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Research article

A scale conjugate neural network learning process for the nonlinear malaria disease model

  • Received: 15 March 2023 Revised: 25 May 2023 Accepted: 29 May 2023 Published: 03 July 2023
  • MSC : 68T07, 92B20, 65L06

  • The purpose of this work is to provide a stochastic framework based on the scale conjugate gradient neural networks (SCJGNNs) for solving the malaria disease model of pesticides and medication (MDMPM). The host and vector populations are divided in the mathematical form of the malaria through the pesticides and medication. The stochastic SCJGNNs procedure has been presented through the supervised neural networks based on the statics of validation (12%), testing (10%), and training (78%) for solving the MDMPM. The optimization is performed through the SCJGNN along with the log-sigmoid transfer function in the hidden layers along with fifteen numbers of neurons to solve the MDMPM. The accurateness and precision of the proposed SCJGNNs is observed through the comparison of obtained and source (Runge-Kutta) results, while the small calculated absolute error indicate the exactitude of designed framework based on the SCJGNNs. The reliability and consistency of the SCJGNNs is observed by using the process of correlation, histogram curves, regression, and function fitness.

    Citation: Manal Alqhtani, J.F. Gómez-Aguilar, Khaled M. Saad, Zulqurnain Sabir, Eduardo Pérez-Careta. A scale conjugate neural network learning process for the nonlinear malaria disease model[J]. AIMS Mathematics, 2023, 8(9): 21106-21122. doi: 10.3934/math.20231075

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  • The purpose of this work is to provide a stochastic framework based on the scale conjugate gradient neural networks (SCJGNNs) for solving the malaria disease model of pesticides and medication (MDMPM). The host and vector populations are divided in the mathematical form of the malaria through the pesticides and medication. The stochastic SCJGNNs procedure has been presented through the supervised neural networks based on the statics of validation (12%), testing (10%), and training (78%) for solving the MDMPM. The optimization is performed through the SCJGNN along with the log-sigmoid transfer function in the hidden layers along with fifteen numbers of neurons to solve the MDMPM. The accurateness and precision of the proposed SCJGNNs is observed through the comparison of obtained and source (Runge-Kutta) results, while the small calculated absolute error indicate the exactitude of designed framework based on the SCJGNNs. The reliability and consistency of the SCJGNNs is observed by using the process of correlation, histogram curves, regression, and function fitness.



    In the last years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives. These kind of equations have gained considerable importance due to their application in various sciences, such as physics, biology, economics, mechanics, chemistry, control theory, engineering, signal and image processing, etc [1,2,3,4,5,6,7].

    Nonlinear coupled systems of fractional order differential equations appear often in investigations connected with disease models [8], anomalous diffusion [9] and ecological models [10]. Unlike the classical derivative operator, one can find a variety of its fractional counterparts such as Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Hilfer, Caputo-Hadamard, etc. Recently, a new class of fractional proportional derivative operators was introduced and discussed in [11,12,13]. Then, the concept of Hilfer type generalized proportional fractional derivative operators was proposed in [14]. For the detailed advantages of the Hilfer derivative, see [15] and a recent application in calcium diffusion in [16].

    Many researchers studied initial and boundary value problems for differential equations and inclusions including different kinds of fractional derivative operators, for instance, see [17,18,19,20,21]. In [22], the authors studied a nonlocal-initial value problem of order in (0,1) involving a ¯ψ-Hilfer generalized proportional fractional derivative of a function with respect to another function. Recently, in [23], the authors investigated the existence and uniqueness of solutions for a nonlocal mixed boundary value problem for Hilfer fractional ¯ψ-proportional type differential equations and inclusions of order in (1,2] of the form

    {Dα,β,σ,ψw+1u(t)=Υ(t,u(t)),t[w1,w2],0w1<w2,u(w1)=0,u(w2)=mj=1ηju(ξj)+ni=1ζiIϕi,σ,ψc+u(θi)+rk=1λkDδk,β,σ,ψc+u(μk), (1.1)

    where Dχ,β,σ,ψc+, denotes the ψ-Hilfer generalized proportional fractional derivative operator of order χ{α,δk}, α,δk(1,2] and type β[0,1], respectively, σ(0,1], ηj,ζi,λkR are given constants, Υ: [w1,w2]×RR is a given continuous function, Iϕi;σ,ψw1+ is the generalized proportional fractional integral operator of order ϕi>0 and ξj,θi,μk(w1,w2), j=1,2,,m, i=1,2,,n, k=1,2,,r, are given points.

    In [24], the authors discussed the existence of solutions for a nonlinear coupled system of (k,ψ)-Hilfer fractional differential equations of different orders in (1,2], complemented with coupled (k,ψ)-Riemann-Liouville fractional integral boundary conditions. In [25] a coupled system of Hilfer type generalized proportional fractional differential equations with nonlocal multi-point boundary conditions of the form

    {(Dδ1,η,σw+1+kDδ11,η,σw+1)τ1(t)=Υ1(t,τ1(t),τ2(t)),t[w1,w2],(Dδ2,η,σw+1+k1Dδ21,η,σw+1)τ2(t)=Υ2(t,τ1(t),τ2(t)),t[w1,w2],τ1(w1)=0,τ1(w2)=mj=1θjτ2(ξj),τ2(w1)=0,τ2(w2)=ni=1εiτ1(λi), (1.2)

    is investigated, in which Dδ1,η,σw+1 and Dδ2,η,σw+1 are the fractional derivatives of Hilfer generalized proportional type of order 1<δ1,δ2<2, the Hilfer parameter 0η1, σ(0,1], k,k1R, Υ1,Υ2: [w1,w2]×RR are continuous functions, w10, θj,εiR, ξj,λi(w1,w2) for i=1,2,3,,n and j=1,2,,m. Existence and uniqueness results are proved by applying classical Banach and Krasnosel'skii fixed-point theorems, and the Leray-Schauder alternative.

    Very recently in [26] the authors established existence and uniqueness results for a class of coupled systems of nonlinear Hilfer-type fractional ¯ψ-proportional differential equations equipped with nonlocal multi-point and integro-multi-strip coupled boundary conditions of the form:

    {Dρ1,φ1,ϑ,¯ψw1+σ(z)=Ψ1(z,σ(z),τ(z)),z[w1,w2],Dρ2,φ2,ϑ,¯ψw1+τ(z) =Ψ2(z,σ(z),τ(z)),z[w1,w2],σ(w1)=0,w2w1¯ψ(s)σ(s)ds=ni=1κiηiξi¯ψ(s)τ(s)ds+mj=1θjτ(ζj),τ(w1) =0,w2w1¯ψ(s)τ(s)ds=ni=1ϕiϵiδi¯ψ(s)σ(s)ds+mj=1ϑjσ(zj), (1.3)

    where Dρκ,φa,ϑ,¯ψw1+ and κ=1,2 denote the Hilfer fractional ¯ψ-proportional derivative operator of the order ρκ(1,2] and type φi[0,1], ϑ(0,1], w1<ζj<ξi<ηi<w2, w1<δj<zi<ϵi<w2, κi,θj,ϕi,ϑjR, i=1,2,,n, j=1,2,,m, ¯ψ: [w1,w2]R is an increasing function with ¯ψ(z)0 for all z[w1,w2] and Ψ1,Ψ2: [w1,w2]×R×RR are continuous functions.

    In this work, motivated by the above mentioned papers, we study a coupled system of ψ-Hilfer generalized proportional sequential fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions, of the form

    {(HDα1,β1,ρ,ψw1k1)(t)+λ1(HDα11,β1,ρ,ψw1k1)(t)=Υ1(t,k1(t),k2(t)),t[w1,w2],(HDα2,β2,ρ,ψw1k2)(t)+λ2(HDα21,β2,ρ,ψw1k2)(t)=Υ2(t,k2(t),k1(t)),t[w1,w2],k1(w1)=0,k1(w2)=ni=1ηik2(ξi)+mj=1ζjpIΦj,ρ,ψk2(θj),k2(w1)=0,k2(w2)=rk=1kk1(ϱk)+ql=1ΘlpIυl,ρ,ψk1(ϑl), (1.4)

    where HDX,βι,ρ,ψw1 denotes the ψ-Hilfer generalized proportional fractional derivative operator of order X{α1,α2} with the parameters βι,ι{1,2}, 1<X2, 0βι1, pIY,ρ,ψw1 is a generalized proportional fractional integral operator of order Y>0, Y{Φj,υl}, λ1,λ2,ηi,ζj,k,ΘlR{0}, ξi,θj,ϱk,ϑl(a,b), i=1,2,,n, j=1,2,,m, k=1,2,,r, l=1,2,,q and Υ1,Υ2: [w1,w2]×R×RR are nonlinear continuous functions.

    Here, we emphasize that problem (1.4) is novel, and its investigation will enhance the scope of the literature on nonlocal Hilfer-type fractional ψ-proportional systems. Note that, when ψ(t)=t, problem (1.4) reduces to a coupled system of Hilfer generalized proportional fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions; while if ρ=1, reduces to a coupled system of ψ-Hilfer fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions. If ψ(t)=t, ζj=0, Θl=0, problem (1.4) is reduced to problem (1.2).

    In solving (1.4), we first convert it into an equivalent fixed point problem, with the help of an auxiliary result based on a linear variant (1.4). Afterward, under different assumptions, we apply different fixed point theorems to establish our results on existence and uniqueness of solutions. For the first result (Theorem 3.1), we apply the Leray-Schauder's alternative to show that there exists at least one solution for the problem (1.4). The second result (Theorem 3.2), relying on Krasnosel'skii's fixed point theorem, shows that the problem (1.4) has at least one solution under different assumptions, and the last result (Theorem 3.3), shows the existence of a unique solution to the problem (1.4) by means of Banach's contraction mapping principle. In Section 4, we illustrate all the obtained theoretical results with the aid of constructed numerical examples. We emphasize that the problem (1.4) is novel and its investigation will enhance the scope of the literature on coupled systems of ψ-Hilfer generalized proportional fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions. The used method is standard, but its configuration in the problem (1.4) is new.

    The structure of the rest of the paper is organized as follows: In Section 2, some necessary definitions and preliminary results related to our problem are presented. Section 3 contains the main results for the problem (1.4), while numerical examples illustrating these results are constructed in Section 4. A brief conclusion closes the paper.

    In this section, we introduce some necessary definitions and preliminary results needed in main results later.

    Definition 2.1. [11,12] Let the functions ϑ0,ϑ1: [0,1]×R[0,) be continuous such that for all tR and for ρ[0,1], we get

    limρ0+ϑ0(ρ,t)=0,limρ0+ϑ1(ρ,t)=1,limρ1ϑ0(ρ,t)=1,limρ1ϑ1(ρ,t)=0

    and

    ϑ0(ρ,t)0,0<ρ1,ϑ1(ρ,t)0,0ρ<1.

    Let also ψ(t) be a strictly positive increasing continuous function. So, the proportional differential operator of order ρ of function Υ(t) with respect to function ψ(t) is defined by

    pDρ,ψΥ1(t)=ϑ1(ρ,t)Υ(t)+ϑ0(ρ,t)Υ(t)ψ(t).

    Moreover, if ϑ0(ρ,t)=ρ and ϑ1(ρ,t)=1ρ, then operator pDρ,ψ becomes

    pDρ,ψΥ(t)=(1ρ)Υ1(t)+ρΥ(t)ψ(t).

    The integral corresponding to the above proportional derivative is defined as

    pI1,ρ,ψw1Υ1(t)=1ρtw1eρ1ρ(ψ(t)ψ(s))Υ(s)ψ(s)ds,

    where

    pI0,ρ,ψw1Υ1(t)=Υ1(t).

    The generalized proportional integral of order n corresponding to proportional derivative pDn,ρ,ψΥ1(t), is given by

    pIn,ρ,ψw1Υ1(t)=1ρnΓ(n)tw1eρ1ρ(ψ(t)ψ(s))(ψ(t)ψ(s))n1Υ(s)ψ(s)ds.

