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Research article

Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means

  • Received: 14 August 2021 Accepted: 20 October 2021 Published: 18 November 2021
  • MSC : 47H09, 47H10, 34A12

  • We first discuss the existence of solutions of the infinite system of (n1,n)-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations

    {Dα0+ui(ρ)+ηfi(ρ,v(ρ))=0,ρ(0,1),Dα0+vi(ρ)+ηgi(ρ,u(ρ))=0,ρ(0,1),u(j)i(0)=v(j)i(0)=0,0jn2,ui(1)=ζ10ui(ϑ)dϑ, vi(1)=ζ10vi(ϑ)dϑ,iN,

    in the sequence space of weighted means c0(W1,W2,Δ), where n3, α(n1,n], η,ζ are real numbers, 0<η<α, Dα0+ is the Riemann-Liouville's fractional derivative, and fi,gi, i=1,2,, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of η such that for each η lying in this interval, the system of (n1,n)-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

    Citation: Majid Ghasemi, Mahnaz Khanehgir, Reza Allahyari, Hojjatollah Amiri Kayvanloo. Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means[J]. AIMS Mathematics, 2022, 7(2): 2680-2694. doi: 10.3934/math.2022151

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  • We first discuss the existence of solutions of the infinite system of (n1,n)-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations

    {Dα0+ui(ρ)+ηfi(ρ,v(ρ))=0,ρ(0,1),Dα0+vi(ρ)+ηgi(ρ,u(ρ))=0,ρ(0,1),u(j)i(0)=v(j)i(0)=0,0jn2,ui(1)=ζ10ui(ϑ)dϑ, vi(1)=ζ10vi(ϑ)dϑ,iN,

    in the sequence space of weighted means c0(W1,W2,Δ), where n3, α(n1,n], η,ζ are real numbers, 0<η<α, Dα0+ is the Riemann-Liouville's fractional derivative, and fi,gi, i=1,2,, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of η such that for each η lying in this interval, the system of (n1,n)-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.



    We are interested to discuss about the existence of positive solutions of the following infinite coupled system of (n1,n)-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations (IBVP for short) in the sequence space of weighted means c0(W1,W2,Δ)

    {Dα0+ui(ρ)+ηfi(ρ,v(ρ))=0,ρ(0,1),Dα0+vi(ρ)+ηgi(ρ,u(ρ))=0,ρ(0,1),u(j)i(0)=v(j)i(0)=0,0jn2,ui(1)=ζ10ui(ϑ)dϑ, vi(1)=ζ10vi(ϑ)dϑ,iN, (1.1)

    where n3, α(n1,n], η,ζ are real numbers, 0<η<α, Dα0+ is the Riemann-Liouville's (R-L's) fractional derivative, and fi,gi, i=1,2,, are continuous and sign-changing. This kind of problems that the nonlinearity in (1.1) may change signs is mentioned as semipositone problems in the literature.

    Fractional differential equations (FDEs) occur in the various fields of biology [16], economy [20,38], engineering [24,32], physical phenomena [5,7,8,16,25], applied science, and many other fields [3,9,14,21]. Hristova and Tersian [18] solved an FDE with a different strategy, and Harjani, Lˊopez, and Sadarangani [17] solved an FDE using a fixed point approach. Now, we intend to solve an FDE by using the technique of measure of noncompactness. On the other hand, we encounter many real world problems, which can be modeled and described using infinite systems of FDEs (see [4,27,34,36,37]). In the theory of infinite system of FDEs, the measure of noncompactness (MNC) plays a significant role, which was introduced by Kuratowski [23] (see recent works [27,35,36]). The MNC has been utilized in sequence spaces for various classes of differential equations, see [2,6,11,12,13,26,29,30,35,36].

    The difference sequence spaces of weighted means λ(u,v,Δ) (λ=c0,c, and l) first have been introduced in [33]. Thereafter, Mursaleen et al. [28] constructed some estimations for the Hausdorff MNC of some matrix operators on these spaces. They also determined several classes of compact operators in such spaces. Motivated by the mentioned papers, in this work, we first discuss the existence of solutions of IBVP (1.1) in the difference sequence space of weighted means c0(W1,W2,Δ). Then, we find an interval of η such that for any η belongs to this interval, IBVP (1.1) has a positive solution. Eventually, we demonstrate an example illustrating the obtained results. Here, we preliminarily collect some definitions and auxiliary facts applied throughout this paper.

    Suppose that (Λ,) is a real Banach space containing zero element. We mean by D(z,r) the closed ball centered at z with radius r. For a nonempty subset U of Λ, the symbol ¯U denotes the closure of U and the symbol ConvU denotes the closed convex hull of U. We denote by MΛ the family of all nonempty, bounded subsets of Λ and by NΛ the family consisting of nonempty relatively compact subsets of Λ.

