    Case 2. We prove that the operator {\bf T} is a contraction mapping on \mathbb{B}_{\varrho}.

    For any y, \omega\in\mathbb{B}_{\varrho}, and for any \wp\in(\varpi_{1}, \varpi_{2}], then by supposition (A_{4}), we have

    \begin{eqnarray} &&\big\vert(\wp-\varpi_{1})^{n-\delta}{\bf T}_{1}y(\wp)-(\wp-\varpi_{1})^{n-\delta}{\bf T}_{1}\omega(\wp) \big\vert\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{1}{\Gamma(n-\delta+\lambda)}\\&&\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n-\delta+\lambda-1}\Big[\mathcal{G}(\ell, y(\ell))-\mathcal{G}(\ell, \omega(\ell))\Big]d\ell\\&&\leq \Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{e^{\frac{\vartheta-1}{\vartheta}(\wp-\varpi)}(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{1}{\Gamma(n-\delta+\lambda)}\\&&\int\limits_{\varpi_{1}}^{\varpi_{2}}e^{\frac{\vartheta-1}{\vartheta}(\varpi_{2}-\ell)}(\varpi_{2}-\ell)^{n-\delta+\lambda-1}L\big\vert y(\ell)-\omega(\ell)\big\vert d\ell\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\wp-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}(\varpi_{2}-\ell)^{\lambda}L\big\| y-\omega\big\|_{\mathbb{C}_{n-\delta}}. \end{eqnarray} (3.42)

    Since \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1, this yields

    \begin{eqnarray} && \big\|{\bf T}y-{\bf T}\omega\big\|_{\mathbb{C}_{n-\delta}}\\&&\leq\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa+\lambda}}{\vartheta^{\vartheta^{n-\delta+\lambda}}\Gamma(\delta-\kappa+1)}\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}L\|y-\omega\|_{\mathbb{C}_{n-\delta}}. \end{eqnarray} (3.43)

    Due to assumption (3.33), which shows that the operator {\bf T} is a contraction mapping.

    Case 3: Now we show that the operator {\bf T}_{2} is completely continuous on \mathbb{B}_{\epsilon}.

    From the continuity of \mathcal{G}, we deduce that the operator {\bf T}_{2}:\mathbb{B}_{\epsilon}\mapsto\mathbb{B}_{\epsilon} is continuous on \mathbb{B}_{\epsilon}. Furthermore, we prove that for all \epsilon > 0 there exists some \epsilon^{\prime} > 0 such that \|{\bf T}_{2}y\|_{\mathbb{C}_{n-\delta}} < \epsilon^{\prime}. In view of case 1, for y\in\mathbb{B}_{\epsilon}, we have that

    \|{\bf T}_{2}y\|_{\mathbb{C}_{n-\delta}}\leq\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{\delta-\kappa+1}\Gamma(\delta-\kappa+1)}\frac{e_{\jmath}}{m_{1}+m_{2}}+\\ \frac{\mathfrak{B}(\delta-n+1, \lambda)(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\Gamma(\lambda)}\Big[L\epsilon+\|\tilde{\mathcal{G}}\|_{\mathbb{C}_{n-\delta}}\Big], (3.44)

    which is free of \wp and y, so there exists

    \begin{eqnarray} \epsilon^{\prime} = \frac{e_{\jmath}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}+\frac{\mathfrak{B}(\delta-n+1, \lambda)(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\Gamma(\lambda)}\Big[L\epsilon+\|\tilde{\mathcal{G}}\|_{\mathbb{C}_{n-\delta}}\Big] \end{eqnarray} (3.45)

    such that \|{\bf T}_{2}(y)\|_{\mathbb{C}_{n-\delta}}\leq\epsilon^{\prime}. Therefore, {\bf T}_{2} is uniformly bounded set on \mathbb{B}_{\epsilon}. Finally, to show that {\bf T}_{2} is equicontinuous in \mathbb{B}_{\epsilon}, for any z\in\mathbb{B}_{\epsilon} and \wp_{1}, \wp_{2}\in(\varpi_{1}, \varpi_{2}] having \wp_{1} < \wp_{2}, we have