    Based on the generalized proportional integral of order n, we can obtain the following general proportional fractional integral and derivative.

    Definition 2.2. [11,12] Let ρ(0,1] and α>0. The fractional proportional integral of order α of the function f with respect to function ψ is defined by

    (pIα,ρ,ψw1Υ)(t)=1ραΓ(α)tw1eρ1ρ(ψ(t)ψ(s))(ψ(t)ψ(s))α1Υ(s)ψ(s)ds.

    Definition 2.3. [11,12] Let ρ(0,1] α>0 and ψ(t) is a continuous function on [w1,w2], ψ(t)>0. The generalized proportional fractional derivative of order α of the function Υ with respect to function ψ is defined by

    (pDα,ρ,ψw1Υ)(t)=pDn,ρ,ψρnαΓ(nα)tw1eρ1ρ(ψ(t)ψ(s))(ψ(t)ψ(s))nα1Υ(s)ψ(s)ds,

    where

    pDn,ρ,ψ=pDρ,ψpDρ,ψpDρ,ψpDρ,ψntimes.

    Below we present the generalized proportional fractional derivatives of order α of function Υ with respect to another function ψ in Hilfer sense.

    Definition 2.4. [27] For ρ(0,1]. Let functions Υ,ψCm([w1,w2],R) and ψ be positive and strictly increasing with ψ(t)0, for all t[w1,w2]. The ψ-Hilfer generalized propotional fractional derivative of order α and type β for Υ with respect to another function ψ is defined by

    (HDα,β,ρ,ψw1Υ)(t)=pIβ(nα),ρ,ψw1[Dn,ρ,ψ(pI(1β)(nα),ρ,ψw1Υ)](t),

    where order n1<α<n and 0β1.

    If γ=α+β(nα), then the ψ-Hilfer generalized proportional derivative HDα,β,ρ,ψw1 is equivalent to

    (HDα,β,ρ,ψw1Υ)(t)=pIβ(nα),ρ,ψw1[Dn,ρ,ψ(pI(1β)(nα),ρ,ψw1Υ)](t)=(pIβ(nα),ρ,ψw1Dγ,ρ,ψw1Υ)(t).

    Lemma 2.1. [27] Let n1<α<nN, 0<ρ1, 0β1 and n1<γ<n such that γ=α+nβαβ. If ΥC([w1,w2],R) and I(nγ,ρ,ψ)w1Cn([w1,w2],R), then

    (pIα,ρ,ψw1HDα,β,ρ,ψw1Υ)(t)=Υ(t)nj=1eρ1ρ(ψ(t)ψ(w1))(ψ(t)ψ(w1))γjργjΓ(γj+1)(pIjγ,ρ,ψw1f)(w1).

    Lemma 2.2. Let 1<α1,α2<2, 0β1,β21, γi=αi+βi(2αi), i=1,2, ¨Λ0 and z,wC([w1,w2],R). Then the pair (k1,k2) is the solution of the coupled system

    {(HDα1,β1,ρ,ψw1k1)(t)+λ1(HDα11,β1,ρ,ψw1k1)(t)=z(t),(HDα2,β2,ρ,ψw1k2)(t)+λ2(HDα21,β2,ρ,ψw1k2)(t)=w(t),k1(w1)=0,k1(w2)=ni=1ηik2(ξi)+mj=1ζjpIΦj,ρ,ψw1k2(θj),k2(w1)=0,k2(w2)=rk=1kk1(ϱk)+ql=1ΘlpIυl,ρ,ψw1k1(ϑl), (2.1)

    if and only if

    k1(t)=eρ1ρ(ψ(t)ψ(w1))¨Λργ11Γ(γ1)(ψ(t)ψ(w1))γ11[Y1(ni=1ηipIα2,ρ,ψw1w(ξi)λ2ni=1ηipI1,ρ,ψw1k2(ξi)+mj=1ζjpIα2+Φj,ρ,ψw1w(θj)λ2mj=1ζjpI1+Φj,ρ,ψw1k2(θj)pIα1,ρ,ψw1z(w2)+λ1pI1,ρ,ψw1k1(w2))+X2(rk=1kpIα1,ρ,ψw1z(ϱk)λ1rk=1kpI1,ρ,ψw1k1(ϱk)+ql=1ΘlpIα1+υl,ρ,ψw1z(ϑl)λ1ql=1ΘlpI1+υl,ρ,ψw1k1(ϑl)pIα2,ρ,ψw1w(w2)+λ2pI1,ρ,ψw1k2(w2))]+pIα1,ρ,ψw1z(t)λ1pI1,ρ,ψw1k1(t) (2.2)

    and

    k2(t)=eρ1ρ(ψ(t)ψ(w1))¨Λργ21Γ(γ2)(ψ(t)ψ(w1))γ21[Y2(ni=1ηipIα2,ρ,ψw1w(ξi)λ2ni=1ηipI1,ρ,ψw1k2(ξi)+mj=1ζjpIα2+Φj,ρ,ψw1w(θj)λ2mj=1ζjpI1+Φj,ρ,ψw1k2(θj)pIα1,ρ,ψw1z(w2)+λ1pI1,ρ,ψw1k1(w2))+X1(rk=1kpIα1,ρ,ψw1z(ϱk)λ1rk=1kpI1,ρ,ψw1k1(ϱk)+ql=1ΘlpIα1+υl,ρ,ψw1z(ϑl)λ1ql=1ΘlpI1+υl,ρ,ψw1k1(ϑl)pIα2,ρ,ψw1w(w2)+λ2pI1,ρ,ψw1k2(w2))]+pIα2,ρ,ψw1w(t)λ2pI1,ρ,ψw1k2(t), (2.3)

    where ¨Λ=X1Y1X2Y2,

    X1=eρ1ρ(ψ(w2)ψ(w1))ργ11Γ(γ1)(ψ(w2)ψ(w1))γ11,X2=ni=1ηieρ1ρ(ψ(ξi)ψ(w1))ργ21Γ(γ2)(ψ(ξi)ψ(w1))γ21+mj=1ζjeρ1ρ(ψ(θj)ψ(w1))ρΦj+γ21Γ(Φj+γ2)(ψ(θj)ψ(w1))Φj+γ21,Y1=rk=1keρ1ρ(ψ(ϱk)ψ(w1))ργ11Γ(γ1)(ψ(ϱk)ψ(w1))γ11+ql=1Θleρ1ρ(ψ(ϑl)ψ(w1))ρυl+γ11Γ(υl+γ1)(ψ(ϑl)ψ(w1))υl+γ11,Y2=eρ1ρ(ψ(w2)ψ(w1))ργ21Γ(γ2)(ψ(w2)ψ(w1))γ21. (2.4)

    Proof. Let the pair (k1,k2) be the solution of the system (2.1). We take the Riemann-Liouville integrals to Eq (2.1),

    {pIα1,ρ,ψw1[(HDα1,β1,ρ,ψw1k1)(t)+λ1(HDα11,β1,ρ,ψw1k1)(t)]=pIα1,ρ,ψw1z(t),pIα2,ρ,ψw1[(HDα2,β2,ρ,ψw1k2)(t)+λ2(HDα21,β2,ρ,ψw1k2)(t)]=pIα2,ρ,ψw1w(t). (2.5)

    Then, applying Lemma 2.1 with n=2 to Eq (2.5), we get

    k1(t)=pIα1,ρ,ψw1z(t)λ1pI1,ρ,ψw1k1(t)+eρ1ρ(ψ(t)ψ(w1))ργ11Γ(γ1)(ψ(t)ψ(w1))γ11(pI1γ1,ρ,ψw1x)(w1)+eρ1ρ(ψ(t)ψ(w1))ργ12Γ(γ11)(ψ(t)ψ(w1))γ12(pI2γ1,ρ,ψw1x)(w1)=pIα1,ρ,ψw1z(t)λ1pI1,ρ,ψw1k1(t)+c0eρ1ρ(ψ(t)ψ(w1))ργ11Γ(γ1)(ψ(t)ψ(w1))γ11+c1eρ1ρ(ψ(t)ψ(w1))ργ12Γ(γ11)(ψ(t)ψ(w1))γ12 (2.6)

    and

    k2(t)=pIα2,ρ,ψw1w(t)λ2pI1,ρ,ψw1k2(t)+eρ1ρ(ψ(t)ψ(w1))ργ21Γ(γ2)(ψ(t)ψ(w1))γ21(pI1γ2,ρ,ψw1y)(w1)+eρ1ρ(ψ(t)ψ(w1))ργ22Γ(γ21)(ψ(t)ψ(w1))γ22(pI2γ2,ρ,ψw1y)(w1)=pIα2,ρ,ψw1w(t)λ2pI1,ρ,ψw1k2(t)+c2eρ1ρ(ψ(t)ψ(w1))ργ21Γ(γ2)(ψ(t)ψ(w1))γ21+c3eρ1ρ(ψ(t)ψ(w1))ργ22Γ(γ21)(ψ(t)ψ(w1))γ22, (2.7)

    where

    c0=(pI1γ1,ρ,ψw1x)(w1),c1=(pI2γ1,ρ,ψw1x)(w1),c2=(pI1γ2,ρ,ψw1y)(w1)andc3=(pI2γ2,ρ,ψw1y)(w1).

    From the conditions k1(w1)=0 and k2(w1)=0 we get c1=0 and c3=0, since γ1[α1,2] and γ2[α2,2] (see [27]), and Eqs (2.6) and (2.7) are reduced to

    k1(t)=pIα1,ρ,ψw1z(t)λ1pI1,ρ,ψw1k1(t)+c0eρ1ρ(ψ(t)ψ(w1))ργ11Γ(γ1)(ψ(t)ψ(w1))γ11, (2.8)
    k2(t)=pIα2,ρ,ψw1w(t)λ2pI1,ρ,ψw1k2(t)+c2eρ1ρ(ψ(t)ψ(w1))ργ21Γ(γ2)(ψ(t)ψ(w1))γ21. (2.9)

    From the boundary conditions

    k1(w2)=ni=1ηik2(ξi)+mj=1ζjpIΦj,ρ,ψw1k2(θj)

    and

    k2(w2)=rk=1kk1(ϱk)+ql=1ΘlpIυl,ρ,ψw1k1(ϑl),

    we get

    pIα1,ρ,ψw1z(w2)λ1pI1,ρ,ψw1k1(w2)+c0eρ1ρ(ψ(w2)ψ(w1))ργ11Γ(γ1)(ψ(w2)ψ(w1))γ11=ni=1ηipIα2,ρ,ψw1w(ξi)λ2ni=1ηipI1,ρ,ψw1k2(ξi)+c2ni=1ηieρ1ρ(ψ(ξi)ψ(w1))ργ21Γ(γ2)(ψ(ξi)ψ(w1))γ21+mj=1ζjpIα2+Φj,ρ,ψw1w(θj)λ2mj=1ζjpI1+Φj,ρ,ψw1k2(θj)+c2mj=1ζjeρ1ρ(ψ(θj)ψ(w1))ρΦj+γ21Γ(Φj+γ2)(ψ(θj)ψ(w1))Φj+γ21 (2.10)

    and

    pIα2,ρ,ψw1w(w2)λ2pI1,ρ,ψw1k2(w2)+c2eρ1ρ(ψ(w2)ψ(w1))ργ21Γ(γ2)(ψ(w2)ψ(w1))γ21=rk=1kpIα1,ρ,ψw1z(ϱk)λ1rk=1kpI1,ρ,ψw1k1(ϱk)+c0rk=1keρ1ρ(ψ(ϱk)ψ(w1))ργ11Γ(γ1)(ψ(ϱk)ψ(w1))γ11+ql=1ΘlpIα1+υl,ρ,ψw1z(ϑl)λ1ql=1ΘlpI1+υl,ρ,ψw1k1(ϑl)+c0ql=1Θleρ1ρ(ψ(ϑl)ψ(w1))ρυl+γ11Γ(υl+γ1)(ψ(ϑl)ψ(w1))υl+γ11. (2.11)

    From Eqs (2.10) and (2.11), by using the notations (2.4) we get the system

    X1c0X2c2=M, (2.12)
    Y2k2c0+Y1c2=N, (2.13)

    where

    M=ni=1ηipIα2,ρ,ψw1w(ξi)λ2ni=1ηipI1w1k2(ξi)+mj=1ζjpIα2+Φj,ρ,ψw1w(θj)λ2mj=1ζjpI1+Φjw1k2(θj)pIα1,ρ,ψw1v(w2)+λ1pI1w1k1(w2),N=rk=1kpIα1,ρ,ψw1v(ϱk)λ1rk=1kpI1w1k1(ϱk)+ql=1ΘlpIα1+υl,ρ,ψw1v(ϑl)λ1ql=1ΘlpI1+υlw1k1(ϑl)pIα2,ρ,ψw1w(w2)+λ2pI1w1k2(w2).

