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Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

  • This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.

    Citation: Madeaha Alghanmi, Shahad Alqurayqiri. Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions[J]. AIMS Mathematics, 2024, 9(6): 15040-15059. doi: 10.3934/math.2024729

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  • This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.



    Differential equations of fractional order have constituted in recent years a significant field of applied research due to their ability to describe many real-world phenomena more accurately than those of integer order. As many processes and changes in nature are affected by delays when they occur, differential equations with finite and infinite delay attracted the scholars to model many applications as delayed differential equations, for example, co-infection of malaria and HIV/AIDS [1,2], predator and prey models [3,4], output feedback controller systems [5], BAM neural networks [6], neural networks with Lévy noise network-based control systems [7], HFMD model [8], etc.

    The coupled systems of different types of fractional differential equations that are associated with different types of initial and boundary conditions play an important role in mathematical modeling, as such systems occur in various problems of applied sciences such as engineering and physical applications. Therefore, there are many applications and studies dealing with coupled systems; for instance, Hu and Zhang [9] investigated the existence of solutions for a coupled system of p-Laplacian fractional differential equations with infinite-point boundary conditions by applying coincidence degree theory. In [10], the authors derive the existence and uniqueness results for a class of coupled systems involving implicit fractional differential equations with periodic boundary conditions. The existence and uniqueness of integrable solutions for a nonlinear coupled system of fractional differ-integral equations with weighted initial conditions were established in [11]. Ahmad et al. [12] obtained, with the aid of Schaefer's and Banach fixed point theorems, the existence results for a nonlinear coupled system involving different orders of both Riemann-Liouville and Caputo generalized fractional derivatives and equipped with Riemann Stieltjes type integral boundary conditions. Aljodi [13] studied a new class of coupled systems of Caputo-Fabrizio differential equations equipped with nonlocal coupled boundary conditions and established the existence and uniqueness results based on Banach and Krasnoselskii fixed point theorems. Zhao [14] discussed the existence criteria for the solutions of a class of coupled systems involving Atangana-Baleanu fractional order differential equations with (p1,p1)-Laplacian operators and proved the stability of the obtained system by means of generalized Ulam-Hyers stability. In [15], the authors investigated a new class of coupled implicit systems involving ϱ-fractional derivatives of different orders and anti-periodic boundary conditions. Zhao [16] established important results related to the stability, existence, and uniqueness of the solutions of a coupled system involving Atangana-Baleanu-Caputo fractional differential equations with a Laplacian operator and impulses using generalized Ulam-Hyers for the stability and by an F-contractive operator and a fixed-point theorem on metric space for the uniqueness.

    The theory of differential equations with infinite delay has gained much attention since the early 1970s, and has developed rapidly since that time. General axioms and theorems were established to deal with this kind of equation. The appropriate selection of the phase space F was very important in the study of differential equations with infinite delay, which was identified by specific axioms, see [17,18,19]. For more details on the theoretical aspects of differential equations with unbounded delay, we refer the reader to the works [17,20,21,22,23]. Although there has recently been considerable work on fractional differential equations with infinite delay; see, for instance [24,25,26,27,28,29,30,31], the studies on coupled systems of delayed differential equations, especially with infinite delay, are limited; for instance, see [3,32,33,34].

    Recently, Liu and Zhao [33] investigated the existence results in a Banach space for a coupled system involving neutral integro-functional differential equations of fractional order between 0 and 1, with infinite delay, by applying standard fixed-point theorems.

