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Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions

  • In this paper, we studied the existence of solutions for a coupled system of nonlinear sequential proportional ψ-Hilfer fractional differential equations with multi-point boundary conditions. By using a Burton's version of the Krasnosel'ski˘i's fixed-point theorem we established sufficient conditions for the existence result. An example illustrating our main result was also provided.

    Citation: Ayub Samadi, Sotiris K. Ntouyas, Jessada Tariboon. Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions[J]. AIMS Mathematics, 2024, 9(5): 12982-13005. doi: 10.3934/math.2024633

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  • In this paper, we studied the existence of solutions for a coupled system of nonlinear sequential proportional ψ-Hilfer fractional differential equations with multi-point boundary conditions. By using a Burton's version of the Krasnosel'ski˘i's fixed-point theorem we established sufficient conditions for the existence result. An example illustrating our main result was also provided.



    Fractional differential equations (FDEs) provide many mathematical models in physics, biology, economics, and chemistry, etc [1,2,3,4]. In fact, it consists of many integrals and derivative operators of non-integer orders, which generalize the theory of ordinary differentiation and integration. Hence, a more general approach is allowed to calculus and one can say that the aim of the FDEs is to consider various phenomena by studying derivatives and integrals of arbitrary orders. For intercalary specifics about the theory of FDEs, the readers are referred to the books of Kilbas et al.[2] and Podlubny [4]. In the literature, several concepts of fractional derivatives have been represented, consisting of Riemann-Liouville, Liouville-Caputo, generalized Caputo, Hadamard, Katugampola, and Hilfer derivatives. The Hilfer fractional derivative [5] extends both Riemann-Liouville and Caputo fractional derivatives. For applications of Hilfer fractional derivatives in mathematics and physics, etc see [6,7,8,9,10,11]. For recent results on boundary value problems for fractional differential equations and inclusions with the Hilfer fractional derivative see the survey paper by Ntouyas [12]. The ψ-Riemann-Liouville fractional integral and derivative operators are discussed in [1], while the ψ-Hilfer fractional derivative is discussed in [13]. Recently, the notion of a generalized proportional fractional derivative was introduced by Jarad et al. [14,15,16]. For some recent results on fractional differential equations with generalized proportional derivatives, see [17,18].

    In [19], an existence result was proved via Krasnosel'ski˘i's fixed-point theorem for the following sequential boundary value problem of the form

    {HDα,ς,ψ[HDβ,ς,ψp(w)ϕ(w,p(w))ni=1Iνi;ψhi(w,p(w))]=Υ(w,p(w)),w[a,b],p(a)=0,HDb,ς,ψp(a)=0,p(b)=τp(ζ), (1.1)

    where HDω,ς,ψ indicates the ψ-Hilfer fractional derivative of order ω{α,β}, with 0<α1, 1<β2, 0ς<1, Iνi;ψ is the ψ-Riemann–Liouville fractional integral of order νi>0, for i=1,2,,n, hiC([0,1]×R,R), for i=1,2,,n, ϕC([0,1]×R,R{0}), ΥC([0,1]×R,R), τR and ζ(a,b). In [16], the consideration of Hilfer-type generalized proportional fractional derivative operators was initiated.

    Coupled systems of fractional order are also significant, as such systems appear in the mathematical models in science and engineering, such as bio-engineering [20], fractional dynamics [21], financial economics [22], etc. Coupled systems of FDEs with diverse boundary conditions have been the focus of many researches. In [23], the authors studied existence and Ulam-Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations. Existence and uniqueness results are derived in [24] for a coupled system of Hilfer-Hadamard fractional differential equations with fractional integral boundary conditions. Recently, in [25] a coupled system of nonlinear fractional differential equations involving the (k,ψ)-Hilfer fractional derivative operators complemented with multi-point nonlocal boundary conditions were discussed. Moreover, Samadi et al. [26] have considered a coupled system of Hilfer-type generalized proportional fractional differential equations.

    In this article, motivated by the above works, we study a coupled system of ψ-Hilfer sequential generalized proportional FDEs with boundary conditions generated by the problem (1.1). More precisely, we consider the following coupled system of nonlinear proportional ψ-Hilfer sequential fractional differential equations with multi-point nonlocal boundary conditions of the form

    {HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))ni=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)Φ2(w,p1(w),p2(w))mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2), (1.2)

    where HDν,ϑ1;ς;ψ denotes the ψ-Hilfer generalized proportional derivatives of order ν{ν1,ν2,ν3,ν4}, with parameters ϑl, 0ϑl1, l{1,2,3,4}, ψ is a continuous function on [t1,t2], with ψ(w)>0, pIη,ς,ψ is the generalized proportional integral of order η>0, η{ηi,ηj}, θ1,θ2R, ξ1,ξ2[t1,t2], Φ1,Φ2C([t1,t2]×R×R,R{0}) and Hi,Gj,Υ1,Υ2C([t1,t2]×R×R,R), for i=1,2,,n and j=1,2,,m.