    Definition 1.1. [1] The function ˜μ:MΛ[0,+) is called an MNC in Λ if for any U,V1,V2MΛ, the properties (i)(v) hold:

    (i) ker˜μ={UMΛ:˜μ(U)=0} and ker˜μNΛ.

    (ii) If V1V2, then ˜μ(V1)˜μ(V2).

    (iii) ˜μ(¯U)=˜μ(ConvU)=˜μ(U).

    (iv) For each ρ[0,1], ˜μ(ρU+(1ρ)V)ρ˜μ(U)+(1ρ)˜μ(V).

    (v) If for each natural number n, Un is a closed set in MΛ, Un+1Un, and limn˜μ(Un)=0, then U=n=1Un is nonempty.

    In what follows, we mean by MY, the family of bounded subsets of the metric space (Y,d).

    Definition 1.2. [10] Suppose that (Y,d) is a metric space. Also, suppose that PMY. The Kuratowski MNC of P, which is denoted by α(P), is the infimum of the set of positive real numbers ε such that P can be covered by a finite number of sets of diameter less than to ε. Indeed,

    α(P)=inf{ε>0:Pnj=1Kj,KjY,diam(Kj)<ε (j=1,,n); nN},

    when diam(Kj)=sup{d(ς,ν):ς,νKj}.

    The Hausdorff MNC (ball MNC) of the bounded set P, which is denoted by χ(P), is defined by

    χ(P)=inf{ε>0:Pnj=1D(yj,rj),yjY,rj<ε (j=1,,n); nN}.

    Here, we quote a result contained in [10].

    Lemma 1.3. Let (Y,d) be a metric space and let P,P1,P2MY. Then

    (i) β(P)=0 if and only if P is totally bounded,

    (ii) P1P2β(P1)β(P2),

    (iii) β(¯P)=β(P),

    (iv) β(P1P2)=max{β(P1),β(P2)}.

    Besides, if Y is a normed space, then

    (v) β(P1+P2)β(P1)+β(P2),

    (vi) for each complex number ρ, β(ρP)=|ρ|β(P).

    Now, we state a version of Darbo's theorem [10], which is fundamental in our work.

    Theorem 1.4. [10] Suppose that ˜μ is an MNC in a Banach space Λ. Also, suppose that DΛ is a bounded, closed, and convex set and that S:DD is a continuous mapping. If a constant κ[0,1) exists such that

    ˜μ(S(X))κ˜μ(X)

    for any nonempty subset X of D, then S has a fixed point in the set D.

    Suppose that J=[0,s] and that Λ is a Banach space. Consider the Banach space C(J,Λ) with the norm

    zC(J,Λ):=sup{z(ρ):ρJ},zC(J,Λ).

    Proposition 1.5. [10] Suppose that ΩC(J,Λ) is equicontinuous and bounded. Then ˜μ(Ω()) is continuous on J and

    ˜μ(Ω)=supρJ˜μ(Ω(ρ)),˜μ(ρ0Ω(ϱ)dϱ)ρ0˜μ(Ω(ϱ))dϱ.

    Definition 1.6. (see [22,31]) Suppose that f:(0,)R is a continuous function. The R-L's fractional derivative of order (>0) is defined as

    D0+f(ȷ)=1Γ(n)(ddȷ)nȷ0f(ς)(ȷς)1n+dς,

    when n=[]+1 and the right-hand side is pointwise defined on (0,+).

    We terminate this section by describing the unique solution of a nonlinear FDE, which will be needed later.

    Lemma 1.7. [39] Let hC[0,1]. Then the BVP

    {D0+u(ρ)+h(ρ)=0,ρ(0,1), 2n1<n,u(j)(0)=0,j[0,n2],u(1)=ζ10u(ϱ)dϱ, (1.2)

    has a unique solution

    u(ρ)=10H(ρ,ϱ)h(ϱ)dϱ,

    when H(ρ,ϱ) is the Green's function of BVP (1.2) defined as

    H(ρ,ϱ)={ρ1(1ϱ)1(ζ+ζϱ)(ζ)(ρϱ)1(ζ)Γ(),0ϱρ1,ρ1(1ϱ)1(ζ+ζϱ)(ζ)Γ(),0ρϱ1.

    The function H(ρ,ϱ) has the following properties:

    ζρ1q(ϱ)H(ρ,ϱ)M0ρ1(ζ)Γ(),H(ρ,ϱ)M0q(ϱ),for ρ,ϱ[0,1],

    where M0=(ζ)(1)++ζ and q(ϱ)=ϱ(1ϱ)1(ζ)Γ().