    Our goal in this work is to extend the previous studies on coupled systems with infinite delay by introducing a new class of the form:

    {CDς0+u1(t)=f(t,u1t,u2t),tΩ:=[0,1],u1(t)=η1(t),t(,0],CDγ0+u2(t)=g(t,u1t,u2t),tΩ:=[0,1],u2=η2(t),t(,0],u1(1)=λ1σ10u2(s)ds,u2(1)=λ2σ20u1(s)ds,σ1,σ2(0,1), (1.1)

    where CDς0+, CDγ0+ are the Caputo derivatives of fractional order ς,γ(1,2], respectively, and λ1, λ2 are constants. f,g:Ω×TR are continuous functions and ηiT such that ηi(0)=0, for i=1,2, where T is denoted as a phase space that is specified in Section 2. The functions uit:(,0]R, which are elements in T, are defined as uit(τ)=ui(t+τ),τ0, for tΩ and ui:(,1]R,i=1,2.

    The remaining content of this article is presented as follows: Some basic materials related to our work are presented, and the integral equation that is equivalent to the solution for the linear variant of problem (1.1) is derived in Section 2. In Section 3, we establish our main results with the aid of Schaefer's theorem and the contraction mapping principle, and we provide some examples to illustrate the obtained results. Finally, the conclusion is presented in Section 4.

    For this paper, we define the phase space (T,.T) as a seminormed linear space of functions that map (,0] into R and satisfy the following fundamental axioms, see [22]:

    (N1) For a function x that maps (,1] into R, such that x0T, and for each t[0,1], the following conditions hold:

    (1) xt is an element in T,

    (2) xtTq(t)sup{|x(s)|:0st}+p(t)x0T, where q,p:[0,)[0,) are two functions independent of x(.) such that q is a continuous, p is a locally bounded, and

    q=sup{|q(t)|:t[0,1]},p=sup{|p(t)|:t[0,1]},

    (3) |x(t)|LxtT, where L0 be a constant.

    (N2) For x(.) satisfies (N1),xt is a continuous T valued function on [0,1],

    (N3) T is a complete space.

    Next, we define the space Ta={x:(,1]R:x|(,0]T and x|[0,1]C(Ω,R)} with a seminorm .Ta defined by xTa=ηT+supsΩ|x(s)|,xTa and x(t)=η(t) for t(,0].

    Definition 2.1. [35] The Riemann-Liouville fractional integral for a function h:[0,)R, of order β>0 is defined by

    Iβ0+h(t)=t0(ts)β1Γ(β)f(s)ds,t>0.

    Definition 2.2. [35] The Caputo fractional derivative of order β for a function h:[0,]R with h(t)ACn[0,) is defined by

    CDβ0+h(t)=1Γ(nβ)t0h(n)(s)(ts)βn+1ds=Inβ0+h(n)(t), t>0,n1<β<n,

    where n=[β]+1.

    Lemma 2.1. [35] Let β>0 and h(t)ACn[0,) or Cn[0,). Then

    (Iβ0+CDβ0+h)(t)=h(t)n1k=0h(k)(0)k!tk,t>0,n1<β<n. (2.1)

    In the following, we prove an auxiliary lemma that is associated with the linear variant of problem (1.1).

    Lemma 2.2. Let h1,h2C(0,1) and u1,u2AC(Ω,R)Ta and

    Λ1=1λ1λ2σ21σ2240. (2.2)

    Then, the unique solution to the problem is:

    {CDς0+u1(t)=h1(t),CDγ0+u2(t)=h2(t),tΩ:=[0,1],u1(t)=η1(t),u2(t)=η2(t),t(,0],u1(1)=λ1σ10u2(s)ds,u2(1)=λ2σ20u1(s)ds,σ1,σ2(0,1), (2.3)

    is given by:

    u1(t)={η1(t),t(,0],1Γ(ς)t0(ts)ς1h1(s)ds+tΛ1{λ1Γ(γ)σ10s0(sτ)γ1h2(τ)dτds1Γ(ς)10(1s)ς1h1(s)ds+λ1λ2σ212Γ(ς)σ20s0(sτ)ς1h1(τ)dτdsλ1σ212Γ(γ)10(1s)γ1h2(s)ds},t[0,1]. (2.4)
    u2(t)={η2(t),t(,0],1Γ(γ)t0(ts)γ1h2(s)ds+tΛ1{λ2Γ(ς)σ20s0(sτ)ς1h1(τ)dτds1Γ(γ)10(1s)γ1h2(s)ds+λ1λ2σ222Γ(γ)σ10s0(sτ)γ1h2(τ)dτdsλ2σ222Γ(ς)10(1s)ς1h1(s)ds},t[0,1]. (2.5)