    By solving the above system, we obtain the constants c0 and c2 as

    c0=Y1M+X2NX1Y1X2Y2andc2=Y2M+X1NX1Y1X2Y2.

    Now substitute the values of c0 and c2 into Eqs (2.8) and (2.9) and yield Eqs (2.2) and (2.3), as desired. We can prove the converse of the lemma by direct computation.

    In this section, we prove the existence and uniqueness results for the problem (1.4) by using three fixed ponit theorems.

    First, we defined the spaces

    X={k1|k1(t)C([w1,w2],R)}

    with the norm

    k1=sup{|k1(t)|,t[w1,w2]},

    and

    Y={k2|k2(t)C([w1,w2],R)}

    with the norm

    k2=sup{|k2(t)|,t[w1,w2]}.

    Then it is well known that (X,) and (Y,) are Banach apaces. Obviously, the product space of X×Y endowed with norm (k1,k2)=k1+k2 for (k1,k2)X×Y is a Banach space.

    In view of the Lemma 2.2, we define an operator T: X×YX×Y by

    T(k1,k2)(t)=(T1(k1,k2)(t),T2(k1,k2)(t)),

    where

    T1(k1,k2)(t)=eρ1ρ(ψ(t)ψ(w1))¨Λργ11Γ(γ1)(ψ(t)ψ(w1))γ11[Y1(ni=1ηipIα2,ρ,ψw1Υ2(ξi,k1(ξi),k2(ξi))λ2ni=1ηipI1,ρ,ψw1k2(ξi)+mj=1ζjpIα2+Φj,ρ,ψw1Υ2(θj,k1(θj),k2(θj))λ2mj=1ζjpI1+Φj,ρ,ψw1k2(θj)pIα1,ρ,ψw1Υ1(w2,k1(w2),k2(w2))+λ1pI1,ρ,ψw1k1(w2))+X2(rk=1kpIα1,ρ,ψw1Υ1(ϱk,k1(ϱk),k2(ϱk))λ1rk=1kpI1,ρ,ψw1k1(ϱk)+ql=1ΘlpIα1+υl,ρ,ψw1Υ1(ϑl,k1(ϑl),k2(ϑl))λ1ql=1ΘlpI1+υl,ρ,ψw1k1(ϑl)pIα2,ρ,ψw1Υ2(w2,k1(w2),k2(w2))+λ2pI1,ρ,ψw1k2(w2))]+pIα1,ρ,ψw1Υ1(t,k1(t),k2(t))λ1pI1,ρ,ψw1k1(t)

    and

    T2(k1,k2)(t)=eρ1ρ(ψ(t)ψ(w1))¨Λργ21Γ(γ2)(ψ(t)ψ(w1))γ21[Y2(ni=1ηipIα2,ρ,ψw1Υ2(ξi,k1(ξi),k2(ξi))λ2ni=1ηipI1,ρ,ψw1k2(ξi)+mj=1ζjpIα2+Φj,ρ,ψw1Υ2(θj,k1(θj),k2(θj))λ2mj=1ζjpI1+Φj,ρ,ψw1k2(θj)pIα1,ρ,ψw1Υ1(w2,k1(w2),k2(w2))+λ1pI1,ρ,ψw1k1(w2))+X1(rk=1kpIα1,ρ,ψw1Υ1(ϱk,k1(ϱk),k2(ϱk))λ1rk=1kpI1,ρ,ψw1k1(ϱk)+ql=1ΘlpIα1+υl,ρ,ψw1Υ1(ϑl,k1(ϑl),k2(ϑl))λ1ql=1ΘlpI1+υl,ρ,ψw1k1(ϑl)pIα2,ρ,ψw1Υ1(w2,k1(w2),k2(w2))+λ2pI1,ρ,ψw1k2(w2))]+pIα2,ρ,ψw1Υ2(t,k1(t),k2(t))λ2pI1,ρ,ψw1k2(t).

    Then, we introduce the following notation for computational convenience.

    Notation 3.1. Let Ai,Bi,Ci for i=1,2 be the constants:

    A1=(ψ(w2)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1((ψ(w2)ψ(w1))α1ρα1Γ(α1+1))+X2(rk=1|k|(ψ(ϱk)ψ(w1))α1ρα1Γ(α1+1)+ql=1|Θl|(ψ(ϑl)ψ(w1))α1+υlρα1+υlΓ(α1+υl+1))]+(ψ(w2)ψ(w1))α1ρα1Γ(α1+1),B1=(ψ(w2)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1(ni=1|ηi|(ψ(ξi)ψ(w1))α2ρα2Γ(α2+1)+mj=1|ζj|(ψ(θj)ψ(w1))α2+Φjρα2+ΦjΓ(α2+Φj+1))+X2((ψ(w2)ψ(w1))α2ρα2Γ(α2+1))],C1=(ψ(w2)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1(|λ2|ni=1|ηi|(ψ(ξi)ψ(w1))ρ+|λ1|(ψ(w2)ψ(w1))ρ+|λ2|mj=1|ζj|(ψ(θj)ψ(w1))1+Φjρ1+ΦjΓ(2+Φj))+X2(|λ1|rk=1|k|(ψ(ϱk)ψ(w1))ρ+|λ2|(ψ(w2)ψ(w1))ρ+|λ1|ql=1|Θl|(ψ(ϑl)ψ(w1))1+υlρ1+υlΓ(1+υl+1))]+|λ1|(ψ(w2)ψ(w1))ρ,A2=(ψ(w2)ψ(w1))γ21|¨Λ|ργ21Γ(γ2)[Y2((ψ(w2)ψ(w1))α1ρ+α1Γ(α1+1))+X1(rk=1|k|(ψ(ϱk)ψ(w1))α1ρα1Γ(α1+1)+ql=1|Θl|(ψ(ϑl)ψ(w1))α1+υlρα1+υlΓ(α1+υl+1)+(ψ(w2)ψ(w1))α2ρα2Γ(α2+1))],B2=(ψ(w2)ψ(w1))γ21|¨Λ|ργ21Γ(γ2)[Y2(ni=1|ηi|(ψ(ξi)ψ(w1))α2ρα2Γ(α2+1)+mj=1|ζj|(ψ(θj)ψ(w1))α2+Φjρα2+ΦjΓ(α2+Φj+1))]+(ψ(w2)ψ(w1))α2ρα2Γ(α2+1),C2=(ψ(w2)ψ(w1))γ21|¨Λ|ργ21Γ(γ2)[Y2(|λ2|ni=1|ηi|(ψ(ξi)ψ(w1))ρ+|λ2|mj=1|ζj|(ψ(θj)ψ(w1))1+Φjρ1+ΦjΓ(2+Φj)+|λ1|(ψ(w2)ψ(w1))ρ)+X1(|λ1|rk=1|k|(ψ(ϱk)ψ(w1))ρ+|λ1|ql=1|Θl|(ψ(ϑl)ψ(w1))1+υlρ1+υlΓ(2+υl)+|λ2|(ψ(w2)ψ(w1))ρ)]+|λ2|(ψ(w2)ψ(w1))ρ.

    Now we prove our first existence result via Leray-Schauder alternative [28].

    Theorem 3.1. Let Υ1,Υ2: [w1,w2]×R2R be continuous functions. Suppose that:

    (H1) There exist ui,vi0 for i=1,2 and u0,v0>0 such that for each k1,k2R, t[w1,w2],

    |Υ1(t,k1,k2)|u0+u1|k1|+u2|k2|,|Υ2(t,k1,k2)|v0+v1|k1|+v2|k2|.

    If

    (A1+A2)u1+(B1+B2)v1+(C1+C2)<1

    and

    (A1+A2)u2+(B1+B2)v2+(C1+C2)<1,

    where the constants Ai,Bi,Ci for i=1,2 are defined in the Notation 3.1, then the problem (1.4) has at lest one solution on [w1,w2].

    Proof. Since f and g are continuous functions, then T is a continuous operator. We prove that TBr is uniformly bounded, where Br is the closed ball

    Br={(k1,k2)X×Y:(k1,k2)r}.

    For all (k1,k2)Br, by (H1) we have

    |Υ1(t,k1,k2)|u0+u1|k1|+u2|k2|u0+u1k1+u2k2u0+(u1+u2)r:=P1

    and similarly

    |Υ2(t,k1,k2)|v0+(v1+v2)r:=P2.

    So, for any k1,k2Br, we have

    |T1(k1,k2)(t)|1|¨Λ|ργ11Γ(γ1)(ψ(t)ψ(w1))γ11[Y1(ni=1|ηi|pIα2,ρ,ψw1|Υ2(ξi,k2(ξi),k1(ξi))|+|λ2|ni=1|ηi|pI1,ρ,ψw1|k2(ξi)|+mj=1|ζj|pIα2+Φj,ρ,ψw1|Υ2(θj,k2(θj),k1(θj))|+|λ2|mj=1|ζj|pI1+Φj,ρ,ψw1|k2(θj)|+pIα1,ρ,ψw1|Υ1(w2,k1(w2),k2(w2))|+|λ1|pI1w1|k1(w2)|)+X2(rk=1|k|pIα1,ρ,ψw1|Υ1(ϱk,k1(ϱk),k2(ϱk))|+|λ1|rk=1|k|pI1,ρ,ψw1|k1(ϱk)|+ql=1|Θl|pIα1+υl,ρ,ψw1|Υ1(ϑl,k1(ϑl),k2(ϑl))|+|λ1|ql=1|Θl|pI1+υl,ρ,ψw1|k1(ϑl)|+pIα2,ρ,ψw1|Υ2(w2,k2(w2),k1(w2))|+|λ2|pI1,ρ,ψw1|k2(w2)|)]+pIα1,ρ,ψw1|Υ1(t,k1(t),k2(t))|+|λ1|pI1,ρ,ψw1|k1(t)|(ψ(w2)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1(P2ni=1|ηi|(ψ(ξi)ψ(w1))α2ρα2Γ(α2+1)+P2mj=1|ζj|(ψ(θj)ψ(w1))α2+Φjρα2+ΦjΓ(α2+Φj+1)+|λ2|k2ni=1|ηi|(ψ(ξi)ψ(w1))ρ+|λ2|k2mj=1|ζj|(ψ(θj)ψ(w1))1+Φjρ1+ΦjΓ(2+Φj)+P1(ψ(w2)ψ(w1))α1ρα1Γ(α1+1)+|λ1|k1(ψ(w2)ψ(w1))ρ)+X2(P1rk=1|k|(ψ(ϱk)ψ(w1))α1ρα1Γ(α1+1)+|λ1|k1rk=1|k|(ψ(ϱk)ψ(w1))ρ+|λ1|k1ql=1|Θl|(ψ(ϑl)ψ(w1))1+υlρ1+υlΓ(1+υl+1)+P1ql=1|Θl|(ψ(ϑl)ψ(w1))α1+υlρα1+υlΓ(α1+υl+1)+P2(ψ(w2)ψ(w1))α2ρα2Γ(α2+1)+|λ2|k2(ψ(w2)ψ(w1))ρ)]+P1(ψ(w2)ψ(w1))α1ρα1Γ(α1+1)+|λ1|k1(ψ(w2)ψ(w1))ρP1A1+P2B1+rC1,

    and hence

    T1(k1,k2)P1A1+P2B1+rC1.