    We emphasize that:

    ● We study a general system involving ψ-Hilfer proportional fractional derivatives.

    ● Our equations contain fractional derivatives of different orders as well as sums of fractional integrals of different orders.

    ● Our system contains nonlocal coupled boundary conditions.

    ● Our system covers many special cases by fixing the parameters involved in the problem. For example, taking ψ(w)=w, it will reduce to a coupled system of Hilfer sequential generalized proportional FDEs with boundary conditions, while if ς=1, it reduces to a coupled system of ψ-Hilfer sequential FDEs. Besides, by taking Φ1,Φ2=1 in the problem (1.2), then we obtain the following new coupled system of the form:

    {HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)ni=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2).

    In obtaining the existence result of the problem (1.2), first the problem (1.2) is converted into a fixed-point problem and then a generalization of Krasnosel'ski˘i's fixed-point theorem due to Burton is applied.

    The structure of this article has been organized as follows: In Section 2, some necessary concepts and basic results concerning our problem are presented. The main result for the problem (1.2) is proved in Section 3, while Section 4 contains an example illustrating the obtained result.

    In this section, we summarize some known definitions and lemmas needed in our results.

    Definition 2.1. [17,18] Let ς(0,1] and ν>0. The fractional proportional integral of order ν of the continuous function F is defined by

    pIν,ς,ψF(w)=1ςνΓ(ν)wt1eς1ς(ψ(w)ψ(s))(ψ(w)ψ(s))ν1F(s)ψ(s)ds,t1>w.

    Definition 2.2. [17,18] Let ς(0,1], ν>0, and ψ(w) is a continuous function on [t1,t2], ψ(w)>0. The generalized proportional fractional derivative of order ν of the continuous function F is defined by

    (pDν,ς,ψF)(w)=(pDn,ς,ψ)ςnνΓ(nν)wt1eς1ς(ψ(w)ψ(s))(ψ(w)ψ(s))nν1F(s)ψ(s)ds,

    where n=[ρ]+1 and [ν] denotes the integer part of the real number ν, where Dn,ς,ψ=Dς,ψDς,ψntimes.

    Now the generalized Hilfer proportional fractional derivative of order ν of function F with respect to another function ψ is introduced.

    Definition 2.3. [27] Let ς(0,1], F,ψCm([t1,t2],R) in which ψ is positive and strictly increasing with ψ(w)0 for all w[t1,t2]. The ψ-Hilfer generalized proportional fractional derivative of order ν and type ϑ for F with respect to another function ψ is defined by

    (HDν,ϑ,ς,ψF)(w)=pIϑ(nν),ς,ψ[pDn,ς,ψ(pI(1ϑ)(nν),ς,ψF)](w),

    where n1<ν<n and 0ϑ1.

    Lemma 2.4. [27] Let m1<ν<m,nN, 0<ς1, 0ϑ1 and m1<γ<m such that γ=ν+mϑνϑ. If FC([t1,t2],R) and pI(mγ,ς,ψ)FCm([t1,t2],R), then

    (pIν,ς,ψHDν,ϑ,ς,ψF)(w)=F(w)nj=1eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γjςγjΓ(γj+1)(pIjγ,ς,ψF)(t1).

    To prove the main result we need the following lemma, which concerns a linear variant of the ψ-Hilfer sequential proportional coupled system (1.2). This lemma plays a pivotal role in converting the nonlinear problem in system (1.2) into a fixed-point problem.

    Lemma 2.5. Let 0<ν1,ν31, 1<ν2,ν42, 0ϑi1, γi=νi+ϑi(1νi), i=1,3 and γj=νj+ϑj(2νj), j=2,4, Θ=M1N2M2N10, ψ is a continuous function on [t1,t2], with ψ(w)>0, and Q1,Q2C([t1,t2],R), Φ1,Φ2C([t1,t2]×R×R,R{0}) and Hi,Gj,Q1,Q2C([t1,t2]×R×R,R), for i=1,2,,n and j=1,2,,m, and pI(1γi,ς,ψ)QjCm([t1,t2],R),i=1,2,3,4,j=1,2. Then the pair (p1,p2) is a solution of the system