    Proof. Applying Iς0+ and Iγ0+ to the first and second differential equations in (2.3), respectively, and then in view of Lemma 2.1, for t[0,1], we find

    u1(t)=1Γ(ς)t0(ts)ς1h1(s)ds+c1+c2t, (2.6)
    u2(t)=1Γ(γ)t0(ts)γ1h2(s)ds+c3+c4t, (2.7)

    where c1,c2,c3,c4R. Using the first boundary conditions: ui(0)=ηi(0)=0,i=1,2 in (2.6) and (2.7), respectively, we get c1=c3=0. Consequently, (2.6) and (2.7) have the form:

    u1(t)=1Γ(ς)t0(ts)ς1h1(s)ds+c2t, (2.8)
    u2(t)=1Γ(γ)t0(ts)γ1h2(s)ds+c4t. (2.9)

    From the second boundary conditions: u1(1)=λ1σ10u2(s)ds,u2(1)=λ2σ20u1(s)ds, together with (2.8) and (2.9), it implies that

    c2λ1σ212c4=λ1Γ(γ)σ10s0(sτ)γ1h2(τ)dτds1Γ(ς)10(1s)ς1h1(s)ds,
    c4λ2σ222c2=λ2Γ(ς)σ20s0(sτ)ς1h1(τ)dτds1Γ(γ)10(1s)γ1h2(s)ds.

    By solving these two equations together, we get

    c2=11λ1λ2σ21σ224{λ1Γ(γ)σ10s0(sτ)γ1h2(τ)dτds1Γ(ς)10(1s)ς1h1(s)ds+λ1λ2σ212Γ(ς)σ20s0(sτ)ς1h1(τ)dτdsλ1σ212Γ(γ)10(1s)γ1h2(s)ds},
    c4=11λ1λ2σ21σ224{λ2Γ(ς)σ20s0(sτ)ς1h1(τ)dτds1Γ(γ)10(1s)γ1h2(s)ds+λ1λ2σ222Γ(γ)σ10s0(sτ)γ1h2(τ)dτdsλ2σ222Γ(ς)10(1s)ς1h1(s)ds}.

    By replacing the values of c2 and c4 in (2.8) and (2.9), respectively, we obtain solutions (2.4) and (2.5). The converse of the lemma can be proved by direct computation.

    To establish our main results for problem (1.1), in view of Lemma 2.2, let us transform problem (1.1) into a fixed-point problem by introducing an operator J:=(J1,J2):ΠΠ as

    J1(u1,u2)(t)={η1(t),t(,0],t0(ts)ς1Γ(ς)f(s,u1s,u2s)ds+tΛ1{λ1Γ(γ)σ10s0(sτ)γ1g(τ,u1τ,u2τ)dτds10(1s)ς1Γ(ς)f(s,u1s,u2s)ds+λ1λ2σ212Γ(ς)σ20s0(sτ)ς1f(τ,u1τ,u2τ)dτdsλ1σ212Γ(γ)10(1s)γ1g(s,u1s,u2s)ds},t[0,1],

    and

    J2(u1,u2)(t)={η2(t),t(,0],t0(ts)γ1Γ(γ)g(s,u1s,u2s)ds+tΛ1{λ2Γ(ς)σ20s0(sτ)ς1f(τ,u1τ,u2τ)dτds10(1s)γ1Γ(γ)g(s,u1s,u2s)ds+λ1λ2σ222Γ(γ)σ10s0(sτ)γ1g(τ,u1τ,u2τ)dτdsλ2σ222Γ(ς)10(1s)ς1f(s,u1s,u2s)ds},t[0,1],

    where Π=Ta×Ta is a seminormed space endowed with the seminorm

    (u1,u2)Π=u1Ta+u2Ta,for(u1,u2)Π.