    In the same way, we can obtain that

    T2(k1,k2)P1A2+P2B2+rC2.

    Consequently,

    T(k1,k2)(A1+A2)P1+(B1+B2)P2+(C1+C2)r.

    Therefore, the set TBr is uniformly bounded.

    Next, it is proven that TBr is equicontinuous. Let (k1,k2)Br and t1,t2[w1,w2] with t1<t2. Then, we have

    |T1(k1,k2)(t2)T1(k1,k2)(t1)|(ψ(t2)ψ(w1))γ11(ψ(t1)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1(ni=1|ηi|pIα2,ρ,ψw1|Υ2(ξi,k2(ξi),k1(ξi))|+|λ2|ni=1|ηi|pI1,ρ,ψw1|k2(ξi)|+mj=1|ζj|pIα2+Φj,ρ,ψw1|Υ2(θj,k2(θj),k1(θj))|+|λ2|mj=1|ζj|pI1+Φj,ρ,ψw1|k2(θj)|+pIα1,ρ,ψw1|Υ1(w2,k1(w2),k2(w2))|+|λ1|pI1,ρ,ψw1|k1(w2)|)+X2(rk=1|k|pIα1,ρ,ψw1|Υ1(ϱk,k1(ϱk),k2(ϱk))|+|λ1|rk=1|k|pI1,ρ,ψw1|k1(ϱk)|+ql=1|Θl|pIα1+υl,ρ,ψw1|Υ1(ϑl,k1(ϑl),k2(ϑl))|+|λ1|ql=1|Θl|pI1+υl,ρ,ψw1|k1(ϑl)|+pIα2,ρ,ψw1|Υ2(w2,k2(w2),k1(w2))|+|λ2|pI1w1|k2(w2)|)]+1ρα1Γ(α1)|t1w1[(ψ(t2)ψ(s))α1(ψ(t1)ψ(s))α1]ψ(s)Υ1(x,k1(s),k2(s)ds|+1ρα1Γ(α1)|t2t1(ψ(t2)ψ(s))α1ψ(s)Υ1(x,k1(s),k2(s)ds|+|λ1||pI1,ρ,ψw1k1(t2)pI1,ρ,ψw1k1(t1)|(ψ(t2)ψ(w1))γ11(ψ(t1)ψ(w1))γ11|¨Λ|ργ11Γ(γ1)[Y1(P2ni=1|ηi|(ψ(ξi)ψ(w1))α2ρα2Γ(α2+1)+|λ2|rni=1|ηi|(ψ(ξi)ψ(w1))ρ+P2mj=1|ζj|(ψ(θj)ψ(w1))α2+Φjρα2+ΦjΓ(α2+Φj+1)+|λ2|rmj=1|ζj|(ψ(θj)ψ(w1))1+Φjρ1+ΦjΓ(2+Φj)+P1(ψ(w2)ψ(w1))α1ρα1Γ(α1+1)+|λ1|r(ψ(w2)ψ(w1))ρΓ(2))+X2(P1rk=1|k|(ψ(ϱk)ψ(w1))α1ρα1Γ(α1+1)+|λ1|rrk=1|k|(ψ(ϱk)ψ(w1))ρ+P1ql=1|Θl|(ψ(ϑl)ψ(w1))α1+ϑlρα1+ϑlΓ(α1+ϑl+1)+|λ1|rql=1|Θl|(ψ(ϑl)ψ(w1))1+ϑlρ1+ϑlΓ(2+ϑl)+P2(ψ(w2)ψ(w1))α2ρα2Γ(α2+1)+|λ2|r(ψ(w2)ψ(w1))ρ)] +P1ρα1Γ(α1+1)[(ψ(t2)ψ(w1))α1(ψ(t1)ψ(w1))α1+2(ψ(t2)ψ(t1))α1]+|λ1|ρr(ψ(t2)ψ(t1)).

    Then, we obtain that

    |T1(k1,k2)(t2)T1(k1,k2)(t1)|0,

    when t2t1, independently of k1 and k2. Similarly,

    |T2(k1,k2)(t2)T2(k1,k2)(t1)|0,

    as t2t1. Therefore TBr is equicontinuous on [w1,w2]. From the above three steps and Arzelaˊ-Ascoli theorem, we conclude that T is completely continuous.

    Let

    U = \{(k_1, k_2) \in X\times Y : (k_1, k_2) = \mu T(k_1, k_2), \, 0\leq \mu \leq 1 \}.

    We prove that U is bounded. Let

    (k_1, k_2) \in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R})

    be any solution of (k_1, k_2) = \mu T(k_1, k_2). For each t\in [\mathtt{w}_1, \mathtt{w}_2] , we have

    k_1(t) = \mu T_1(k_1, k_2), \; k_2(t) = \mu T_2(k_1, k_2).

    Then

    \begin{eqnarray*} \nonumber |k_1(t)|& = & \mu |T_1(k_1, k_2)(t)|\\ \nonumber &\leq &|T_1(k_1, k_2)(t)|\\\nonumber &\leq & \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[ Y_1\Big( (v_0+v_1|k_1|+v_2|k_2|)\sum\limits_{i = 1}^{n} |\eta_{i}|\; \frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))^{\alpha_2}}{\rho^{\alpha_2}\Gamma(\alpha_2 +1)} \\\nonumber &&+ (v_0+v_1|k_1|+v_2|k_2| )\sum\limits_{j = 1}^{m} |\zeta_{j}|\; \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{\alpha_{2}+\Phi_{j}}}{\rho^{\alpha_{2}+\Phi_{j}}\Gamma(\alpha_{2}+\Phi_{j}+1)}\\\nonumber && + |\lambda_{2}|\|k_2\| \sum\limits_{i = 1}^{n} |\eta_{i}|\; \frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))}{\rho} + |\lambda_{2}|\|k_2\|\sum\limits_{j = 1}^{m}|\zeta_{j}| \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{1+\Phi_{j}}}{\rho^{1+\Phi_{j}}\Gamma(2+\Phi_{j})}\\\nonumber &&+(u_0+u_1|k_1|+u_2|k_2|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\; \|k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg)\\\nonumber &&+X_2\Big( (u_0+u_1|k_1|+u_2|k_2|)\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} \\\nonumber &&+ |\lambda_{1}|\; \|k_1\|\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))}{\rho} +|\lambda_{1}|\; \|k_1\|\sum\limits_{l = 1}^{q} |\Theta_{l}|\; \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{1+\upsilon_{l}}}{\rho^{1+\upsilon_{l}}\Gamma(2+\upsilon_{l})} \\\nonumber &&+(u_0+u_1|k_1|+u_2|k_2|)\sum\limits_{l = 1}^{q}|\Theta_{l}|\; \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{\alpha_{1}+\upsilon_{l}}}{\rho^{\alpha_{1}+\upsilon_{l}}\Gamma(\alpha_{1}+\upsilon_{l}+1)} \\\nonumber &&+(v_0+v_1|k_1|+v_2|k_2|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{2}}}{ \rho^{\alpha_{2}}\Gamma(\alpha_{2}+1)} + |\lambda_{2}|\|k_2\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg) \Bigg]\\\nonumber && +(u_0+u_1|k_1|+u_2|k_2|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\; \|k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \\ & \leq& (u_0+u_1\|k_1\|+u_2\|k_2\|)\mathcal{A}_1+ (v_0+v_1\|k_1\|+v_2\|k_2\|)\mathcal{B}_1+(\|k_1\| +\|k_2\|)\mathcal{C}_1, \end{eqnarray*}

    and thus

    \begin{eqnarray} \|k_1\|\le (u_0+u_1\|k_1\|+u_2\|k_2\|)\mathcal{A}_1+ (v_0+v_1\|k_1\|+v_2\|k_2\|)\mathcal{B}_1+(\|k_1\| +\|k_2\|)\mathcal{C}_1.\; \; \; \; \; \end{eqnarray} (3.1)

    Similarly, we have

    \begin{eqnarray} \|k_2\| \leq (u_0+u_1\|k_1\|+u_2\|k_2\|)\mathcal{A}_2+ (v_0+v_1\|k_1\|+v_2\|k_2\|)\mathcal{B}_2+(\|k_1\| +\|k_2\|)\mathcal{C}_2. \; \; \; \; \; \end{eqnarray} (3.2)

    Thus we obtain

    \begin{eqnarray*} \label{Contra} \|k_1\| + \|k_2\| &\leq& (\mathcal{A}_1+\mathcal{A}_2 ) u_0+(\mathcal{B}_1+\mathcal{B}_2 )v_0 +[(\mathcal{A}_1 + \mathcal{A}_2 )u_1 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_1+(\mathcal{C}_1 +\mathcal{C}_2)]\|k_1\|\\ &&+[(\mathcal{A}_1 + \mathcal{A}_2 )u_2 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_2+(\mathcal{C}_1 +\mathcal{C}_2 )]\|k_2\|. \end{eqnarray*}

    This imply that,

    \begin{eqnarray*} \|(k_1, k_2)\| \leq \frac{(\mathcal{A}_1+\mathcal{A}_2 ) u_0+(\mathcal{B}_1+\mathcal{B}_2 )v_0}{1- P^{*}}, \end{eqnarray*}

    where

    P^{*} = \max\{(\mathcal{A}_1 + \mathcal{A}_2 )u_1 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_1+(\mathcal{C}_1 +\mathcal{C}_2), (\mathcal{A}_1 + \mathcal{A}_2 )u_2 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_2+(\mathcal{C}_1 +\mathcal{C}_2 ) \}.

    Then, the set U is bounded. Therefore, by Leray-Schauder alternative the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2].

    Now, we prove the second existence of results by applying Krasnosel'skii point theorem [29].

    Theorem 3.2. Suppose \Upsilon_1, \Upsilon_2 : [\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R} are continuous functions. In addition we assume that:

    (H_2) There exist positive functions \varphi_1, \varphi_2 \in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R}^{+}), such that

    \begin{eqnarray*} |\Upsilon_1(t, k_1, k_2)| \leq \varphi_1(t), \, \, |\Upsilon_2(t, k_1, k_2)| \leq \varphi_2(t),\; {\text{for all}} \; t \in [\mathtt{w}_1, \mathtt{w}_2]. \end{eqnarray*}

    If

    \begin{equation} \mathcal{C}_1 +\mathcal{C}_2 < 1, \end{equation} (3.3)

    then, the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2].