    \begin{eqnarray} &&\big\vert(\wp_{2}-\varpi_{1})^{n-\delta}{\bf T}_{2}y(\wp_{2})-(\wp_{1}-\varpi_{1})^{n-\delta}{\bf T}_{2}y(\wp_{1})\big\vert\\&& = \frac{e_{\jmath}}{m_{1}+m_{2}}\bigg\vert\sum\limits_{\kappa = 1}^{n}\frac{(\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\delta}}{\vartheta^{\delta-\kappa}\Gamma(\delta-\kappa+1)}+\frac{(\wp_{2}-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\\&& \int\limits_{\varpi_{1}}^{\wp_{2}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{2}-\ell)}(\wp_{2}-\ell)^{\lambda-1}\mathcal{G}(\ell, y(\ell))d\ell\\&&\quad-\frac{(\wp_{1}-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp_{1}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{1}-\ell)}(\wp_{1}-\ell)^{\lambda-1}\mathcal{G}(\ell, y(\ell))d\ell\bigg\vert\\&&\leq\frac{e_{\jmath}}{m_{1}+m_{2}}\sum\limits_{\kappa = 1}^{n}\frac{\big\vert (\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\delta}\big\vert }{\vartheta^{\delta-\kappa}\Gamma(\delta-\kappa+1)}\\&&\quad+\bigg\vert\frac{(\wp_{2}-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp_{2}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{2}-\ell)}(\wp_{2}-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-n}\|\mathcal{G}\|_{\mathbb{C}_{n-\delta, \psi[\varpi_{1}, \varpi_{2}]}}d\ell\\&&\quad-\frac{(\wp_{1}-\varpi_{1})^{n-\delta}}{\vartheta^{\lambda}\Gamma(\lambda)}\int\limits_{\varpi_{1}}^{\wp_{1}}e^{\frac{\vartheta-1}{\vartheta}(\wp_{1}-\ell)}(\wp_{1}-\ell)^{\lambda-1}(\ell-\varpi_{1})^{\delta-n}\|\mathcal{G}\|_{\mathbb{C}_{n-\delta, \psi[\varpi_{1}, \varpi_{2}]}}d\ell\bigg\vert\\&& = \sum\limits_{\kappa = 1}^{n}\frac{e_{\jmath}}{m_{1}+m_{2}}\frac{\big\vert (\wp_{2}-\varpi_{1})^{n-\kappa}-(\wp_{1}-\varpi_{1})^{n-\delta}\big\vert }{\vartheta^{\delta-\kappa}\Gamma(\delta-\kappa+1)}+\frac{\mathfrak{B}(\delta-n+1)}{\vartheta^{\lambda}}\\&& \|\mathcal{G}\|_{\mathbb{C}_{n-\delta;\psi[\varpi_{1}, \varpi_{2}]}}\big\vert(\wp_{2}-\varpi_{1})^{\lambda}-(\wp_{1}-\varpi_{1})^{\lambda}\big\vert .\\ \end{eqnarray} (3.46)

    Since \big\vert e^{\frac{\vartheta-1}{\vartheta}\wp}\big\vert < 1. It is noting that the right hand side of the aforesaid variant is free of y. So,

    \begin{eqnarray} \big\vert(\wp_{2}-\varpi_{1})^{n-\delta}{\bf T}_{2}y(\wp_{2})-(\wp_{1}-\varpi_{1})^{n-\delta}{\bf T}_{2}y(\wp_{1})\big\vert\mapsto 0, \; as\; \vert \wp_{2}-\wp_{1}\vert\mapsto 0. \end{eqnarray} (3.47)

    This shows that {\bf T}_{2} is equicontinuous on \mathbb{B}_{\epsilon}. According to Arzela-Ascoli Theorem, observed that ({\bf T}_{2}\mathbb{B}_{\epsilon}) is relatively compact. By Kransnoselskii's fixed point theorem, the problem (1.1) has at least one solution.