    For t[0,1], let us assume the solution (u1,u2)Π, that satisfies (2.4) and (2.5), to be a decomposition of two functions (v1,v2) and (ˉw1,ˉw2), such that (u1,u2)(t)=(v1,v2)(t)+(w1,w2)(t), which implies (u1t,u2t)=(v1t,v2t)+(ˉw1t,ˉw2t).

    The function (v1,v2)(.):(,1]×(,1]R2 is defined by

    (v1,v2)(t)={(η1,η2)(t),t(,0],(0,0),t(0,1], (3.1)

    which yield (v10,v20)=(η1,η2). Also, the function (ˉw1,ˉw2)(.):(,1]×(,1]R2 is defined by

    (¯w1,¯w2)(t)={(0,0),t(,0],(w1,w2)(t),t(0,1], (3.2)

    where the function (w1,w2)(.) satisfies

    w1(t)=1Γ(ς)t0(ts)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds+tΛ1{λ1Γ(γ)σ10s0(sτ)γ1g(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτds1Γ(ς)10(1s)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds+λ1λ2σ212Γ(ς)σ20s0(sτ)ς1f(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτdsλ1σ212Γ(γ)10(1s)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds}, (3.3)

    and

    w2(t)=1Γ(γ)t0(ts)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds+tΛ1{λ2Γ(ς)σ20s0(sτ)ς1f(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτds1Γ(γ)10(1s)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds+λ1λ2σ222Γ(γ)σ10s0(sτ)γ1g(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτdsλ2σ222Γ(ς)10(1s)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds}. (3.4)

    Thus, for every (w1,w2)Π, (w1,w2)(0)=(0,0).

    Now, set Ta={wTa such that w0=0} and introduce a seminorm .Ta on Ta by

    wTa=w0T+supt[0,1]|w(t)|=supt[0,1]|w(t)|,wTa,

    which means that .Ta is indeed a norm in Ta and consequently, the space (Ta,.Ta) is a Banach space. Furthermore, consider the Banach space Π=Ta×Ta with the norm

    (w1,w2)Π=w1Ta+w2Ta,

    for (w1,w2)Π, and define the operator S=(S1,S2):ΠΠ by

    S(w1(t),w2(t)):=(S1(w1(t),w2(t)),S2(w1(t),w2(t))), (3.5)

    where

    S1(w1(t),w2(t))=1Γ(ς)t0(ts)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds+tΛ1{λ1Γ(γ)σ10s0(sτ)γ1g(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτds1Γ(ς)10(1s)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds+λ1λ2σ212Γ(ς)σ20s0(sτ)ς1f(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτdsλ1σ212Γ(γ)10(1s)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds},t(0,1], (3.6)

    and

    S2(w1(t),w2(t))=1Γ(γ)t0(ts)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds+tΛ1{λ2Γ(ς)σ20s0(sτ)ς1f(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτds1Γ(γ)10(1s)γ1g(s,v1s+ˉw1s,v2s+ˉw2s)ds+λ1λ2σ222Γ(γ)σ10s0(sτ)γ1g(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)dτdsλ2σ222Γ(ς)10(1s)ς1f(s,v1s+ˉw1s,v2s+ˉw2s)ds},t(0,1]. (3.7)

    Clearly, if the operator S has a fixed point, then J has a fixed point, and vice versa.

    Further, to establish our main results, we introduce the following hypotheses:

    (C1) There exist continuous nonnegative functions αi,φiC([0,1],R+),i=1,2,3, such that

    |f(t,u1,u2)|α1(t)+α2(t)||u1||T+α3(t)||u2||T,(u1,u2)T,tΩ,
    |g(t,u1,u2)|φ1(t)+φ2(t)||u1||T+φ3(t)||u2||T,(u1,u2)T,tΩ.