    Proof. First, we separate the operator T as

    T_1(k_1, k_2)(t) = T_{11}(k_1, k_2)(t) + T_{12}(k_1, k_2)(t),
    T_{2}(k_1, k_2)(t) = T_{21}(k_1, k_2)(t) +T_{22}(k_1, k_2)(t)\

    with

    \begin{eqnarray*} \nonumber T_{11}(k_1, k_2)(t) & = & \frac{e^{\frac{\rho-1}{\rho} (\psi(t) -\psi(\mathtt{w}_1))}}{{\ddot\Lambda}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}(\psi(t)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}\Bigg[ Y_1\Big( \sum\limits_{i = 1}^{n} \eta_{i}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} \Upsilon_2(\xi_i, k_1(\xi_i), k_2(\xi_i)) \\ &&+\sum\limits_{j = 1}^{m} \zeta_{j}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{2}+\Phi_{j}, \rho, \psi}} \Upsilon_2(\theta_{j}, k_1(\theta_{j}), k_2(\theta_{j})) -{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} \Upsilon_1(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2)) \bigg)\\ \nonumber &&+X_2\Big( \sum\limits_{k = 1}^{r} \aleph_{k}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} \Upsilon_1(\varrho_{k}, k_1(\varrho_{k}), k_2(\varrho_{k}) )\\ &&+\sum\limits_{l = 1}^{q}\Theta_{l}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{1}+\upsilon_{l}, \rho, \psi}} \Upsilon_1(\vartheta_{l}, k_1(\vartheta_{l}), k_2(\vartheta_{l})) -{^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} \Upsilon_2(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2)) \bigg) \Bigg]\\ &&+{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} \Upsilon_1(t, k_1(t), k_2(t)), \; \; t\in [\mathtt{w}_1, \mathtt{w}_2], \\ T_{12}(k_1, k_2)(t) & = & \frac{e^{\frac{\rho-1}{\rho} (\psi(t) -\psi(\mathtt{w}_1))}}{{\ddot\Lambda}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}(\psi(t)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}\Bigg[Y_1\Big(- \lambda_{2}\sum\limits_{i = 1}^{n} \eta_{i}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_2(\xi_i) \\ \nonumber &&- \lambda_{2}\sum\limits_{j = 1}^{m} \zeta_{j}\; {^{p}I_{\mathtt{w}_1}^{1+\Phi_{j}, \rho, \psi}} k_2(\theta_{j}) + \lambda_{1}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_1(\mathtt{w}_2) \bigg)\\ \nonumber &&+X_2\Big(- \lambda_{1}\sum\limits_{k = 1}^{r} \aleph_{k}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_1(\varrho_{k}) - \lambda_{1}\sum\limits_{l = 1}^{q} \Theta_{l}\; {^{p}I_{\mathtt{w}_1}^{1+\upsilon_{l}, \rho, \psi }} k_1(\vartheta_{l}) \\ \nonumber &&+ \lambda_{2}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_2(\mathtt{w}_2)\bigg) \Bigg]- \lambda_{1}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_1(t), \; \; t\in [\mathtt{w}_1, \mathtt{w}_2], \\ \nonumber T_{21}(k_1, k_2)(t)& = &\frac{e^{\frac{\rho-1}{\rho} (\psi(t) -\psi(\mathtt{w}_1))} }{\ddot{\Lambda}\rho^{\gamma_{2}-1} \Gamma(\gamma_{2})}(\psi(t)-\psi(\mathtt{w}_1))^{\gamma_{2}-1}\Bigg[ Y_2\big(\sum\limits_{i = 1}^{n} \eta_{i}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} \Upsilon_2(\xi_i, k_1(\xi_i), k_2(\xi_i)) \\ \nonumber && +\sum\limits_{j = 1}^{m} \zeta_{j}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{2}+\Phi_{j}, \rho, \psi}} \Upsilon_2(\theta_{j}, k_1(\theta_{j}), k_2(\theta_{j})) -{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} \Upsilon_1(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2)) \bigg)\\ \nonumber && +X_1\Big( \sum\limits_{k = 1}^{r} \aleph_{k}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} \Upsilon_1(\varrho_{k}, k_1(\varrho_{k}), k_2(\varrho_{k})) +\sum\limits_{l = 1}^{q}\Theta_{l}\; {^{p}I_{\mathtt{w}_1}^{\alpha_{1}+\upsilon_{l}, \rho, \psi}} \Upsilon_1(\vartheta_{l}, k_1(\vartheta_{l}), k_2(\vartheta_{l})) \\ \nonumber &&\; -{^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} \Upsilon_1(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2)) \bigg) \Bigg]+{^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} \Upsilon_2(t, k_1(t), k_2(t)), \; \; t\in [\mathtt{w}_1, \mathtt{w}_2], \\ \nonumber T_{22}(k_1, k_2)(t) & = & \frac{e^{\frac{\rho-1}{\rho} (\psi(t) -\psi(\mathtt{w}_1))} }{\ddot{\Lambda}\rho^{\gamma_{2}-1} \Gamma(\gamma_{2})}(\psi(t)-\psi(\mathtt{w}_1))^{\gamma_{2}-1}\Bigg[ Y_2\big(- \lambda_{2}\sum\limits_{i = 1}^{n} \eta_{i}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_2(\xi_i)\\ &&- \lambda_{2}\sum\limits_{j = 1}^{m} \zeta_{j}\; {^{p}I_{\mathtt{w}_1}^{1+\Phi_{j}, \rho, \psi}} k_2(\theta_{j}) + \lambda_{1}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_1(\mathtt{w}_2) \bigg)\\ \nonumber &&+X_1\Big( - \lambda_{1}\sum\limits_{k = 1}^{r} \aleph_{k}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_1(\varrho_{k})- \lambda_{1}\sum\limits_{l = 1}^{q} \Theta_{l}\; {^{p}I_{\mathtt{w}_1}^{1+\upsilon_{l}, \rho, \psi }} k_1(\vartheta_{l})\\ &&+ \lambda_{2}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} k_2(\mathtt{w}_2)\bigg) \Bigg]- \lambda_{2}\; {^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}k_2(t)}, \; \; t\in [\mathtt{w}_1, \mathtt{w}_2]. \end{eqnarray*}

    We claim that TB_r \subset B_{r} where

    B_r = \{(k_1, k_2)\in X\times Y: \| (k_1, k_2) \|\leq r \}.

    We set

    \sup\limits_{t\in[\mathtt{w}_1, \mathtt{w}_2]} \varphi_i(t) = \|\varphi_i \|

    for i = 1, 2 and choose

    \begin{equation} \nonumber r \geq \frac{\| \varphi_1 \|(\mathcal{A}_1+\mathcal{A}_2)+\| \varphi_2 \|(\mathcal{B}_1+\mathcal{B}_2)}{1-(\mathcal{C}_1 +\mathcal{C}_2)}. \end{equation}

    Let (k_1, k_2), (\bar k_1, \bar k_2)\, \in B_{r}. Acconding the proof of Theorem 3.1, the following inequalities are obtained

    \begin{eqnarray*} |T_{11}(k_1, k_2)(t) + T_{12}(\bar k_1, \bar k_2)(t)| &\leq& \| \varphi_1 \|\mathcal{A}_1+\| \varphi_2 \|\mathcal{B}_1+r\mathcal{C}_1, \\ \nonumber |T_{21}(k_1, k_2)(t) + T_{22}(\bar k_1, \bar k_2)(t)| &\leq& \| \varphi_1 \|\mathcal{A}_2+\| \varphi_2 \|\mathcal{B}_2+r\mathcal{C}_2, \end{eqnarray*}

    and therefore

    \| T_1(k_1, k_2) + T_{2}(k_1, k_2) \|\leq \| \varphi_1 \|(\mathcal{A}_1+\mathcal{A}_2)+\| \varphi_2 \|(\mathcal{B}_1+\mathcal{B}_2)+ r(\mathcal{C}_1+\mathcal{C}_2 )\leq r.

    Hence,

    T_{11}(k_1, k_2)(t) +T_{12}(\bar k_1, \bar k_2)(t) \subset B_r

    and

    T_{21}(k_1, k_2)(t) +T_{22}(\bar k_1, \bar k_2)(t) \subset B_r.

    Consider the operators T_{11} and T_{21}. By continuity of f and g, T_{11} and T_{21} are continuous operators. For any (k_1, k_2) \in B_r, we have

    \begin{eqnarray*} |T_{11}(k_1, k_2)(t)| &\le& \frac{(\psi(t)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[ Y_1\Big( \sum\limits_{i = 1}^{n} |\eta_{i}|{^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} |\Upsilon_2(\xi_i, k_1(\xi_i), k_2(\xi_i))| \\ &&+\sum\limits_{j = 1}^{m} |\zeta_{j}|{^{p}I_{\mathtt{w}_1}^{\alpha_{2}+\Phi_{j}, \rho, \psi}} |\Upsilon_2(\theta_{j}, k_1(\theta_{j}), k_2(\theta_{j}))| +{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} |\Upsilon_1(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2))| \bigg)\\ \nonumber &&+X_2\Big( \sum\limits_{k = 1}^{r} |\aleph_{k}|{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} |\Upsilon_1(\varrho_{k}, k_1(\varrho_{k}), k_2(\varrho_{k}) )|+\sum\limits_{l = 1}^{q}|\Theta_{l}| {^{p}I_{\mathtt{w}_1}^{\alpha_{1}+\upsilon_{l}, \rho, \psi}} |\Upsilon_1(\vartheta_{l}, k_1(\vartheta_{l}), k_2(\vartheta_{l}))| \\ &&+{^{p}I_{\mathtt{w}_1}^{\alpha_{2}, \rho, \psi}} |\Upsilon_2(\mathtt{w}_2, k_1(\mathtt{w}_2), k_2(\mathtt{w}_2))| \bigg) \Bigg] +{^{p}I_{\mathtt{w}_1}^{\alpha_{1}, \rho, \psi}} |\Upsilon_1(t, k_1(t), k_2(t))|\\ &\leq& \| \varphi_1 \|\mathcal{A}_1+\| \varphi_2 \|\mathcal{B}_1. \end{eqnarray*}

    In a similar way, we can get

    \begin{eqnarray*} |T_{21}(k_1, k_2)(t)| \leq \| \varphi_1\| \mathcal{A}_2+ \| \varphi_2 \| \mathcal{B}_2. \end{eqnarray*}

    Hence, we obtain that

    \begin{eqnarray*} \|(T_{11}, T_{21})(k_1, k_2)\| \leq \| \varphi_1\| (\mathcal{A}_1+ \mathcal{A}_2) +\| \varphi_2 \|( \mathcal{B}_1+ \mathcal{B}_2), \end{eqnarray*}

    which yields that (T_{11}, T_{12})B_r is uniformly bounded. For any t_{1}, \, t_{2} \in [\mathtt{w}_1, \mathtt{w}_2], t_2 > t_1 and for all (k_1, k_2) \in B_r the operators (T_{11}, T_{12})B_r are equicontinuous by the proof of Theorem 3.1.

    Lastly, it is proven that the operators T_{12} and T_{22} are contraction mappings. For all (k_1, k_2), (\bar k_1, \bar k_2) \in B_{r}, we have:

    \begin{eqnarray*} && |T_{12}(\bar k_1, \bar k_2)(t) - T_{12}(k_1, k_2)(t)| \leq \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[ Y_1\Big( |\lambda_{2}|\sum\limits_{i = 1}^{n} |\eta_{i}|{^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} |\bar k_2(\xi_i)-k_2(\xi_i) | \\ \nonumber &&\; \; \; +| \lambda_{2}|\sum\limits_{j = 1}^{m} |\zeta_{j}|{^{p}I_{\mathtt{w}_1}^{1+\Phi_{j}, \rho, \psi}} |\bar k_2(\theta_{j})-k_2(\theta_{j})| +| \lambda_{1}|{^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}}| \bar k_1(\mathtt{w}_2)-k_1(\mathtt{w}_2)| \bigg)\\ \nonumber &&\; \; \; +X_2\Big( |\lambda_{1}|\; \sum\limits_{k = 1}^{r} |\aleph_{k}|{^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} |\bar k_1(\varrho_{k})-k_1(\varrho_{k})| +|\lambda_{1}|\sum\limits_{l = 1}^{q}| \Theta_{l}|{^{p}I_{\mathtt{w}_1}^{1+\upsilon_{l}, \rho, \psi }} |\bar k_1(\vartheta_{l})-k_1(\vartheta_{l})| \\ \nonumber &&\; \; \; + |\lambda_{2}|{^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} |\bar k_2(\mathtt{w}_2)-k_2(\mathtt{w}_2)| \bigg) \Bigg]+ |\lambda_{1}|{^{p}I_{\mathtt{w}_1}^{1, \rho, \psi}} |\bar k_1(t)-k_1(t)|\\ && \leq \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[ Y_1\Big( |\lambda_{2}|\|\bar k_2-k_2\| \sum\limits_{i = 1}^{n} |\eta_{i}|\frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))}{\rho}\\ \nonumber &&\; \; \; + |\lambda_{2}|\|\bar k_2-k_2\|\sum\limits_{j = 1}^{m}|\zeta_{j}| \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{1+\Phi_{j}}}{\rho^{1+\Phi_{j}}\Gamma(2+\Phi_{j})} +|\lambda_{1}|\| \bar k_1-k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg)\\ \nonumber &&\; \; \; +X_2\Big( |\lambda_{1}|\|\bar k_1-k_1\|\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))}{\rho} +|\lambda_{1}|\|\bar k_1-k_1\|\sum\limits_{l = 1}^{q} |\Theta_{l}| \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{1+\upsilon_{l}}}{\rho^{1+\upsilon_{l}}\Gamma(1+\upsilon_{l}+1)}\\ \nonumber &&\; \; \; +|\lambda_{2}|\|\bar k_2-k_2\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg) \Bigg] + |\lambda_{1}|\|\bar k_1-k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho}\\ &&\leq \mathcal{C}_1 (\| \bar k_1-k_1\| + \|\bar k_2-k_2\|). \end{eqnarray*}

    Additionally, we also obtain that

    \begin{eqnarray*} && |T_{21}(\bar k_1, \bar k_2)(t) - T_{21}(k_1, k_2)(t)| \leq \mathcal{C}_2(\| \bar k_1-k_1\| + \|\bar k_2-k_2\|). \end{eqnarray*}

    Combining the above inequalities, we have

    \begin{eqnarray*} \|(T_{11}, T_{21})(k_1, k_2) \| \leq (\mathcal{C}_1 + \mathcal{C}_{2})\Big(\| \bar k_1-k_1\| +\|\bar k_2-k_2\|\big). \end{eqnarray*}

    We have (T_{11}, T_{21}) is a contraction. Consequently, by Krasnosel'skii's fixed point theorem, the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2].