    Suppose that S is the space of complex or real sequences. Any vector subspace of S is said to be a sequence space. We denote by c the space of convergent sequences and by c0 the space of null sequences.

    A complete linear metric sequence space is called an FK space if it has the property that convergence implies coordinatewise convergence. Moreover, a normed FK space is called a BK space. It is known the spaces c0 and c are BK spaces with the norm z=supkN|zk| (see [12]).

    Suppose that X and Y are sequence spaces. We denote by (X,Y) the class of infinite matrices B that map X into Y. We denote by B=(bmk)m,k=0 an infinite complex matrix and by Bm its mth row. Then we can write

    Bm(x)=k=0bmkxk and B(x)=(Bm(x))m=0.

    Thus B(X,Y) if and only if Bm(x) converges for all m and all xX and B(x)Y.

    The set

    XB={xS:B(x)X} (2.1)

    is called the matrix domain of B in X; see [19]. An infinite matrix Y=(ynl) is said to be a triangle if ynn0 and ynl=0 for each l>n. The matrix domain of a triangle Y, XY, shares many properties with the sequence space X. For instance, if X is a BK space, then XY is a BK space with the norm ZXY=YZX for each ZXY; see [15].

    Now, let W=(wk) be a sequence. The difference sequence of W is denoted by ΔW=(wkwk1). Suppose that W1=(w1k) and W2=(w2k) are the sequences of real numbers such that w1k0 and w2k0 for all k. Also, consider the triangle Y=(ynl) defined by

    (ynl)={w1n(w2lw2l+1),ln,w1nw2n,l=n,0,l>n.

    The difference sequence space of weighted means c0(W1,W2,Δ) is defined as the matrix domain of the triangle Y in the space c0. Evidently, c0(W1,W2,Δ) is a BK space with the norm defined by

    x=Y(x)=supm|Ym(x)|,xc0(W1,W2,Δ).

    Now, we describe the Hausdorff MNC χ in the space c0(W1,W2,Δ). For this purpose, we quote the following two theorems.

    Theorem 2.1. [26] Suppose that PMc0. Also, suppose that Pm:c0c0 is the operator defined by Pm(z)=(z0,z1,,zm,0,0,). Then

    χ(P)=limmsupzP(IPm)(z),

    when I is the identity operator.

    Theorem 2.2. [19] Let X be a normed sequence space. Also, let χY and χ denote the Hausdorff MNC on MXY and MX, the family of bounded sets in XY and X, respectively. Then

    χY(P)=χ(Y(P)),

    where PMXY.

    Combining these two facts gives us the following theorem.

    Theorem 2.3. Let PMc0(W1,W2,Δ). Then the Hausdorff MNC χ on the space c0(W1,W2,Δ) can be defined as the following form:

    χY(P)=χ(Y(P))=limmsupxP(IPm)(Y(x)).

    In this section, we first make some sufficient conditions to discuss the existence of solutions of IBVP (1.1) in the space c0(W1,W2,Δ). Then, we give an interval of η such that any η belongs to this interval and the infinite system (1.1) has a positive solution. Eventually, we demonstrate an example to present the effectiveness of the obtained result.

    Here, we consider some assumptions.

    (A1) Let J1=[0,1], let fi,giC(J1×R,R), and let the function K:J1×C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))c0(W1,W2,Δ)×c0(W1,W2,Δ) be defined by

    (ϱ,U,V)K(U,V)(ϱ)=((fi(ϱ,V(ϱ)))i=1,(gi(ϱ,U(ϱ)))i=1),

    such that the family of functions (K(U,V)(ϱ)) is equicontinuous at each point of the space C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)).

    (A2) For each kN and U=(ui)C(J1,c0(W1,W2,Δ)), the following inequalities hold:

    fk(ϱ,U(ϱ))pk(ϱ)uk(ϱ),
    gk(ϱ,U(ϱ))qk(ϱ)uk(ϱ),

    where pk,qk:J1R+=[0,+) are mappings and the families {pk} and {qk} are equibounded.

    (A3) Let fi, giC(J1×R+,R) and let a function θL1(J1,(0,+)) exist such that fi(ρ,Z(ρ))θ(ρ) and gi(ρ,Z(ρ))θ(ρ), for each iN, ρJ1, and nonnegative sequence (Z(ρ)) in c0(W1,W2,Δ).

    (A4) For any iN and ρJ1, let fi(ρ,U0(ρ))>0, where U0(ρ)=(u0i(ρ)) and u0i(ρ)=0 for all i and all ρ. Also, the sequence (fi(ρ,U0(ρ))) is equibounded.