    (C2) There exist constants i>0,χi>0,i=1,2, such that

    |f(t,u1,u2)f(t,u1,u2)|1||u1u1||T+2||u2u2||T,(u1,u2),(u1,u2)T,tΩ,
    |g(t,u1,u2)g(t,u1,u2)|χ1||u1u1||T+χ2||u2u2||T,(u1,u2),(u1,u2)T,tΩ.

    In the following, for brevity, we use the notations:

    Λ2=1Γ(ς+1)+1|Λ1|Γ(ς+1)+|λ2|σ222|Λ1|Γ(ς+1)+|λ2|σς+12|Λ1|Γ(ς+2)[1+|λ1|σ212], (3.8)
    Λ3=1Γ(γ+1)+1|Λ1|Γ(γ+1)+|λ1|σ212|Λ1|Γ(γ+1)+|λ1|σγ+11|Λ1|Γ(γ+2)[1+|λ2|σ222], (3.9)

    and

    Φ=min{1q(¯α2Λ2+¯φ2Λ3),1q(¯α3Λ2+¯φ3Λ3)},

    where ¯αi=sup{|αi(t)|:tΩ} and ¯φi=sup{|φi(t)|:tΩ},i=1,2,3.

    The aim of our first result is to provide sufficient criteria that ensure the existence of solutions for problem (1.1) in view of Schaefer's theorem [36].

    Lemma 3.1. (Schaefer)[36]. For a Banach space B, assume that P:BB is a continuous and compact mapping on B. Then P has a fixed point ˉuB, if the set of all solutions of the equation u=ρPu, for 0<ρ<1, is bounded.

    Theorem 3.1. Let f,g:Ω×TR be continuous functions, and condition (C1) holds true. Then problem (1.1) has at least one solution on (,1], if the following inequalities are satisfied:

    q(¯α2Λ2+¯φ2Λ3)<1andq(¯α3Λ2+¯φ3Λ3)<1, (3.10)

    where Λ2,Λ3 are respectively introduced by (3.8) and (3.9).

    Proof. We start the proof by showing that the operator S:ΠΠ defined by (3.5) is continuous and maps any bounded subset of Π into a relatively compact subset of Π; that is, S is completely continuous. Clearly, the continuity of the operator S:ΠΠ follows the continuity of the functions f and g. Now, let us consider the bounded set Bˉr={(w1,w2):(w1,w2)<ˉr}Π. Then, positive constants Mf and Mg can be found such that

    |f(s,v1s+ˉw1s,v2s+ˉw2s)|¯α1+¯α2||v1s+ˉw1s||T+¯α3||v2s+ˉw2s||T¯α1+¯α2(qsupsΩ|w1(s)|+pη1T)+¯α3(qsupsΩ|w2(s)|+pη2T)¯α1+p(¯α2η1T+¯α3η2T)+q(¯α2w1Ta+¯α3w2Ta)¯α1+p(¯α2η1T+¯α3η2T)+αqˉr=Mf.

    Simmilarly

    |g(s,v1s+ˉw1s,v2s+ˉw2s)|¯φ1+p(¯φ2η1T+¯φ3η2T)+φqˉr=Mg,

    where ||v1s+ˉw1s||T||v1s||T+||ˉw1s||Tqsup{|w1(s)|:sΩ}+pη1T=qw1Ta+pη1T, ||v2s+ˉw2s||T||v2s||T+||ˉw2s||Tqsup{|w2(s)|:sΩ}+pη1T=qw2Ta+pη2T, and α=max{ˉα2,ˉα3},φ=max{ˉφ2,ˉφ3}.