    {HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))ni=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Q1(w),w[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)Φ2(w,p1(w),p2(w))mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Q2(w),w[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2),

    if and only if

    p1(w)=pIν2,ς,ψΦ1(w,p1(w),p2(w))(ni=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1,ς,ψQ1(w))+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2){N2[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))
    ×(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))×(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))]+M2[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))×(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2))]} (2.1)

    and

    p2(w)=pIν4,ς,ψΦ2(w,p1(w),p2(w))(mj=1pIˉηj,ς,ψGj(w,p1(w),p2(w))+pIν3,ς,ψQ2(w))
    +eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41Θςγ41Γ(γ4){N1[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))pIν2,ς,ψΦ2(t2,p1(t2),p2(t2))×(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))]+M1[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))×(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2))]}, (2.2)

    where

    M1=eς1ς(ψ(t2)ψ(t1))(ψ(t2)ψ(t1))γ21ςγ21Γ(γ2),M2=θ1eς1ς(ψ(ξ1)ψ(t1))(ψ(ξ1)ψ(t1))γ41ςγ41Γ(γ4),N1=θ2eς1ς(ψ(ξ2)ψ(t1))(ψ(ξ2)ψ(t1))γ21ςγ21Γ(γ2),N2=eς1ς(ψ(t2)ψ(t1))(ψ(t2)ψ(t1))γ41ςγ41Γ(γ4). (2.3)

    Proof. Due to Lemma 2.4 with m=1, we get

    HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))ni=1pIηi,ς,ψHi(w,p1(w),p2(w))=pIν1;ς;ψQ1(w)+c0eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ11ςγ11Γ(γ1),HDν4,ϑ4;ς;ψp2(w)Φ2(w,p1(w),p2(w))mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))=pIν3;ς;ψQ2(w)+d0eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ31ςγ31Γ(γ3), (2.4)

    where c0,d0R. Now applying the boundary conditions

    HDν2,ϑ2;ς;ψp1(t1)=HDν4,ϑ4;ς;ψp1(t1)=0,

    we get c0=d0=0. Hence

    HDν2,ϑ2;ς;ψp1(w)=Φ1(w,p1(w),p2(w))(ni=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w)),HDν4,ϑ4;ς;ψp2(w)=Φ2(w,p1(w),p2(w))(mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w)). (2.5)

    Now, by taking the operators pIν2,ς,ψ and pIν4,ς,ψ into both sides of (2.5) and using Lemma 2.4, we get

    p1(w)=pIν2;ς;ψΦ1(w,p1(w),p2(w))(ni=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w))+c1eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21ςγ21Γ(γ2)+c2eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ22ςγ22Γ(γ21),p2(w)=pIν4;ς;ψΦ2(w,p1(w),p2(w))(mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w))+d1eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41ςγ41Γ(γ4)+d2eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ42ςγ42Γ(γ41). (2.6)

    Applying the conditions p1(t1)=p2(t1)=0 in (2.6), we get c2=d2=0 since γ2[ν2,2] and γ4[ν4,2]. Thus we have

    p1(w)=pIν2;ς;ψ(Φ1(w,p1(w),p2(w))(ni=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w)))+c1eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21ςγ21Γ(γ2),p2(w)=pIν4;ς;ψ(Φ2(w,p1(w),p2(w))(mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w)))+d1eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41ςγ41Γ(γ4). (2.7)

    In view of (2.7) and the conditions p1(t2)=θ1p2(ξ1) and p2(t2)=θ2p1(ξ2), we get

    pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))+c1eς1ς(ψ(t2)ψ(t1))(ψ(t2)ψ(t1))γ21ςγ21Γ(γ2)=θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))+d1θ1eς1ς(ψ(ξ1)ψ(t1))(ψ(ξ1)ψ(t1))γ41ςγ41Γ(γ4), (2.8)

    and

    pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+Iν3,ς,ψQ2(t2))+d1eς1ς(ψ(t2)ψ(t1))(ψ(t2)ψ(t1))γ41ςγ21Γ(γ2)=θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))+c1θ2eς1ς(ψ(ξ2)ψ(t1))(ψ(ξ2)ψ(t1))γ21ςγ21Γ(γ2). (2.9)

    Due to (2.3), (2.8), and (2.9), we have

    c1M1d1M2=M,c1N1+d1N2=N, (2.10)

    where

    M=θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2)),N=θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2)).

    By solving the above system, we conclude that

    c1=1Θ[N2M+M2N],d1=1Θ[M1N+N1M].

    Replacing the values c1 and d1 in Eq (2.7), we obtain the solutions (2.1) and (2.2). The converse is obtained by direct computation. The proof is complete.

    Let Y=C([t1,t2],R)={p:[t1,t2]Ris continuous}. The space Y is a Banach space with the norm p=supw[t1,t2]|p(w)|. Obviously, the space (Y×Y,(p1,p2)) is also a Banach space with the norm (p1,p2)=p1+p2.