    Consider the fractional differential equation with boundary condition which encompasses the Hilfer- GPF derivative of the form

    \begin{eqnarray} \left \{ \begin{array}{cc} \mathcal{D}_{\varpi_{1}^{+}}^{\lambda, \zeta;\vartheta}y(\wp) = \wp^{-\frac{1}{6}}+\frac{\wp^{5/6}}{16}\sin y(\wp), \, \wp\in\mathbb{J} = [0, 2], \lambda\in(0, 1), \zeta\in[0, 1]\\ \mathcal{I}_{\varpi_{1}^{+}}^{1-\delta;\vartheta}\Big[\frac{1}{3}y(0^{+})+\frac{2}{3}y(2^{-})\Big] = \frac{2}{5}, \lambda\leq\delta = \lambda+\zeta-\lambda\zeta, \\ \end{array} \right. \end{eqnarray} (4.1)

    By comparison (1.1) with (2.9), we have \lambda = \frac{1}{2}, \zeta = \frac{1}{3}, \delta = \frac{2}{3}, m_{1} = \frac{1}{3}, m_{2} = \frac{2}{3}, \vartheta = 1 and e_{1} = \frac{2}{5}. It is clear that \wp^{\frac{1}{3}}\mathcal{G}(\wp, y(\wp)) = \wp^{\frac{1}{6}}+\frac{\wp^{7/6}}{16}\sin y(\wp)\in\mathbb{C}([0, 2]), So \mathcal{G}(\wp, y(\wp))\in\mathbb{C}_{\frac{1}{3}}. Thus, it follows that, for any y\in\mathbb{R}^{+} and \wp\in\mathbb{J},

    \begin{eqnarray} \big\vert \mathcal{G}(\wp, y(\wp))\big\vert&&\leq\wp^{\frac{1}{6}}\bigg(1+\frac{\wp^{2/3}}{16}\big\vert\wp^{1/3}y(\wp)\big\vert\bigg)\\&&\leq\bigg(1+\frac{1}{16}\big\|y\|_{\mathbb{C}_{\frac{1}{3}}}\bigg). \end{eqnarray} (4.2)

    Hence, the assumption (A_{1}) is fulfilled having \mathcal{M} = 1 and \mathfrak{m} = \frac{1}{16}. It is easy to verify that the assumption (A_{2}) is hold too. In fact, by simple computations, we obtain

    \begin{eqnarray} \mathcal{G}: = \frac{\mathfrak{m}\mathcal{M}_{1}\Gamma(\delta)}{\vartheta^{\lambda}\Gamma(\lambda+1)}\Big[(\varpi_{2}-\varpi_{1})^{\lambda}+(\varpi_{2}-\varpi_{1})^{\lambda+1-\delta}\Big]\approx-0.03510 < 1. \end{eqnarray} (4.3)

    Hence, all suppositions of Theorem 3.1 implies that the problem (1.1) has a unique solution in \mathbb{C}_{\frac{1}{3}}^{\frac{2}{3}}([0, 2]).

    Also, assume that \mathcal{G}(\wp, y(\wp)) = \wp^{-\frac{1}{6}}+\frac{\wp^{5/6}}{16}\sin y(\wp). Thus \big\vert \mathcal{G}(\wp_{1}, y(\wp))\leq \wp^{-\frac{1}{6}}+\frac{\wp^{5/6}}{16}\big\vert = \eta(\wp)\in\mathbb{C}_{1-\delta}([0, 2]). So, (A_{3}) is satisfied. Therefore, in view of Theorem 3.2, we conclude that problem (1.1) has a solution in \mathbb{C}_{1/3}^{2/3}([0, 2]).