    Banach's fixed point theorem [30] is applied to obtain our uniqueness and existence result.

    Theorem 3.3. Let \Upsilon_1, \Upsilon_2 : [\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R} such that:

    (H_3) There exist positive constants L_1, L_2, such that, for all t\in [\mathtt{w}_1, \mathtt{w}_2] and o_i, \bar{o}_i \in \mathbb{R}, i = 1, 2, we have

    \begin{eqnarray*} |\Upsilon_1(t, o_2, \bar{o}_2) - \Upsilon_1(t, o_1, \bar{o}_1)| &\leq& L_1 (|o_2-o_1|+|\bar{o}_2-\bar{o}_1|), \\ |\Upsilon_2(t, o_2, \bar{o}_2) - \Upsilon_2(t, o_1, \bar{o}_1)| &\leq& L_2 (|o_2-o_1|+|\bar{o}_2-\bar{o}_1|). \end{eqnarray*}

    Then, the problem (1.4) has a unique solution, provided that

    \begin{eqnarray} L_1(\mathcal{A}_1+\mathcal{A}_2)+L_2(\mathcal{B}_1+\mathcal{B}_2)+\mathcal{C}_1+\mathcal{C}_2 < 1. \end{eqnarray} (3.4)

    Proof. Let

    \sup\limits_{t\in[\mathtt{w}_1, \mathtt{w}_2]} |\Upsilon_1(t, 0, 0)| = M_1 < \infty,
    \sup\limits_{t\in[\mathtt{w}_1, \mathtt{w}_2]} |\Upsilon_2(t, 0, 0)| = M_2 < \infty

    and

    B_r = \{(k_1, k_2)\in X\times Y: \| (k_1, k_2) \|\leq r \}

    with

    \begin{eqnarray*} \label{rBanach} r \geq \frac{M_1(\mathcal{A}_1+\mathcal{A}_2)+M_2(\mathcal{B}_1+\mathcal{B}_2)}{1- [L_1(\mathcal{A}_1+\mathcal{A}_2)+L_2(\mathcal{B}_1+\mathcal{B}_2)+\mathcal{C}_1+\mathcal{C}_2]}. \end{eqnarray*}

    For all (k_1, k_2) \in B_r and t\in [\mathtt{w}_1, \mathtt{w}_2]. By applying (H_3), we obtain the following inequalities

    \begin{eqnarray*} |\Upsilon_1(t, k_1(t), k_2(t))| &\leq& |\Upsilon_1(t, k_1(t), k_2(t))-\Upsilon_1(t, 0, 0)|+|\Upsilon_1(t, 0, 0)|\\ &\leq& L_1 (|k_1(t)|+|k_2(t)|) + M_1\leq L_1(\|k_1\|+\|k_2\|)\le L_1r +M_1, \\ |\Upsilon_2(t, k_1(t), k_2(t))| &\leq& |\Upsilon_2(t, k_1(t), k_2(t))-\Upsilon_2(t, 0, 0)|+|\Upsilon_2(t, 0, 0)|\\ &\leq& L_2 (|k_1(t)|+|k_2(t)|) + M_2\leq L_2r +M_2. \end{eqnarray*}

    We have

    \begin{eqnarray*} |T_1(k_1, k_2)(t)| &\leq & \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[ Y_1\Big((L_2r +M_2) \sum\limits_{i = 1}^{n} |\eta_{i}|\; \frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))^{\alpha_2}}{\rho^{\alpha_2}\Gamma(\alpha_2 +1)} \\ &&+(L_2r +M_2)\sum\limits_{j = 1}^{m} |\zeta_{j}|\; \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{\alpha_{2}+\Phi_{j}}}{\rho^{\alpha_{2}+\Phi_{j}}\Gamma(\alpha_{2}+\Phi_{j}+1)}\\ && + |\lambda_{2}|\|k_2\| \sum\limits_{i = 1}^{n} |\eta_{i}|\; \frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))}{\rho} + |\lambda_{2}|\|k_2\|\sum\limits_{j = 1}^{m}|\zeta_{j}| \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{1+\Phi_{j}}}{\rho^{1+\Phi_{j}}\Gamma(2+\Phi_{j})} \\ &&+(L_1r +M_1) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\; \|k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg)\\ &&+X_2 \big( (L_1r +M_1)\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} \\ &&+ |\lambda_{1}|\; \|k_1\|\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))}{\rho} +|\lambda_{1}|\; \|k_1\|\sum\limits_{l = 1}^{q} |\Theta_{l}|\; \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{1+\upsilon_{l}}}{\rho^{1+\upsilon_{l}}\Gamma(1+\upsilon_{l}+1)} \\ &&+(L_1r +M_1)\sum\limits_{l = 1}^{q}|\Theta_{l}|\; \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{\alpha_{1}+\upsilon_{l}}}{\rho^{\alpha_{1}+\upsilon_{l}}\Gamma(\alpha_{1}+\upsilon_{l}+1)} \\ &&+(L_2r +M_2) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{2}}}{ \rho^{\alpha_{2}}\Gamma(\alpha_{2}+1)} + |\lambda_{2}|\|k_2\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg) \Bigg]\\ && +(L_1r +M_1) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\; \|k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \\ & \leq& (L_1r +M_1)\mathcal{A}_1+ (L_2r +M_2)\mathcal{B}_1+r\mathcal{C}_1. \end{eqnarray*}

    Thus,

    \| T_1(k_1, k_2)\| \leq (L_1r +M_1)\mathcal{A}_1+ (L_2r +M_2)\mathcal{B}_1+r\mathcal{C}_1.

    We also have

    \| T_2(k_1, k_2)\| \leq (L_1r +M_1)\mathcal{A}_2+ (L_2r +M_2)\mathcal{B}_2+r\mathcal{C}_2.

    Therefore

    \begin{eqnarray*} \| T(k_1, k_2)\| \leq M_1(\mathcal{A}_1+\mathcal{A}_2)+M_2(\mathcal{B}_1+\mathcal{B}_2) + [L_1(\mathcal{A}_1+\mathcal{A}_2)+L_2(\mathcal{B}_1+\mathcal{B}_2)+\mathcal{C}_1+\mathcal{C}_2]r. \end{eqnarray*}

    Hence, T(B_r) \subset B_r. Now, it is shown that (T_1, T_2) is a contraction mapping. For all (k_1, k_2), (\bar k_1, \bar k_2)\in B_r, we have

    \begin{eqnarray*} \begin{aligned} |T_{1}(\bar k_1, \bar k_2)(t) - T_{1}(k_1, k_2)(t)| & \leq \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\gamma_{1}-1}}{{|\ddot\Lambda|}\rho^{\gamma_{1}-1} \Gamma(\gamma_{1})}\Bigg[Y_1\Big(L_2 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|) \sum\limits_{i = 1}^{n} |\eta_{i}|\frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))^{\alpha_2}}{\rho^{\alpha_2}\Gamma(\alpha_2 +1)} \\\nonumber &\; \; \; \; +L_2 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|)\sum\limits_{j = 1}^{m} |\zeta_{j}| \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{\alpha_{2}+\Phi_{j}}}{\rho^{\alpha_{2}+\Phi_{j}}\Gamma(\alpha_{2}+\Phi_{j}+1)}\\\nonumber &\; \; \; \; + |\lambda_{2}|\|\bar k_2-k_2\| \sum\limits_{i = 1}^{n} |\eta_{i}|\frac{(\psi(\xi_i)-\psi(\mathtt{w}_1))}{\rho} + |\lambda_{2}|\|\bar k_2-k_2\|\sum\limits_{j = 1}^{m}|\zeta_{j}| \frac{(\psi(\theta_{j})-\psi(\mathtt{w}_1))^{1+\Phi_{j}}}{\rho^{1+\Phi_{j}}\Gamma(2+\Phi_{j})}\\\nonumber &\; \; \; \; +L_1 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\| \bar k_1-k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg)\\\nonumber &\; \; \; \; +X_2 \big( L_1 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|)\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} \\\nonumber &\; \; \; \; + |\lambda_{1}|\|\bar k_1-k_1\|\sum\limits_{k = 1}^{r} |\aleph_{k}| \frac{(\psi(\varrho_{k})-\psi(\mathtt{w}_1))}{\rho} +|\lambda_{1}| \|\bar k_1-k_1\|\sum\limits_{l = 1}^{q} |\Theta_{l}|\; \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{1+\upsilon_{l}}}{\rho^{1+\upsilon_{l}}\Gamma(2+\upsilon_{l})} \\\nonumber &\; \; \; \; +L_1 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|)\sum\limits_{l = 1}^{q}|\Theta_{l}| \frac{(\psi(\vartheta_{l})-\psi(\mathtt{w}_1))^{\alpha_{1}+\upsilon_{l}}}{\rho^{\alpha_{1}+\upsilon_{l}}\Gamma(\alpha_{1}+\upsilon_{l}+1)} \\\nonumber &\; \; \; \; +L_2 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{2}}}{ \rho^{\alpha_{2}}\Gamma(\alpha_{2}+1)} + |\lambda_{2}|\|\bar k_2-k_2\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \bigg) \Bigg]\\\nonumber &\; \; \; \; + L_1 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|) \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))^{\alpha_{1}}}{\rho^{\alpha_{1}}\Gamma(\alpha_{1}+1)} + |\lambda_{1}|\| \bar k_1-k_1\| \frac{(\psi(\mathtt{w}_2)-\psi(\mathtt{w}_1))}{\rho} \\ &\leq\big[ L_1 \mathcal{A}_1+L_2 \mathcal{B}_1+\mathcal{C}_1\big] (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|). \end{aligned} \end{eqnarray*}

    By a similar argument, we have

    \begin{eqnarray*} |T_{2}(\bar k_1, \bar k_2)(t) - T_{2}(k_1, k_2)(t)| &\leq& L_1 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|)\mathcal{A}_2 +L_2 (\|\bar k_1-k_1\|+\|\bar k_2-k_2\|)\mathcal{B}_2 \\ &&+(\|\bar k_1-k_1\| +\|\bar k_2-k_2\|)\mathcal{C}_2. \end{eqnarray*}

    Hence, we obtain that

    \begin{eqnarray*} \|T(\bar k_1, \bar k_2)-T(k_1, k_2) \| \leq [L_1(\mathcal{A}_1+ \mathcal{A}_2)+L_2(\mathcal{B}_1 +\mathcal{B}_2)+\mathcal{C}_1+\mathcal{C}_2 ](\|\bar k_1-k_1\| +\|\bar k_2-k_2\|). \end{eqnarray*}

    By assumption (3.4), the operator T is a contraction mapping. By Banach's fixed point theorem the problem (1.4) has a unique solution.