    (A5) There exists σ>0 such that gi(ρ,Z(ρ))>0, where iN and (ρ,Z(ρ))J1×([0,σ]). Put

    P:=supkNsupϱJ1|pk(ϱ)|,

    and

    Q:=supkNsupϱJ1|qk(ϱ)|.

    Theorem 3.1. Assume that IBVP (1.1) fulfills the hypotheses (A1), (A2) and M0|η|(αζ)Γ(α)(Q+P)<1, then it has at least one solution.

    Proof. Let (U,V)=((ui),(vi)) be in C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)) and satisfy the initial conditions of IBVP (1.1) and let each ui and vi be continuous on J1. We define the operator T:C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))C(I,c0(W1,W2,Δ)×c0(W1,W2,Δ)) by

    T(U,V)(ρ)=((η10H(ρ,ϱ)fi(ϱ,V(ϱ))dϱ)i=1,(η10H(ρ,ϱ)gi(ϱ,U(ϱ))dϱ)i=1).

    Note that the product space C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)) is equipped with the norm

    (U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))=UC(J1,c0(W1,W2,Δ))+VC(J1,c0(W1,W2,Δ))

    for each (U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)). Then, using our assumptions for any ρJ1, we can write

    T(U,V)(ρ)c0(W1,W2,Δ)×c0(W1,W2,Δ)=(η10H(ρ,ϱ)fi(ϱ,V(ϱ))dϱ)c0(W1,W2,Δ)+(η10H(ρ,ϱ)gi(ϱ,U(ϱ))dϱ)c0(W1,W2,Δ)=|η|supn|k=1Ynk10H(ρ,ϱ)fk(ϱ,V(ϱ))dϱ|+|η|supn|k=1Ynk10H(ρ,ϱ)gk(ϱ,U(ϱ))dϱ|M0|η|(αζ)Γ(α)(supn|k=1Ynk10pk(ϱ)vk(ϱ)dϱ|+supn|k=1Ynk10qk(ϱ)uk(ϱ)dϱ|)M0|η|(αζ)Γ(α)(P+Q)(UC(J1,c0(W1,W2,Δ))+VC(J1,c0(W1,W2,Δ)))=M0|η|(P+Q)(αζ)Γ(α)(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)).

    Accordingly, we obtain

    T(U,V)C(J1,c0(W1,W2,Δ)×c0(W1,W2,Δ))M0|η|(P+Q)(αζ)Γ(α)(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)).

    It implies that

    r(M0|η|(αζ)Γ(α)(P+Q)r. (3.1)

    Let r0 denote the optimal solution of inequality (3.1). Take

    D=D((U0,U0),r0)={(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)):(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))r0, u(j)i(0)=v(j)i(0)=0,j[0,n2], ui(1)=ζ10ui(ϱ)dϱ,vi(1)=ζ10vi(ϱ)dϱ}.

    Clearly, D is bounded, closed, and convex and T is bounded on D. Now, we prove that T is continuous. Let (U1,V1) be a point in D and let ε be an arbitrary positive number. Employing assumption (A1), there exists δ>0 such that if (U2,V2)D and (U1,V1)(U2,V2)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))δ, then

    K((U1,V1))K((U2,V2))C(J1,c0(W1,W2,Δ)×c0(W1,W2,Δ))(αζ)Γ(α)εM0|η|.

    Therefore, for any ρ in I, we get

    T(U1,V1)(ρ)T(U2,V2)(ρ)c0(W1,W2,Δ)×c0(W1,W2,Δ)=((η10H(ρ,ϱ)fi(ϱ,V1(ϱ))dϱ),(η10H(ρ,ϱ)gi(ϱ,U1(ϱ))dϱ))((η10H(ρ,ϱ)fi(ϱ,V2(ϱ))dϱ),(η10H(ρ,ϱ)gi(ϱ,U2(ϱ))dϱ))c0(W1,W2,Δ)×c0(W1,W2,Δ)=(η10H(ρ,ϱ)(fi(ϱ,V1(ϱ))fi(ϱ,V2(ϱ))dϱ)c0(W1,W2,Δ)+(η10H(ρ,ϱ)(gi(ϱ,U1(ϱ))gi(ϱ,U2(ϱ))dϱ)c0(W1,W2,Δ)=|η|supn|k=1Ynk10H(ρ,ϱ)(fk(ϱ,V1(ϱ))fk(ϱ,V2(ϱ)))dϱ|+|η|supn|k=1Ynk10H(ρ,ϱ)(gk(ϱ,U1(ϱ))gk(ϱ,U2(ϱ)))dϱ|M0|η|(αζ)Γ(α)(supn|k=1Ynksupρ[0,1](fk(ϱ,V1(ϱ))fk(ϱ,V2(ϱ)))|+supn|k=1Ynksupρ[0,1](gk(ϱ,U1(ϱ))gk(ϱ,U2(ϱ)))|)=M0|η|(αζ)Γ(α)K(U1,V1)K(U2,V2)C(I,c0(W1,W2,Δ)×c0(W1,W2,Δ))ε.