    Due to Lemma 2.5, we define an operator V:Y×YY×Y by

    V(p1,p2)(w)=(V1(p1,p2)(w)V2(p1,p2)(w)), (3.1)

    where

    V1(p1,p2)(w)=pIν2,ς,ψΦ1(w,p1(w),p2(w))(ni=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1,ς,ψΥ1(w,p1(w),p2(w)))+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2){N2[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψΥ2(ξ1,p1(ξ1),p2(ξ1)))pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψΥ1(t2,p1(t2),p2(t2)))]+M2[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψΥ1(ξ2,p1(ξ1),p2(ξ2)))pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψΥ2(t2,p1(t2),p2(t2)))]},w[t1,t2],

    and

    V2(p1,p2)(w)=pIν4,ς,ψΦ2(w,p1(w),p2(w))(mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3,ς,ψΥ2(w,p1(w),p2(w)))+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41Θςγ41Γ(γ4){N1[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(mj=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψΥ2(ξ1,p1(ξ1),p2(ξ1))))pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(ni=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψΥ1(t2,p1(t2),p2(t2)))]+M1[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(ni=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψΥ1(ξ2,p1(ξ2),p2(ξ2))))pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(mj=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψΥ2(t2,p1(t2),p2(t2)))]},w[t1,t2].

    To prove our main result we will use the following Burton's version of Krasnosel'ski˘i's fixed-point theorem.

    Lemma 3.1. [28] Let S be a nonempty, convex, closed, and bounded set of a Banach space (X,) and let A:XX and B:SX be two operators which satisfy the following:

    (i) A is a contraction,

    (ii) B is completely continuous, and

    (iii) x=Ax+By,ySxS.

    Then there exists a solution of the operator equation x=Ax+Bx.

    Theorem 3.2. Assume that:

    (H1) The functions Φk:[t1,t2]×R2R{0}, Υk:[t1,t2]×R2R for k=1,2 and hi,gj:[t1,t2]×R2R for i=1,2,,n,j=1,2,,m, are continuous and there exist positive continuous functions ϕk, ωk:[t1,t2]R, k=1,2, hi:[t1,t2]R, gj:[t1,t2]R i=1,2,,nj=1,2,,m, with bounds ϕk, ωk, k=1,2, and hi, i=1,2,,m, gj,j=1,2,,m, respectively, such that

    |Φ1(w,u1,u2)Φ1(w,¯u1,¯u2)|ϕ1(w)(|u1¯u1|+|u2¯u2|),|Φ2(w,u1,u2)Φ2(w,¯u1,¯u2)|ϕ2(w)(|u1¯u1|+|u2¯u2|),|Υ1(w,u1,u2)Υ1(w,¯u1,¯u2|ω1(w)(|u1¯u1|+|u2¯u2|),|Υ2(w,u1,u2)Υ2(w,¯u1,¯u2|ω2(w)(|u1¯u1|+|u2¯u2|),|Hi(w,u1,u2)Hi(w,¯u1,¯u2)|hi(w)(|u1¯u1|+|u2¯u2|),|Gj(w,u1,u2)Gj(w,¯u1,¯u2)|gj(w)(|u1¯u1|+|u2¯u2|), (3.2)

    for all w[t1,t2] and ui,¯uiR, i=1,2.

    (H2) There exist continuous functions Fk,Lk,k=1,2, λi,μj,i=1,2,,n,j=1,2,,m such that

    |Φ1(w,u1,u2)|F1(w),|Φ2(w,u1,u2)|F2(w),|Hi(w,u1,u2)|λi(w),|Gj(w,u1,u2)|μj(w),|Υ1(w,u1,u2)|L1(w),|Υ2(w,u1,u2)|L2(w), (3.3)

    for all w[t1,t2] and u1,u2R.

    (H3) Assume that

    K:={(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)]+(N1+M1|θ2|)(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)}×[F1ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)ϕ1]+{(N2|θ1|+M2)(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)+(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)[1+(N1|θ1|+M1)(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)]}×[F2mj=1gj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)+mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)ϕ2]<1,

    where Fk=supt[t1,t2]|Fk(t)|, Lk=supt[t1,t2],k=1,2, λi=supt[t1,t2], i=1,2,,n, and μj=supt[t1,t2], j=1,2,,m.

    Then the ψ-Hilfer sequential proportional coupled system (1.2) has at least one solution on [t1,t2].

    Proof. First, we consider a subset S of Y×Y defined by S={(p1,p2)Y×Y:(p1,p2)r}, where r is given by

    r=R1+R2 (3.4)

    where

    R1=[1+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)(N2+M2|θ2|)]F1(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)×(ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1))+[N2|θ1|+M2](ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)F2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)×(mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)+mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1))

    and

    R2=[1+(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)(N1|θ1|+M1)]F2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)×(mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)+mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1))+[N1+M1|θ2|](ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)×(ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)).