    Finally, if \mathcal{G}(\wp, y(\wp)) = \wp^{-\frac{1}{6}}+\frac{\wp^{5/6}}{16}\sin y(\wp), then for y, \omega\in\mathbb{R}^{+} and \wp\in\mathbb{J}, we have

    \begin{eqnarray*} \Big\vert \mathcal{G}(\wp, y(\wp))-\mathcal{G}(\wp, \omega(\wp))\Big\vert\leq\frac{1}{16}\big\vert y-\omega\big\vert. \end{eqnarray*}

    Therefore, the assumption (A_{4}) is fulfilled having L = \frac{1}{16}. Clearly, assumption (A_{5}) and inequality (3.33) are holds. In fact, simple computations yields

    \begin{eqnarray} \mathcal{Q}&&: = \bigg[\Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}+\frac{\mathfrak{B}(\delta-n, \lambda+1)}{\vartheta^{\delta}\Gamma(\delta-n)}\bigg]\\&&\quad\times\frac{\Gamma(\delta-n)(\varpi_{2}-\varpi_{1})^{\lambda}}{\vartheta^{\lambda}\mathfrak{B}(\delta-n, 1)\Gamma(\lambda+1)}L\approx 0.1456 < 1, \end{eqnarray} (4.4)

    and

    \begin{eqnarray} \Big\vert\frac{m_{2}}{m_{1}+m_{2}}\Big\vert\sum\limits_{\kappa = 1}^{n}\frac{(\varpi_{2}-\varpi_{1})^{n-\kappa+\lambda}}{\vartheta^{n-\delta+\lambda}\Gamma(\delta-\kappa+1)}\frac{\Gamma(\delta-n+1)}{\vartheta^{\lambda}\Gamma(\lambda+1)}L\approx0.0495 < 1 \end{eqnarray} (4.5)

    of Theorem 3.3, shows that problem (1.1) has a solution in \mathbb{C}_{\frac{1}{3}}^{\frac{2}{3}}([0, 2]).

    In this approach, we have established certain existence consequences for the solution of BVP for Hilfer- FDEs depend on the lessening of FDEs to integral equations. The proposed scheme with the fixed point assertions unifies the existing results in the frame of RL and Caputo GPF sense, respectively. Besides that, the analysis's comprehensive improvements are dependent on various techniques such as Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Also, the Hilffer GPF -derivative comprise two parameters and a proportionality index \vartheta.

    \bullet If \vartheta\mapsto 1 and \lambda = [0, 1], then the contemplated problem converted to RL and Caputo fractional derivative [8]. If \vartheta\in(0, 1) and \zeta = 0, 1 we recaptures the RL and Caputo GPF -derivative [25], respectively (see Figure 1).

    Figure 1.  Plot of y(\wp) , for the RL fractional derivatives (\zeta = 0, \vartheta = 1) , and GPF -derivatives (\zeta = 0, \vartheta\, \in \, (0, 1)) .

    \bullet Clearly, if \vartheta, \zeta\in(0, 1), then the newly employed derivatives amalgamate the existing ones in the adjustment of Hilfer, RL and GPF -derivative, (see Figure 2).

    Figure 2.  Graph of y(\wp) , for the RL fractional derivatives (\zeta = 0, \vartheta = 1) , GPF -derivatives (\zeta = 0, \vartheta = 0.8) and Hilfer GPF -derivatives (\zeta\, \in \, (0, 1), \vartheta\, \in \, (0, 1)) .

    \bullet If \vartheta\mapsto 1 and \zeta, \lambda\in[0, 1], then the formulation for this problem enjoys Hilfer factional derivative [8], (see Figure 3).

    Figure 3.  Plot of y(\wp) , for the Hilfer fractional derivatives (\vartheta = 1) and Hilfer GPF -derivatives (\vartheta\, \in \, (0, 1)) .

    Moreover, a stimulative example is presented to show the efficacy of the established outcomes. We hope that the testified outcomes here will have a considerable impact for more parameters on the stability and other qualitative features of differential equations in the areas of interest of applied sciences.

    The authors declare that there is no conflict of interests.



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