    Accordingly, we get

    T(U1,V1)T(U2,V2)C(J1,c0(W1,W2,Δ)×c0(W1,W2,Δ))ε.

    Thus, F is continuous.

    Next, we show that T(U,V) is continuous on the open interval (0,1). To this aim, let ρ1(0,1) and ε>0 be arbitrary. By applying the continuity of H(ρ,ϱ) with respect to ρ, we are able to find δ=δ(ρ1,ε)>0 such that if |ρρ1|<δ, then

    |H(ρ,ϱ)H(ρ1,ϱ)|<ε|η|(P+Q)(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)).

    We can write

    T(U,V)(ρ)T(U,V)(ρ1)c0(W1,W2,Δ)×c0(W1,W2,Δ)=(η10(H(ρ,ϱ)H(ρ1,ϱ))fi(ϱ,V(ϱ))dϱ)c0(W1,W2,Δ)+(η10(H(ρ,ϱ)H(ρ1,ϱ))gi(ϱ,U(ϱ))dϱ)c0(W1,W2,Δ)=|η|supn|k=1Ynk10(H(ρ,ϱ)H(ρ1,ϱ))fk(ϱ,V(ϱ))dϱ|+|η|supn|k=1Ynk10(H(ρ,ϱ)H(ρ1,ϱ))gk(ϱ,U(ϱ))dϱ||η|Pε(P+Q)|η|(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))(supn|k=1Ynksupρ[0,1]Vk(ρ)|)+|η|Qε(P+Q)|η|(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))(supn|k=1Ynksupρ[0,1]Uk(ρ)|)(P+Q)ε(P+Q)(U,V)C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ))×(UC(J1,c0(W1,W2,Δ))+VC(J1,c0(W1,W2,Δ)))=ε.

    Eventually, we are going to show that T:DD fulfills the conditions of Theorem 1.4. Due to Proposition 1.5 and Theorem 2.3, for any nonempty subset X1×X2D, we obtain

    ˜μ(T(X1×X2))=supρJ1sup(U,V)X1×X2˜μ(T(U,V)(ρ))=supρ[0,1]sup(U,V)X1×X2˜μ((η10H(ρ,ϱ)fi(ϱ,V(ϱ))dϱ),(η10H(ρ,ϱ)gi(ϱ,U(ϱ))dϱ))=|η|supρ[0,1]limrsupVX2supn>r|k=1Ynk10H(ρ,ϱ)fk(ϱ,V(ϱ))dϱ|+|η|supρ[0,1]limrsupUX1supn>r|k=1Ynk10H(ρ,ϱ)gk(ϱ,U(ϱ))dϱ|M0|η|P(αζ)Γ(α)supρ[0,1]limrsupVX2supn>r|k=1Ynkvk(ρ)|+M0|η|Q(αζ)Γ(α)supρ[0,1]limrsupUX1supn>r|k=1Ynkuk(ρ)|=M0|η|(αζ)Γ(α)(P+Q)˜μ(X1×X2).

    Using Theorem 1.4, we conclude that T has a fixed point in D, and hence IBVP (1.1) admits at least one solution in C(J1,c0(W1,W2,Δ))×C(J1,c0(W1,W2,Δ)).

    We are now in a position to discuss about the existence of positive solutions of IBVP (1.1) in the space c0(W1,W2,Δ). To this end, consider the following IBVP

    {Dα0+xi(ρ)=η(fi(ρ,(yi(ρ)K(ρ)))+θ(ρ)),ρ(0,1),Dα0+yi(ρ)=η(gi(ρ,(xi(ρ)K(ρ)))+θ(ρ)),ρ(0,1),x(j)i(0)=y(j)i(0)=0,j[0,n2],xi(1)=ζ10xi(ϑ)dϑ, yi(1)=ζ10yi(ϑ)dϑ,iN, (3.2)

    where

    Z(ρ)={Z(ρ),Z(ρ)0,0,Z(ρ)<0,

    and K(ρ)=η10H(ρ,ϑ)θ(ϑ)dϑ, which is the solution of the BVP

    {Dα0+K(ρ)=ηθ(ρ),ρ(0,1),K(j)(0)=0,j[0,n2],K(1)=ζ10K(ϑ)dϑ.

    We are going to show that there exists a solution (x,y)=((xi),(yi)) for IBVP (1.1) with xi(ρ)K(ρ) and yi(ρ)K(ρ) for each iN and for each ρ[0,1].