    Let us define the operators:

    Hi(p1,p2)(w)=ni=1pIηi,ς,ψHi(w,p1(w),p2(w)),w[t1,t2],
    Gj(p1,p2)(w)=mj=1pI¯ηj,ς,ψGj(w,p1(w),p2(w)),w[t1,t2],
    Y1(p1,p2)(w)=pIν1,ς,ψΥ1(w,p1(w),p2(w)),w[t1,t2],
    Y2(p1,p2)(w)=pIν3,ς,ψΥ2(w,p1(w),p2(w)),w[t1,t2],

    and

    F1(p1,p2)(w)=Φ1(w,p1(w),p2(w)),w[t1,t2],
    F2(p1,p2)(w)=Φ2(w,p1(w),p2(w)),w[t1,t2].

    Then we have

    |Hi(¯p1,¯p2)(w)Hi(p1,p2)(w)|ni=1pIηi,ς,ψ|Hi(w,¯p1(w),¯p2(w))Hi(w,p1(w),p2(w))|ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)(¯p1p1+¯p2p2)

    and

    |Hi(p1,p2)(w)|ni=1pIηi,ς,ψ|Hi(w,p1(w),p2(w))|ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1).

    Also, we obtain

    |Gj(¯p1,¯p2)(w)Gj(p1,p2)(w)|mj=1pI¯ηj,ς,ψ|Gj(w,¯p1(w),¯p2(w))Gj(w,p1(w),p2(w))|mj=1gj(ψ(t2)ψ(t1))¯ηjςηiΓ(¯ηj+1)(¯p1p1+¯p2p2)

    and

    |Gj(p1,p2)(w)|mj=1pI¯ηi,ς,ψ|Hi(w,p1(w),p2(w))|mj=1μj(ψ(t2)ψ(t1))¯ηiς¯ηiΓ(¯ηi+1).

    Moreover, we have

    |Y1(¯p1,¯p2)(w)Y1(p1,p2)(w)|pIν1,ς,ψ|Υ1(w,¯p1(w),¯p2(w))Υ1(w,p1(w),p2(w))|ω1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)(¯p1p1+¯p2p2),
    |Y1(p1,p2)(w)|pIν1,ς,ψ|Υ1(w,p1(w),p2(w))|L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1),

    and

    |Y2(¯p1,¯p2)(w)Y2(p1,p2)(w)|pIν3,ς,ψ|Υ2(w,¯p1(w),¯p2(w))Υ2(w,p1(w),p2(w))|ω2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)(¯p1p1+¯p2p2),
    |Y2(p1,p2)(w)|pIν1,ς,ψ|Υ2(w,p1(w),p2(w))|L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1).

    Finally, we get

    |F1(¯p1,¯p2)(w)F1(p1,p2)(w)||Φ1(w,¯p1(w),¯p2(w))Φ1(w,p1(w),p2(w))|ϕ1(¯p1p1+¯p2p2),
    |F1(p1,p2)(w)||Φ1(w,p1(w),p2(w))|F1,

    and

    |F2(¯p1,¯p2)(w)F2(p1,p2)(w)||Φ2(w,¯p1(w),¯p2(w))Φ2(w,p1(w),p2(w))|ϕ2(¯p1p1+¯p2p2),
    |F2(p1,p2)(w)||Φ2(w,p1(w),p2(w))|F2.

    Now we split the operator V as

    V1(p1,p2)(w)=V1,1(p1,p2)(w)+V1,2(p1,p2)(w),V2(p1,p2)(w)=V2,1(p1,p2)(w)+V2,2(p1,p2)(w),

    with

    V1,1(p1,p2)(w)=pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)]+M2[θ2pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)]},V1,2(p1,p2)(w)=pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)]+M2[θ2pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)]},V2,1(p1,p2)(w)=pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41Θςγ41Γ(γ4)×{N1[θ1pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)]+M1[θ2pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)]},

    and

    V2,2(p1,p2)(w)=pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ41Θςγ41Γ(γ4)×{N1[θ1pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)]+M1[θ2pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)]}.