    Example 4.1. Consider the following coupled system of \psi -Hilfer genneralized proportional fractional differential equation,

    \begin{equation} \begin{aligned} \left\{\begin{array}{llll} (^{H}D_{0.1}^{\frac{3}{2}, \frac{1}{2}, \frac{3}{4}, \frac{\log(t)}{9}} k_1)(t)+ \frac{1}{12}(^{H}D_{0.1}^{\frac{3}{2} -1, \frac{1}{2}, \frac{3}{4}, \frac{\log (t)}{9}} k_1 )(t) = \Upsilon_1(t, k_1(t), k_2(t)), \; t\in\left[\frac{1}{10}, 3\right], \\[0.5cm] (^{H}D_{0.1}^{\frac{5}{4}, \frac{1}{4}, \frac{3}{4}, \frac{\log(t)}{9}} k_2)(t)+ \frac{1}{15}(^{H}D_{0.1}^{\frac{5}{4} -1, \frac{1}{4}, \frac{3}{4}, \frac{\log(t)}{9}} k_2 )(t) = \Upsilon_2(t, k_2(t), k_1(t)), \; \; t\in\left[\dfrac{1}{10}, 3\right], \\[0.5cm] k_1\left(\frac{1}{10}\right) = 0, \; k_1(3) = \frac{2}{21}\; k_2\left(\frac{3}{11}\right)+\frac{4}{23}\; k_2\left(\frac{7}{11}\right)+\dfrac{4}{23}\; {^{p}I_{0.1}^{\frac{1}{2}, \frac{3}{4}, \frac{\log(t)}{9}}} k_2\left(\dfrac{10}{11}\right)+\dfrac{4}{25}\; {^{p}I_{0.1}^{\frac{3}{2}, \frac{3}{4}, \frac{\log(t)}{9}}} k_2\left(\dfrac{8}{12}\right), \\[0.6cm] k_2\left(\frac{1}{10}\right) = 0, \; k_2(3) = \frac{1}{31}\; k_1\left(\frac{1}{6}\right)+\frac{3}{41}\; k_1\left(\frac{5}{11}\right)+\dfrac{4}{23}\; {^{p}I_{0.1}^{\frac{4}{3}, \frac{3}{4}, \frac{\log(t)}{9}}} k_1\left(\dfrac{2}{3}\right). \end{array}\right. \end{aligned} \end{equation} (4.1)

    Here, we take

    \psi(t) = \frac{\log(t)}{9}, \; \mathtt{w}_1 = 1/10, \; \mathtt{w}_2 = 3, \; \alpha_{1} = 3/2, \; \alpha_{2} = 5/4, \; \beta_{1} = 1/2, \; \beta_{2} = 1/4, \; \rho = 3/4,
    \lambda_{1} = 1/12, \; \lambda_{2} = 1/15, \; \eta_1 = 2/21, \; \eta_2 = 4/23, \; \xi_1 = 3/11, \; \xi_2 = 7/11,
    \zeta_1 = 4/23, \; \zeta_2 = 6/25, \; \theta_1 = 10/11, \; \theta_2 = 8/12, \; \Phi_1 = 1/2, \; \Phi_2 = 3/2,
    \aleph_1 = 1/31, \; \aleph_2 = 3/41, \; \varrho_1 = 1/6, \; \varrho_2 = 5/11, \; \upsilon_1 = 4/3, \; \vartheta_1 = 2/3.

    From the above data, we obtain

    k_1\approx 1.1096, \; \bar k_1 \approx 0.9391, \; k_2\approx 0.2570, \; \bar k_2\approx 0.7302, \; \ddot{\Lambda} \approx-0.4005,
    \mathcal{A}_1\approx0.3385, \; \mathcal{A}_2\approx 3.6140 , \; \mathcal{B}_1\approx0.4068 , \; \mathcal{B}_2\approx 0.6287 , \; \mathcal{C}_1\approx0.0891, \; \mathcal{C}_2\approx 0.3720.

    {\rm{(i)}} In order to illustrate Theorem 3.1, consider the functions f and g, defined by

    \begin{eqnarray} \Upsilon_1(t, k_1, k_2) & = & \frac{\cos^{2} k_1}{4} + \frac{|k_1|^{3}e^{-k_2^{2}} }{32\sqrt[5]{t}(1+2|k_1(t)|+|k_1(t)|^{2})} +\frac{e^{-10t}k_2^{7} \sin^{2} k_1 }{25(1+k_2^{6})}, \\ \Upsilon_2(t, k_1, k_2) & = & \frac{1}{12}e^{-|k_2|} +\frac{3k_1\sin^{2} |k_1k_2|}{10+2t^{2}} +\frac{3k_2\cos^{4}k_1}{\sqrt{(t+4)^{3}}}. \end{eqnarray} (4.2)

    Then, we have

    \begin{eqnarray*} |\Upsilon_1(t, k_1, k_2)| &\leq & \frac{1}{4} + \frac{1}{32\sqrt[5]{t}}|k_1| + \frac{1}{25}|k_2| , \\ |\Upsilon_2(t, k_1, k_2)| &\leq & \frac{1}{12} + \frac{3}{10+2t^{2}} |k_1|+ \frac{3}{\sqrt{(t+4)^{3}}}|k_2|. \end{eqnarray*}

    Thus, (H_1) is satisfied with

    u_0 = 1/4 , \; u_1 = 0.05, \; u_2 = 1/25, \; v_0 = 1/12 , \; v_1 = 0.2994, \; v_2 = 0.3613 .

    Then

    (\mathcal{A}_1 + \mathcal{A}_2 )u_1 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_1+(\mathcal{C}_1 +\mathcal{C}_2)\approx0.9687 < 1

    and

    (\mathcal{A}_1 + \mathcal{A}_2 )u_2 +(\mathcal{B}_{1}+\mathcal{B}_2 )v_2+(\mathcal{C}_1 +\mathcal{C}_2 )\approx0.9933 < 1.

    Thus, all assumptions of Theorem 3.1 are satisfied. Hence, the proplem (4.1), with f and g, given by (4.2), has at least one solution on [1/10, 3].

    {\rm{(ii)}} Consider now the following functions f and g ,

    \begin{eqnarray} \Upsilon_1(t, k_1, k_2)& = & \tan^{-1} \Big(\sqrt{1+k_1^{2}}\Big)+ \frac{ t^{3}+4t}{2} + \frac{\cos^{2}(k_2)}{5}\, , \\ \Upsilon_2(t, k_1, k_2)& = & \frac{(k_1k_2)^{4}}{1+(k_1k_2)^{4}}+ \sin t+\frac{e^{-|k_1|^{3}}}{3}. \end{eqnarray} (4.3)

    It is obvious to check that the above functions satisfy

    \begin{eqnarray*} |\Upsilon_1(t, k_1, k_2)| &\leq& \frac{5\pi+2}{10} + \frac{1}{2} t^{3}+2t: = \varphi_1(t) \, , \\ |\Upsilon_2(t, k_1, k_2) |&\leq& \sin t+\frac{4}{3} : = \varphi_2 (t). \end{eqnarray*}

    Then we find that \mathcal{C}_1+\mathcal{C}_2 \approx0.4610 < 1. Hence, by Theorem 3.2, the coupled system (4.1), with f and g, given by (4.3), has at lest one solution on an interval [1/10, 3] .

    \text{(iii)} To illustrate Theorem 3.3, we consider the functions f and g as

    \begin{eqnarray} \Upsilon_1(t, k_1, k_2) & = & \frac{1}{2}+ (\log t^{2}) + \frac{2(k_1)^{2}+|k_1|}{5(1+4|k_1|)} + \frac{\sin |k_2|}{10t+11}\, , \\ \Upsilon_2(t, k_1, k_2) & = & \pi+\frac{t^{2}+2t}{3} + \frac{\tan^{-1} |k_1|}{100e^{2\log 3t}+1} +\frac{k_2^{2}+|k_2|}{(7+2^{10t})(1+|k_2|)}. \end{eqnarray} (4.4)

    We have

    \begin{eqnarray*} |\Upsilon_1(t, \bar k_1, \bar k_2)- \Upsilon_1(t, k_1, k_2)| &\le& \frac{1}{10}|\bar k_1-k_1| +\frac{1}{12}|\bar k_2-k_2|\, , \\ |\Upsilon_2(t, \bar k_1, \bar k_2)- \Upsilon_2(t, k_1, k_2)| &\le& \frac{1}{10}|\bar k_1-k_1| + \frac{1}{9}|\bar k_2-k_2|, \end{eqnarray*}

    and therefore the Lipschitz condition for f and g is satisfied with L_1 = 1/10 and L_2 = 1/9. In addition, we find that

    \begin{eqnarray*} L_1(\mathcal{A}_1+\mathcal{A}_2)+L_2(\mathcal{B}_1+\mathcal{B}_2)+\mathcal{C}_1+\mathcal{C}_2\approx 0.9714 < 1. \end{eqnarray*}

    Thus, by Theorem 3.3, problem (4.1), with f and g, given by (4.4), has a unique solution on [1/10, 3].

    Example 4.2. We investigate the behavior of solutions by replacing the values of proportional constant \rho by 0.1, 0.2, \cdots, 0.9, in the following coupled linear system of \psi -Hilfer generalized proportional fractional differential equations of the form:

    \begin{equation} \begin{aligned} \left\{\begin{array}{llll} (^{H}D_{0}^{\frac{3}{2}, \frac{1}{2}, \rho, t^{2}} k_1)(t)+ \frac{1}{10}(^{H}D_{0}^{\frac{1}{2} , \frac{1}{2}, \rho, t^{2}} k_1 )(t) = e^{\frac{\rho-1}{\rho}t^{2}}\cdot (t^{2})^{-\frac{1}{2}}, \; \; t\in\left(0, 1\right], \\[0.5cm] (^{H}D_{0}^{\frac{11}{10}, \frac{1}{2}, \rho, t^{2}} k_2)(t)+ \dfrac{1}{5}(^{H}D_{0}^{\frac{1}{10} , \frac{1}{2}, \rho, t^{2}} k_2 )(t) = e^{\frac{2\rho-2}{\rho}t^{2}}\cdot (t^{2})^{-\frac{1}{2}}, \; t\in\left(0, 1\right], \\[0.5cm] k_1\left(0\right) = 0, \qquad k_1(1) = \frac{1}{50}\; k_2\left(\frac{1}{5}\right) +\dfrac{1}{5}\; {^{p}I_{0}^{\frac{3}{2}, \rho, t^{2}}} k_2\left(\frac{1}{20}\right), \\[0.6cm] k_2\left(0\right) = 0, \qquad k_2(1) = k_1\left(\frac{7}{10}\right)+\dfrac{1}{2}\; {^{p}I_{0}^{\frac{11}{5}, \rho, t^{2}}} k_1\left(\dfrac{3}{25}\right). \end{array}\right. \end{aligned} \end{equation} (4.5)

    Here, we set

    \psi(t) = t^{2}, \; \mathtt{w}_1 = 0, \; \mathtt{w}_2 = 1, \; \alpha_{1} = 3/2, \; \alpha_{2} = 11/10, \; \beta_{1} = \beta_{2}: = \beta = 1/2,
    \lambda_{1} = 1/10, \; \lambda_{2} = 1/5, \; \eta_1 = 1/50, \; \xi_1 = 1/5, \; \zeta_1 = 1/5, \; \theta_1 = 1/20,
    \Phi_1 = 3/2, \; \aleph_1 = 1, \; \varrho_1 = 7/10, \; \Theta_1 = 1/2, \; \upsilon_1 = 11/5, \; \vartheta_1 = 3/25.