    Accordingly, (U,V) is a nonnegative solution of IBVP (1.1), where U(ρ)=(xi(ρ)K(ρ)) and V(ρ)=(yi(ρ)K(ρ)). Indeed, for any iN and each ρ(0,1), we have

    {Dα0+xi(ρ)=Dα0+ui(ρ)+(Dα0+K(ρ))=η(fi(ρ,v(ρ))+θ(ρ)), Dα0+yi(ρ)=Dα0+vi(ρ)+(Dα0+K(ρ))=η(gi(ρ,u(ρ))+θ(ρ)).

    It implies that

    \begin{equation*} \begin{cases} -D^{\alpha}_{0_+}u_i(\rho) = \eta(f_i(\rho, v(\rho)), &\ \\ -D^{\alpha}_{0_+}v_i(\rho) = \eta(g_i(\rho, v(\rho)). \end{cases} \end{equation*}

    Therefore, we concentrate our attention to the study of IBVP (3.2). We know that (3.2) is equal to

    x_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta, (y_i(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta,
    \begin{equation} y_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta, (x_i(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta. \end{equation} (3.3)

    In view of (3.3), we get

    \begin{equation} x_i(\rho) = \eta\int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta,(\eta\int_{0}^{1} H(\vartheta,\varsigma)g_i(\varsigma, (x_i(\varsigma)-K(\varsigma))^*)d\varsigma)^*)+\theta(\vartheta))d\vartheta. \end{equation} (3.4)

    In what follows, we demonstrate our main result.

    Theorem 3.2. Assume that IBVP (1.1) fulfills the hypotheses (A1) (A5) and \frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(Q+P) < 1 . Then there exists a positive real constant \widetilde{\eta} such that for each 0 < \eta\leq \widetilde{\eta} , IBVP (1.1) has at least one positive solution.

    Proof. Take any \delta\in(0, 1) . Regarding assumptions (A4) and (A5) , we are able to find 0 < \varepsilon < \min\{1, \sigma\} such that for each i\in\mathbb{N} , \rho\in J_1 and the nonnegative sequence \mathfrak{Z} in C(J_1, c_0(W_1, W_2, \Delta)) with \|\mathfrak{Z}\|_{C(J_1, c_0(W_1, W_2, \Delta))} < \varepsilon , we have

    f_i(\rho,\mathfrak{Z}(\rho))\geq \delta f_i(\rho,U^0(\rho)),\qquad g_i(\rho,\mathfrak{Z}(\rho)) > 0.

    Suppose that

    0 < \eta < \widetilde{\eta}: = \min\Big\{\frac{\varepsilon}{2\Upsilon\widetilde{f}(\varepsilon)},\frac{1}{Q\Upsilon}\Big\},

    where \widetilde{f}(\varepsilon) = \max\{f_i(\rho, \mathfrak{Z}(\rho))+\theta(\rho), \quad i\in\mathbb{N}, \ 0\leq \rho\leq1, \ 0\leq\|\mathfrak{Z}\|_{C(J_1, c_0(W_1, W_2, \Delta))}\leq\varepsilon\} and \Upsilon = \int_{0}^{1}M_0q(\vartheta)d\vartheta . Since {\lim_{\varsigma\rightarrow0}}\frac{\widetilde{f}(\varsigma)}{\varsigma} = +\infty and \frac{\widetilde{f}(\varepsilon)}{\varepsilon} < \frac{1}{2\Upsilon\eta}, then there exists R_0\in(0, \varepsilon) such that \frac{\widetilde{f}(R_0)}{R_0} = \frac{1}{2\Upsilon\eta}. Let

    D_0 = \{x = (x_i)\in C(J_1,c_0(W_1,W_2,\Delta)):\ \|x-K\|_{C(J_1,c_0(W_1,W_2,\Delta))} < R_0,\ x_{i}^{(j)}(0) = 0,
    0\leq j\leq n-2,\ x_i(1) = \zeta\int_{0}^{1}x_i(\vartheta)d\vartheta,\ \text{ for all } i\in\mathbb{N}\}

    Now, for any x\in D_0 and \rho\in J_1 , we have

    \begin{align*} \|(\eta&\int_{0}^{1} H(\rho,\vartheta)(g_i(\rho,(x_i(\rho)-K(\rho))^*))d\vartheta)\|_{c_0(W_1,W_2,\Delta)}\\ = &\sup\limits_{n\in\mathbb{N}}|{\sum\limits_{k = 1}^\infty} Y_{nk}\eta\int_{0}^{1} H(\rho,\vartheta)(g_k(\vartheta,(x_k(\vartheta)-K(\vartheta))^*))d\vartheta|\\ \leq& \eta \int_{0}^{1} M_0q(\vartheta) Q|{\sum\limits_{k = 1}^\infty} Y_{nk}(x_k(\vartheta)-K(\vartheta))^*d\vartheta| \\ = &\eta \int_{0}^{1} M_0q(\vartheta) Q\|x-K\|_{C(J_1,c_0(W_1,W_2,\Delta))}\\ \leq&\eta\int_{0}^{1} M_0q(\vartheta) QR_0d\vartheta \\ < &R_0 < \varepsilon. \end{align*}