    In the following, we will show that the operators V1 and V2 fulfill the assumptions of Lemma 3.1. We divide the proof into three steps:

    Step 1. The operators V1,1 and V2,1 are contraction mappings. For all (p1,p2),(¯p1,¯p2)Y×Y we have

    |V1,1(¯p1,¯p2)(w)V1,1(p1,p2)(w)|(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)F1(p1,p2)(w)Hi(p1,p2)(w)|
    +(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2){N1|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w)
    F2(p1,p2)(w)Gj(p1,p2)(w)|+(N2+M2|θ2|)(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)F1(p1,p2)(w)Hi(p1,p2)(w)|
    +M2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w)F2(p1,p2)(w)Gj(p1,p2)(w)|}(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)]
    ×|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)F1(p1,p2)(w)Hi(p1,p2)(w)|+(N2|θ1|+M2)(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)ψ(t1))γ21Θςγ21×|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w)F2(p1,p2)(w)Gj(p1,p2)(w)|
    (ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)]×[|F1(¯p1,¯p2)(w)||Hi(¯p1,¯p2)(w)Hi(p1,p2)(w)|+|Hi(p1,p2)(w)||F1(¯p1,¯p2)(w)F1(p1,p2)(w)]+(N2|θ1|+M2)(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)ψ(t1))γ21Θςγ21×[|F2(¯p1,¯p2)(w)||Gj(¯p1,¯p2)(w)Gj(p1,p2)(w)|
    +|Gj(p1,p2)(w)||F2(¯p1,¯p2)(w)F2(p1,p2)(w)|](ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)]×[F1ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)(¯p1p1+¯p2p2)+ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)ϕ1(¯p1p1+¯p2p2)]
    +(N2|θ1|+M2)(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)ψ(t1))γ21Θςγ21×[F2mj=1gj(ψ(t2)ψ(t1))¯ηjςηiΓ(¯ηj+1)(¯p1p1+¯p2p2)+mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)ϕ2(¯p1p1+¯p2p2)]
    ={(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)]×[F1ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)ϕ1]+(N2|θ1|+M2)(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)×[F2mj=1gj(ψ(t2)ψ(t1))¯ηjς¯ηiΓ(¯ηj+1)+mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)ϕ2]}×(¯p1p1+¯p2p2).

    Similarly we can find

    |V2,1(¯p1,¯p2)(w)V2,1(p1,p2)(w)|{(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)[1+(N1|θ1|+M1)(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)]×[F2mj=1gj(ψ(t2)ψ(t1))¯ηjςηiΓ(¯ηj+1)+nj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)ϕ2]+(N1+M1|θ2|)(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)×[F1ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)ϕ1]}×(¯p1p1+¯p2p2).

    Consequently, we get

    (V1,1,V2,1)(¯p1,¯p2)(V1,1,V2,1)(p1,p2)K(¯p1p1+¯p2p2),

    which means that (V1,1,V2,1) is a contraction.

    Step 2. The operator V2=(V1,2,V2,2) is completely continuous on S. For continuity of V1,2, take any sequence of points (pn,qn) in S converging to a point (p,q)S. Then, by the Lebesgue dominated convergence theorem, we have

    limnV1,2(pn,qn)(w)=pIν2,ς,ψlimnF1(pn,qn)(w)limnY1(pn,qn)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2)×{N2[θ1pIν4,ς,ψlimnF2(pn,qn)(w)limnY2(pn,qn)(w)pIν2,ς,ψlimnF1(pn,qn)(w)limnY1(pn,qn)(w)]+M2[θ2pIν2,ς,ψlimnF1(pn,qn)(w)limnY1(pn,qn)(w)pIν4,ς,ψlimnF2(pn,qn)(w)limnY2(pn,qn)(w)]}=pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)+eς1ς(ψ(w)ψ(t1))(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p,q)(w)Y2(p,q)(w)pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)]+M2[θ2pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)pIν4,ς,ψF2(p,q)(w)Y2(p,q)(w)]}=V1,2(p,q)(w),

    for all w[t1,t2]. Similarly, we prove limnV2,2(pn,qn)(w)=V2,2(p,q)(w) for all w[t1,t2]. Thus V2(pn,qn)=(V1,2(pn,qn),V2,2(pn,qn)) converges to V2(p,q) on [t1,t2], which shows that V2 is continuous.

    Next, we show that the operator (V1,2,V2,2) is uniformly bounded on S. For any (p1,p2)S we have

    |V1,2(p1,p2)(w)|pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|+(ψ(w)ψ(t1))γ21Θςγ21Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(p1,p2)(w)Y2(p1,p2)(w)|+pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|+pIν4,ς,ψ|F2(p1,p2)(w)Y2(p1,p2)(w)|]}(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)×{N2|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)}:=Λ1.

    Similarly we can prove that

    |V2,2(p1,p2)(w)|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)+(ψ(t2)ψ(t1))γ41Θςγ21Γ(γ2)×{N1|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)+N1(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M1|θ2|F1ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M1(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)}:=Λ2.