    Then, for any (w1,w2)Bˉr, tΩ, we have

    |S1(w1(t),w2(t))|1Γ(ς)t0(ts)ς1|f(s,v1s+ˉw1s,v2s+ˉw2s)|ds+t|Λ1|{|λ1|Γ(γ)σ10s0(sτ)γ1|g(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)|dτds+1Γ(ς)10(1s)ς1|f(s,v1s+ˉw1s,v2s+ˉw2s)|ds+|λ1||λ2|σ212Γ(ς)σ20s0(sτ)ς1|f(τ,v1τ+ˉw1τ,v2τ+ˉw2τ)|dτds+|λ1|σ212Γ(γ)10(1s)γ1|g(s,v1s+ˉw1s,v2s+ˉw2s)|ds}1Γ(ς)t0(ts)ς1Mfds+t|Λ1|{|λ1|Γ(γ)σ10s0(sτ)γ1Mgdτds+1Γ(ς)10(1s)ς1Mfds+|λ1||λ2|σ212Γ(ς)σ20s0(sτ)ς1Mfdτds+|λ1|σ212Γ(γ)10(1s)γ1Mgds)Mf{t0(ts)ς1Γ(ς)ds+t|Λ1|Γ(ς)10(1s)ς1ds+t|λ1||λ2|σ212|Λ1|Γ(ς)σ20s0(sτ)ς1dτds}+Mg{t|λ1||Λ1|Γ(γ)σ10s0(sτ)γ1dτds+t|λ1|σ212|Λ1|Γ(γ)10(1s)γ1ds}Mf{1Γ(ς+1)+1|Λ1|Γ(ς+1)+|λ1||λ2|σ21σς+122|Λ1|Γ(ς+2)}+Mg{|λ1|σγ+11|Λ1|Γ(γ+2)+|λ1|σ212|Λ1|Γ(γ+1)}.

    In the same way,

    \begin{eqnarray*} |\mathfrak S_2(w_1(t),w_2(t))| &\le& \mathcal{M}_f\Bigg\{\frac{|\lambda_2|\sigma_2^{\varsigma+1}}{|\Lambda_1|\Gamma(\varsigma+2)}+\dfrac{|\lambda_2|\sigma_2^{2}}{2|\Lambda_1|\Gamma(\varsigma+1)}\Bigg\}\\&&+\mathcal{M}_g\Bigg\{\frac{1}{\Gamma(\gamma+1)}+\frac{1}{|\Lambda_1|\Gamma(\gamma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_2^2\sigma_1^{\gamma+1}}{2|\Lambda_1|\Gamma(\gamma+2)}\Bigg\}. \end{eqnarray*}

    Therefore, for any (w_1, w_2)\in \mathfrak B_{\bar{r}}, we get

    \begin{eqnarray*} \|\mathfrak S(w_1,w_2)\|_{\Pi'}& = &\|\mathfrak S_1(w_1,w_2)\|_{\Pi'}+\|\mathfrak S_2(w_1,w_2)\|_{\Pi'}\\ &\leq& \mathcal{M}_f\Lambda_2+\mathcal{M}_g\Lambda_3, \end{eqnarray*}

    which yields that the operator \mathfrak S is uniformly bounded.

    Now, to prove that \mathfrak S is equicontinuous on {\Pi'} , take 0 < t_{1} < t_{2} < 1 , and (w_1, w_2)\in\mathfrak B_{\bar{r}} . Then