    Thus, using (3.4), we deduce that

    \begin{align*} x_i(\rho) = &\eta \int_{0}^{1} H(\rho,\vartheta)(f_i(\vartheta,(\eta\int_{0}^{1} H(\rho,\varsigma)g_i(\varsigma, (x_i(\varsigma)-K(\varsigma)^*)d\varsigma)^*)+\theta(\vartheta)))d\vartheta \\ \geq& \eta \int_{0}^{1} H(\rho,\vartheta)(\delta f_i(\vartheta,U^0(\vartheta))+\theta(\vartheta))d\vartheta \\ = &\eta(\delta\int_{0}^{1} H(\rho,\vartheta)f_i(\vartheta,U^0(\vartheta))d\vartheta+\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta) d\vartheta) \\ > &\eta\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta) d\vartheta = K(\rho), \end{align*}

    for any \rho\in J_1 , and any i\in\mathbb{N}.

    Thanks to relation (3.3), we get

    \begin{eqnarray*} y_i(\rho)& = &\eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta,(x(\vartheta)-K(\vartheta))^*)+\theta(\vartheta))d\vartheta\\ & = & \eta\int_{0}^{1} H(\rho,\vartheta)(g_i(\vartheta,x(\vartheta)-K(\vartheta))+\theta(\vartheta))d\vartheta \\& > &\eta\int_{0}^{1} H(\rho,\vartheta)\theta(\vartheta)d\vartheta = K(\rho), \end{eqnarray*}

    for any \rho\in J_1 .

    Thus, if 0 < \eta\leq\widetilde{\eta} , then (x, y) is a positive solution of IBVP (3.2) with x_i(\rho)\geq K(\rho) and y_i(\rho)\geq K(\rho) for each i\in\mathbb{N} and for each \rho\in J_1.

    Let U(\rho) = (u_i(\rho)) = (x_i(\rho)-K(\rho)) and let V(\rho) = (v_i(\rho)) = (y_i(\rho)-K(\rho)) . Then (U, V) is a nonnegative solution of IBVP (1.1).

    Example 3.3. Consider the following IBVP of FDEs

    \begin{equation} \begin{cases} D^{\frac{39}{2}}_{0_+}u_i(\rho)+\frac{1}{40} {\sum_{j = i}^{+\infty}}\frac{e^{-2\rho}\big(\arctan^2(v_j(\rho)+1)+\frac{\pi}{2}\sin^2(v_j(\rho)-1)\big)\cos(\rho)}{j(j+1)(\rho+1)} = 0,& 0 < \rho < 1, \\ D^{\frac{39}{2}}_{0_+}v_i(\rho)+\frac{1}{40}{\sum_{j = i}^{+\infty}} \frac{e^{-5\rho}\big(1+u_j(\rho)+\sin^2(u_j(\rho)-1)\big)}{j^2\cosh(\rho) (2\rho+3)} = 0,& 0 < \rho < 1, \\u_i^{(j)}(0) = v_{i}^{(j)}(0) = 0,& 0\leq j\leq 18, \\ u_{i}(1) = 19.4\int_0^1 u_i(\vartheta)d\vartheta, \ v_{i}(1) = 19.4\int_0^1 v_i(\vartheta)d\vartheta, & i\in\mathbb{N},\\ \end{cases} \end{equation} (3.5)

    in the space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) . By taking \alpha = \frac{39}{2} , \eta = \frac{1}{40}, \zeta = 19.4 ,

    f_i(\rho,V(\rho)) = {\sum\limits_{j = i}^{+\infty}}\frac{e^{-2\rho}\big(\arctan^2(v_j(\rho)+1)+ \frac{\pi}{2}\sin^2(v_j(\rho)-1)\big)\cos(\rho)}{j(j+1)(\rho+1)},

    and

    g_i(\rho,U(\rho)) = {\sum\limits_{j = i}^{+\infty}} \frac{e^{-5\rho}\big(1+u_j(\rho)+\sin^2(u_j(\rho)-1)\big)}{j^2\cosh(\rho) (2\rho+3)},