    Therefore V1,2+V2,2Λ1+Λ2,(p1,p2)S, which shows that the operator (V1,2,V2,2) is uniformly bounded on S. Finally we show that the operator (V1,2,V2,2) is equicontinuous. Let τ1<τ2 and (p1,p2)S. Then, we have

    |V1,2(p1,p2)(τ2)V1,2(p1,p2)(τ1)||1ςν2Γ(ν2)τ1t1ψ(s)[(ψ(τ2)ψ(s))ν21(ψ(τ1)ψ(s))ν21]×|F1(p1,p2)(s)Y1(p1,p2)(s)|ds+1ςν2Γ(ν2)τ2τ1ψ(s)(ψ(τ2)ψ(s))ν21|F1(p1,p2)(s)Y1(p1,p2)(s)|ds|+|(ψ(τ2)ψ(t1))γ21(ψ(τ1)ψ(t1))γ21|Θςγ21Γ(γ2)W1ςν2Γ(ν2+1)ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)[|(ψ(τ2)ψ(t1))ν2(ψ(τ1)ψ(t1))ν2|+2(ψ(τ2)ψ(τ1))ν2]+|(ψ(τ2)ψ(t1))γ21(ψ(τ1)ψ(t1))γ21|Θςγ21Γ(γ2)W,

    where

    W=N2|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|F1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1).

    As τ2τ10, the right-hand side of the above inequality tends to zero, independently of (p1,p2). Similarly we have |V2,2(p1,p2)(τ2)V2,2(p1,p2)(τ1)|0 as τ2τ10. Thus (V1,2,V2,2) is equicontinuous. Therefore, it follows by the Arzelá-Ascoli theorem that (V1,2,V2,2) is a completely continuous operator on S.

    Step 3. We show that the third condition (iii) of Lemma 3.1 is fulfilled. Let (p1,p2)Y×Y be such that, for all (¯p1,¯p2)S

    (p1,p2)=(V1,1(p1,p2),V2,1(p1,p2))+(V1,2(¯p1,¯p2,V2,2(¯p1,¯p2)).

    Then, we have

    |p1(w)||V1,1(p1,p2)(w)|+|V1,2(¯p1,¯p2)(w)|pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(p1,p2)(w)Gj(p1,p2)(w)|+pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|+pIν4,ς,ψ|F2(p1,p2)(w)Gj(p1,p2)(w)|]}+pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(¯p1,¯p2)(w)Y2(¯p1,¯p2)(w)|+pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|+pIν4,ς,ψ|F2(¯p1,¯p2)(w)Y2(¯p1,¯p2)(w)|]}(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)×{N2|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)F2+N2(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+M2|θ2|F1ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+M2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)}+(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)×{N2|θ1|(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|F1ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1)}=[1+(ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)(N2+M2|θ2|)]F1(ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)×(ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1))+[N2|θ1|+M2](ψ(t2)ψ(t1))γ21Θςγ21Γ(γ2)F2(ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)×(mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)+mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1))=R1.

    In a similar way, we find

    |p2(w)||V2,1(p1,p2)(w)|+|V2,2(¯p1,¯p2)(w)|[1+(ψ(t2)ψ(t1))γ41Θςγ41Γ(γ4)(N1|θ1|+M1)](ψ(t2)ψ(t1))ν4ςν4Γ(ν4+1)F2×(mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηΓ(¯η+1)+mj=1L2(ψ(t2)ψ(t1))ν3ςν3Γ(ν3+1))+[N1+M1|θ2|](ψ(t2)ψ(t1))ν2ςν2Γ(ν2+1)F1(ψ(t2)ψ(t1))γ41ςγ41Γ(γ4)×(ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)+ni=1L1(ψ(t2)ψ(t1))ν1ςν1Γ(ν1+1))=R2.

    Adding the previous inequalities, we obtain

    p1+p2R1+R2=r.

    As (p1,p2)=p1+p2, we have that (p1,p2)r and so condition (iii) of Lemma 3.1 holds.

    By Lemma 3.1, the ψ-Hilfer sequential proportional coupled system (1.2) has at least one solution on [t1,t2]. The proof is finished.

    Let us consider the following coupled system of nonlinear sequential proportional Hilfer fractional differential equations with multi-point boundary conditions:

    {HD13,15;37;logw[HD54,25;37;logwp1(w)Φ1(w,p1(w),p2(w))2i=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w[12,72],HD23,35;37;logw[HD74,45;37;logwp1(w)Φ2(w,p1(w),p2(w))2j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w[12,72],p1(12)=HD54,25;37;logwp1(12)=0,p1(72)=25p2(32),p2(12)=HD74,45;37;logwp2(12)=0,p2(72)=23p1(52), (4.1)

    where

    2i=1pIηi,ς,ψHi(w,p1,p2)=2i=1pI2(i+1)5,37,logw(|p1|(w+i2)(i+|p1|)+|p2|(w+i3)(i+|p2|)),2j=1pI¯ηj,ς,ψGj(w,p1,p2)=2j=1pI2(j+1)7,37,logw(|p1|(w2+j2)(j+|p1|)+|p2|(w2+j3)(j+|p2|)),Φ1(w,p1,p2)=1100(10w+255)(|p1|1+|p1|+|p2|1+|p2|+12),Φ2(w,p1,p2)=25(2w+99)2(|p1|1+|p1|+|p2|1+|p2|+14),Υ1(w,p1,p2)=1w+2(|p1|3+|p1|)+12(w+1)sin|p2|+13,Υ2(w,p1,p2)=1w2+4(12tan1|p1|+|p2|2+|p2|)+15.