    By using integrating factor technique, we can obtain

    \begin{eqnarray} k_1(t) & = & \frac{2 \Gamma(\beta)}{\rho^{\alpha_1}\Gamma(\alpha_1+\beta-1)}\int_{0}^{t} e^{[\frac{1-\rho+\lambda_{1}}{\rho}](-t^{2}+s^{2})+[\frac{\rho-1}{\rho}](s^{2})} \cdot s^{2\alpha_1-2}ds\\ &&+\frac{2c_1}{\rho^{\gamma_1} \Gamma(\gamma_1)}\int_{0}^{t} e^{[1-\frac{1}{\rho}](t^{2})+\frac{\lambda_{1}}{\rho}(s^{2}-t^{2})} \cdot s^{2\gamma_1-1}ds, \end{eqnarray} (4.6)
    \begin{eqnarray} k_2(t) & = & \frac{2 \Gamma(\beta)}{\rho^{\alpha_2}\Gamma(\alpha_2+\beta-1)}\int_{0}^{t} e^{[\frac{1-\rho+\lambda_{2}}{\rho}](-t^{2}+s^{2})+[\frac{\rho-1}{\rho}](s^{2})} \cdot s^{2\alpha_2-2}ds\\ &&+\frac{2c_3}{\rho^{\gamma_2} \Gamma(\gamma_2)}\int_{0}^{t} e^{[1-\frac{1}{\rho}](t^{2})+\frac{\lambda_{2}}{\rho}(s^{2}-t^{2})} \cdot s^{2\gamma_2-1}ds, \end{eqnarray} (4.7)

    where \gamma_{1} = 7/4 , \gamma_{2} = 31/20 and

    \begin{eqnarray} c_1 = \frac{R(M-O)+P(Q-S)}{PT-RN}, \qquad c_3 = \frac{T(M-O)+N(Q-S)}{PT-RN}, \qquad PT-RN\neq 0, \end{eqnarray} (4.8)

    with the canstants M, N, O, P, Q, R, S and T are defined by

    \begin{eqnarray*} \nonumber M & = & \frac{2\Gamma(\beta)}{\rho^{\alpha_1}\Gamma(\alpha_1+\beta-1)}\int_{0}^{1}e^{[\frac{1-\rho-\lambda_{1}}{\rho}](-s^{2})+[\frac{\rho-1}{\rho}](s^{2})} \cdot s^{2\alpha_1 +2\beta-3} ds, \\ N & = & \frac{2}{\rho^{\gamma_1} \Gamma(\gamma_1)}\int_{0}^{1} e^{[1-\frac{1}{\rho}] +\frac{\lambda_{1}}{\rho}(s^{2})} \cdot s^{2\gamma_1 -1} ds, \\ O & = & \frac{\Gamma(\beta)}{\Gamma(\alpha_2 +\beta -1)} \Bigg[\frac{2\eta_1}{\rho^{\alpha_2}}\int_{0}^{\xi_1} e^{[\frac{1-\rho+\lambda_{2}}{\rho}](\xi_1^{2}+s^{2}) + 2[\frac{\rho-1}{\rho}](s^{2})} \cdot s^{2\alpha_2+2\beta-3} ds\\\nonumber &&+ \frac{4\xi_1}{\rho^{\Phi_1+\alpha_2} \Gamma(\Phi_1)} \int_{0}^{\theta_1} \int_{0}^{r} e^{[\frac{\rho-1}{\rho}](\theta_1^{2}+r^{2})+[\frac{1-\rho+\lambda_{2}}{\rho}](-r^{2}+s^{2})+2[\frac{\rho-1}{\rho}]s^{2}} \cdot r(\theta_1^{2}-r^{2})^{\Phi_{1} -1} \cdot s^{2\alpha_2+2\beta-3} dsdr \Bigg], \\ P& = & \frac{2}{\rho^{\gamma_2}\Gamma(\gamma_2)} \Bigg[ \eta_1 \int_{0}^{\xi_1} e^{[1-\frac{1}{\rho}]\xi_1^{2}+\frac{\lambda{1}}{\rho}[s^{2}-\xi_{1}^{2}]} \cdot s^{2\gamma_2 -1}ds \\ &&+ \frac{2\zeta_1}{\rho^{\Phi_1}\Gamma(\Phi_1)}\int_{0}^{\theta_1}\int_{0}^{r} e^{[\frac{\rho-1}{\rho}](\theta_1^{2} +r^{2}) +(1-\frac{1}{\rho})r^{2} +\frac{\lambda_1}{\rho} (s^{2}-r^{2})} \cdot r(\theta_{1}^{2} -r^{2})^{\Phi_1-1}\cdot s^{2\gamma_2-1} \Bigg]dsdr, \\ Q & = & \frac{2\Gamma(\beta)}{\rho^{\alpha_2} \Gamma(\alpha_{2}+\beta-1)}\int_{0}^{1} e^{[\frac{1-\rho+\lambda_{2}}{\rho}](-s^{2}) + 2[\frac{\rho-1}{\rho}]s^{2}} \cdot s^{2\alpha_2+2\beta-3} ds, \\ R & = & \frac{2}{\rho^{\gamma_2}\Gamma(\gamma_2)}\Bigg[ \int_{0}^{1} e^{[1-\frac{1}{\rho}] +\frac{\lambda_{1}}{\rho}(s^{2})} \cdot s^{2\gamma_2-1} ds\\ &&+ \frac{2\Theta_1}{\rho^{\upsilon_1}\Gamma(\upsilon_1)}\int_{0}^{\vartheta_1}\int_{0}^{r} e^{[\frac{\rho-1}{\rho}](\vartheta_{1}^{2}-r^{2})+[1-\frac{1}{\rho}]r^{2}+\frac{\lambda_{1}}{\rho}(s^{2}-r^{2})}\cdot r(\vartheta_1^{2} -r^{2})^{\upsilon_1-1} \cdot s^{2\gamma_2-1} dsdr \Bigg], \\ S & = & \frac{\Gamma(\beta)}{\Gamma(\alpha_1+\beta-1)} \Bigg[ \int_{0}^{\varrho_1} e^{[\frac{1-\rho+\lambda_{1}}{\rho}](-\varrho_1^{2}+s^{2})+ [\frac{\rho-1}{\rho}]s^{2}} \cdot s^{2\alpha_{1}+2\beta-3} ds\\ &&+\frac{2\Theta_1}{\rho^{\upsilon_1+\alpha_2-1}\Gamma(\upsilon_1)} \int_{0}^{\vartheta_{1}} \int_{0}^{r} e^{[\frac{\rho-1}{\rho}](\vartheta_{1}^{2}-r^{2}) +[\frac{1-\rho+\lambda_{2}}{\rho}](-r^{2}-s^{2})+2[\frac{\rho-1}{\rho}]s^{2}} \cdot r(\vartheta_{1}^{2} -r^{2})^{\upsilon_1-1} \cdot s^{2\alpha_2+2\beta-3} dsdr \Bigg], \\ T & = & \frac{2}{\rho^{\gamma_1}\Gamma(\gamma_1)} \int_{0}^{\varrho_1} e^{[1-\frac{1}{\rho}]\varrho_1^{2} +\frac{\lambda_{1}}{\rho}s^{2}}\cdot s^{2\gamma_1-1}ds. \end{eqnarray*}

    For finding the analytic solutions, we use two constants c_1 and c_3 from Table 1 and substitute them in Eqs (4.6) and (4.7), respectively.

    Table 1.  Table of two constants c_1 and c_3 with varying the values of \rho .
    No. ρ c1 c3
    1 0.1 - 0.2753 - 0.9029
    2 0.2 - 0.3394 - 1.0376
    3 0.3 - 0.7134 - 1.1597
    4 0.4 - 1.0612 - 1.3129
    5 0.5 - 1.4429 - 1.4882
    6 0.6 - 1.8446 - 1.6497
    7 0.7 - 2.2518 - 1.7895
    8 0.8 - 2.6547 - 1.9073
    9 0.9 - 3.0472 - 2.0051

     | Show Table
    DownLoad: CSV

    Next, using the Matlab program, we can find the approximate analytical solutions of k_1(t) and k_2(t) with different values of \rho as 0.1, 0.2, 0.3, \cdots, 0.9 . Two graphs of k_1(t) and k_2(t) can be drawn.

    From Figure 1, if the value \rho is increasing then the value of k_1(t) is decreasing for each point t\in [0, 1] . From the Figure 2, we see that if the value of \rho increases, then the value of k_2(t) decreases for each t\in [0, 1] . The lower and upper bounds for the above two curves correspond to \rho = 0.1 and \rho = 0.9 , respectively, when the value of t increases.

    Figure 1.  The graph of solutions k_1(t) with different values of \rho .
    Figure 2.  The figraph of solutions k_2(t) with varying values of \rho .

    In this paper, we investigated a coupled system of \psi -Hilfer fractional proportional differential equations supplemented with nonlocal integro-multipoint boundary conditions. We rely on standard fixed point theorems, Banach, Krasnosel'skii and Leray-Schauder alternative to establish the desired existence and uniqueness results. The obtained theoretical results are well illustrated by numerical examples. Our results are new and contribute significialy to the existing results in the literature concerning \psi -Hilfer fractional proportional nonlocal integro-multi-point coupled systems.

    Our results are novel and contribute to the existing literature on nonlocal systems of nonlinear \psi -Hilfer generalized fractional proportional differential equations. Note that the results presented in this paper are wider in scope and produced a variety of new results as special cases. For instance, fixing the parameters in the nonlocal integro-multi-point \psi Hilfer generalized proportional fractional system in (1.4), we obtained some new results as special cases associated with the following:

    ● Nonlocal \psi -Hilfer generalized proportional fractional systems of order in (1, 2] if \zeta_j = 0, \; \Theta_{l} = 0, j = 1, 2, \cdots, m, l = 1, 2, \cdots, q.

    ● Integro-multi-point nonlocal \psi -Hilfer generalized proportional fractional systems of order in (1, 2] if \eta_i = 0, \; \aleph_{k} = 0, i = 1, 2, \cdots, n, k = 1, 2, \cdots, r.

    ● Nonlocal Integro-multi-point Hilfer generalized proportional fractional systems of order in (1, 2] if \psi (t) = t.

    ● Integro-multi-point nonlocal Hilfer generalized fractional systems of order in (1, 2] if \rho = 1.

    Furthermore, additional new results can be recorded as special cases for different combinations of the parameters \zeta_j, \Theta_{l} , j = 1, 2, \cdots, m , l = 1, 2, \cdots, q , \eta_i = 0 , \aleph_{k} = 0 , i = 1, 2, \cdots, n , k = 1, 2, \cdots, r involved in the system (1.4). For example, by taking all values where \eta_j = 0 , j = 1, 2, \cdots, m, we obtain the results for a coupled system of nonlinear \psi -Hilfer generalized proportional fractional differential equations supplemented by the following nonlocal boundary conditions:

    \begin{eqnarray*} k_1(\mathtt{w}_1)& = &0, \qquad k_1(\mathtt{w}_2) = \sum\limits_{j = 1}^{m} \zeta_{j}\; {^{p}I^{\Phi_{j}, \rho, \psi}} k_2(\theta_{j}), \\ k_2(\mathtt{w}_1)& = &0, \qquad k_2(\mathtt{w}_2) = \sum\limits_{k = 1}^{r} \aleph_{k}\; k_1(\varrho_{k}) + \sum\limits_{l = 1}^{q} \Theta_{l}\; {^{p}I^{\upsilon_{l}, \rho, \psi}} k_1(\vartheta_{l}). \end{eqnarray*}

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

    This research was funded by National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok with Contract No. KMUTNB-FF-66-11.

    The authors declare that there are no conflicts of interest.



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