    system (3.5) is a special case of IBVP (1.1). Clearly, f_i, g_i\in C(J_1\times \mathbb{R}_+^\infty, \mathbb{R}) for each i\in\mathbb{N} . It can be easily verified that condition (A1) holds. Indeed, suppose that (U, V), (U^1, V^1)\in C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) and that \varepsilon > 0 is arbitrary. Now if \|(U, V)-(U^1, V^1)\|_{C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta))}\leq \frac{6\varepsilon}{\pi^2+12\pi} , then for each \rho\in [0, 1] , we obtain

    \begin{align*} \|K&(U,V)(\rho)-K(U^1,V^1)(\rho)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|\big((f_i(\rho,V(\rho))-f_i(\rho,V^1(\rho))), (g_i(\rho,U(\rho))-g_i(\rho,U^1(\rho)))\big)\|_{c_0(W_1,W_2,\Delta)\times c_0(W_1,W_2,\Delta)}\\ = &\|(f_i(\rho,V(\rho))-f_i(\rho,V^1(\rho)))\|_{c_0(W_1,W_2,\Delta)}+\|(g_i(\rho,U(\rho))-g_i(\rho,U^1(\rho)))\|_{c_0(W_1,W_2,\Delta)}\\ = &\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}Y_{ni} {\sum\limits_{j = i}^{+\infty}}\frac{e^{-2\rho}\cos(\rho)}{j(j+1)(\rho+1)}\big((\arctan^2(v_j(\rho)+1)-\arctan^2(v_j^1(\rho)+1))\\ &+\frac{\pi}{2}(\sin^2(v_j(\rho)-1)-\sin^2(v_j^1(\rho)-1))\big)|\\ &+\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}Y_{ni} {\sum\limits_{j = i}^{+\infty}}\frac{e^{-5\rho}}{j^2\cosh(\rho) (2\rho+3)}\\&\qquad\qquad\big((1+u_j(\rho)-1-u_j^1(\rho))+ (\sin^2(u_j(\rho)-1)-\sin^2(u_j^1(\rho)-1))\big)|\\ \leq&\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}2\pi Y_{ni}(v_j(\rho)-v_j^1(\rho))|+\sup\limits_{n}|\sum\limits_{i = 1}^{\infty}\frac{\pi^2}{6}Y_{ni}(u_j(\rho)-u_j^1(\rho))|\\ \leq&(\frac{\pi^2+12\pi}{6})(\|U(\rho)-U^1(\rho)\|_{c_0(W_1,W_2,\Delta)}+\|V(\rho)-V^1(\rho)\|_{c_0(W_1,W_2,\Delta)})\\ = &(\frac{\pi^2+12\pi}{6})\|(U,V)-(U^1,V^1)\|_{C(J_1,c_0(W_1,W_2,\Delta))\times C(J_1,c_0(W_1,W_2,\Delta))}\\ \leq&\varepsilon. \end{align*}

    Also, we get

    f_i(\rho,V(\rho))\leq \pi v_i(\rho),\qquad g_i(\rho,U(\rho))\leq \frac{\pi^2}{9} u_i(\rho).

    For each natural number i and \rho\in[0, 1] , we put p_i(\rho) = \pi and q_i(\rho) = \frac{\pi^2}{9}. Thus (p_i(\rho)) and (q_i(\rho)) are equibounded on the interval I . Moreover, P = \pi and Q = \frac{\pi^2}{9}. Note that

    f_i(\rho,V(\rho))+\theta(\rho) > 0,\ \mbox{and}\ g_i(\rho,U(\rho))+\theta(\rho) > 0,

    where \theta(\rho) = \tan(\rho) for each \rho\in I . Evidently, f_i(\rho, U^0(\rho)) > 0 , the sequence (f_i(\rho, U^0(\rho))) is equibounded, and g_i(\rho, U(\rho)) > 0. Moreover, \frac{M_0|\eta|}{(\alpha-\zeta)\Gamma(\alpha)}(P+Q) = \frac{101.875\times\sqrt{\pi}}{18.5\times17.5\times\cdots\times1.5\times9} < 1 . Therefore, we conclude from Theorem 3.2 that (3.5) has a positive solution (U, V) in the space C(J_1, c_0(W_1, W_2, \Delta))\times C(J_1, c_0(W_1, W_2, \Delta)) .

    Mursaleen et al. [28] constructed a measure of noncompactness in the difference sequence space of weighted means \lambda(u, v, \Delta) . Also, a fractional differential equation was studied by Yuan [39]. Now, in this work, we discuss the existence of solutions of the infinite coupled system of (n-1, n) -type semipositone boundary value problem of nonlinear fractional differential Eq (1.1) in the difference sequence space of weighted means c_0(W_1, W_2, \Delta) .

    We would like to thank the referees for their useful comments and suggestions which have significantly improved the paper.

    The authors declare that they have no conflict of interest.



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