    Next, we can choose ν1=1/3, ν2=5/4, ν3=2/3, ν4=7/4, ϑ1=1/5, ϑ2=2/5, ϑ3=3/5, ϑ4=4/5, ς=3/7, ψ(w):=logw=logew, t1=1/2, t2=7/2, θ1=2/5, and θ2=2/3. Then, we have γ1=7/15, γ2=31/20, γ3=13/15, γ4=39/20, M10.1930945138, M20.2307306625, N10.1816223751, N20.3208292984, and Θ0.02004452646. Now, we analyse the nonlinear functions in the fractional integral terms. We have

    |Hi(w,p1,p2)Hi(w,¯p1,¯p2)|1i(w+i2)(|p1¯p1|+|p2¯p2|)

    and

    |Gj(w,p1,p2)Gj(w,¯p1,¯p2)|1j(w2+j2)(|p1¯p1|+|p2¯p2|),

    from which hi(w)=1/(i(w+i2)) and gj(w)=1/(j(w2+j2)), respectively. Both of them are bounded as

    |Hi(w,p1,p2)|2w+i2and|Gj(w,p1,p2)|2w2+j2.

    Therefore λi(w)=2/(w+i2) and μj=2/(w2+j2). Moreover, we have

    ni=1hi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)3.021061781,
    ni=1λi(ψ(t2)ψ(t1))ηiςηiΓ(ηi+1)7.281499952,
    mj=1gj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηi+1)2.776491121

    and

    mj=1μj(ψ(t2)ψ(t1))¯ηjς¯ηjΓ(¯ηi+1)7.220966978.

    For the two non-zero functions Φ1 and Φ2 we have

    |Φ1(w,p1,p2)Φ1(w,¯p1,¯p2)|1100(10w+255)(|p1¯p1|+|p2¯p2|),|Φ2(w,p1,p2)Φ2(w,¯p1,¯p2)|25(2w+99)2(|p1¯p1|+|p2¯p2|),
    |Φ1(w,p1,p2)|140(10w+255),and|Φ2(w,p1,p2)|910(2w+99)2,

    from which we get ϕ1=1/26000, ϕ2=1/25000, F1=1/10400, F2=9/100000, by setting ϕ1(w)=1/(100(10w+255)), ϕ2(w)=2/(5(2w+99)2), F1(w)=1/(40(10w+255)), and F2(w)=9/(10(2w+99)2), respectively.

    Finally, for the nonlinear functions of the right sides in problem (4.1) we have

    |Υ1(w,p1,p2)Υ1(w,¯p1,¯p2)|12(w+1)(|p1¯p1|+|p2¯p2|),|Υ2(w,p1,p2)Υ2(w,¯p1,¯p2)|12(w2+4)(|p1¯p1|+|p2¯p2|),

    which give ω1(w)=1/(2(w+1)), ω2(w)=1/(2(w2+4)) and

    |Υ1(w,p1,p2)|1w+2+12(w+1)+13:=L1(w),

    and

    |Υ2(w,p1,p2)|1w2+4(π4+1)+15:=L2(w).

    Therefore, using all of the information to compute a constant K in assumption (H3) of Theorem 3.2, we obtain

    K0.9229566975<1.

    Hence, the given coupled system of nonlinear proportional Hilfer-type fractional differential equations with multi-point boundary conditions (4.1), satisfies all assumptions in Theorem 3.2. Then, by its conclusion, there exists at least one solution (p1,p2)(w) to the problem (4.1) where w[1/2,7/2].

    In this paper, we have presented the existence result for a new class of coupled systems of ψ-Hilfer proportional sequential fractional differential equations with multi-point boundary conditions. The proof of the existence result was based on a generalization of Krasnosel'ski˘i's fixed-point theorem due to Burton. An example was presented to illustrate our main result. Some special cases were also discussed. In future work, we can implement these techniques on different boundary value problems equipped with complicated integral multi-point boundary conditions.

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

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

    Professor Sotiris K. Ntouyas is an editorial board member for AIMS Mathematics and was not involved in the editorial review or the decision to publish this article. The authors declare no conflicts of interest.



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