    \begin{eqnarray*} &&|{\mathfrak S_1}(w_1(t_{2}),w_2(t_{2}))-{\mathfrak S_1}(w_1(t_{1}),w_2(t_{1}))|\\ & = &\Bigg|\frac{1}{\Gamma(\varsigma)}\int_{0}^{t_1}\Big((t_2-s)^{\varsigma-1}-(t_1-s)^{\varsigma-1}\Big)f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})ds\\ &&+\frac{1}{\Gamma(\varsigma)}\int_{t_1}^{t_2}(t_2-s)^{\varsigma-1}f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})ds\\ &&+\frac{t_2-t_1}{\Lambda_1}\Bigg\{\dfrac{\lambda_1}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}g(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})d\tau ds\\ &&-\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})ds\\ &&+\dfrac{\lambda_1\lambda_2\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}f(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})d\tau ds\\ &&-\dfrac{\lambda_1\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}g(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})ds\Bigg\}\Bigg|\\ &\le &\frac{1}{\Gamma(\varsigma)}\int_{0}^{t_1}\big|(t_2-s)^{\varsigma-1}-(t_1-s)^{\varsigma-1}\big|\; \mathcal{M}_f\; ds+\frac{1}{\Gamma(\varsigma)}\int_{t_1}^{t_2}\big|(t_2-s)^{\varsigma-1}\big|\; \mathcal{M}_f\; ds\\ &&+\frac{|t_2-t_1|}{|\Lambda_1|}\Bigg(\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}\; \mathcal{M}_g\; d\tau ds+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}\; \mathcal{M}_f\; ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}\; \mathcal{M}_f\; d\tau ds+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}\; \mathcal{M}_g\; ds\Bigg)\\ &\le &\mathcal{M}_f\Big[\frac{2(t_2-t_1)^\varsigma+|t_2^\varsigma-t_1^\varsigma|}{\Gamma{(\varsigma+1)}}+\frac{(t_2-t_1)}{|\Lambda_1|}\Big(\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma{(\varsigma+2)}}\sigma_2^{\varsigma+1}+\frac{1}{\Gamma(\varsigma+1)}\Big)\Big]\\ &&+\mathcal{M}_g\frac{(t_2-t_1)}{|\Lambda_1|}\Big(\frac{|\lambda_1|}{\Gamma(\gamma+2)}\sigma_1^{\gamma+1}+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma{(\gamma+1)}}\Big). \end{eqnarray*}

    Likewise, we can find that

    \begin{eqnarray*} &&|{\mathfrak S_2}(w_1(t_{2}),w_2(t_{2}))-{\mathfrak S_2}(w_1(t_{1}),w_2(t_{1}))|\\ &\leq& \mathcal{M}_g\Big[\frac{2(t_2-t_1)^\gamma+|t_2^\gamma-t_1^\gamma|}{\Gamma{(\gamma+1)}}+\frac{(t_2-t_1)}{|\Lambda_1|}\Big(\dfrac{|\lambda_1||\lambda_2|\sigma_2^2}{2\Gamma{(\gamma+2)}}\sigma_1^{\gamma+1}+\frac{1}{\Gamma(\gamma+1)}\Big)\Big]\\ &&+\mathcal{M}_f\frac{(t_2-t_1)}{|\Lambda_1|}\Big(\frac{|\lambda_2|}{\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}+\dfrac{|\lambda_2|\sigma_2^2}{2\Gamma{(\varsigma+1)}}\Big). \end{eqnarray*}

    According to the above inequalities, we show that |{\mathfrak S_1}(w_1(t_{2}), w_2(t_{2}))-{\mathfrak S_1}(w_1(t_{1}), w_2(t_{1}))|\to 0 as t_1\to t_2 independently of (w_1, w_2)\in \mathfrak B_{\bar{r}} . Hence, all the hypotheses of the Arzelá-Ascoli theorem are satisfied, and consequently, we conclude that the operator \mathfrak S:{\Pi'}\to {\Pi'} is completely continuous.

    Finally, let us define the set \Psi by

    \Psi = \{(w_1,w_2)\in {\Pi'}|(w_1,w_2) = \xi \mathfrak S(w_1,w_2),0 < \xi < 1\}.

    Then, we need to prove that \Psi is bounded. Let (w_1, w_2)\in \Psi , then (w_1, w_2) = \xi\mathfrak S(w_1, w_2), \; 0 < \xi < 1 . For any t \in \Omega , we get

    \begin{eqnarray*} w_1(t) = \xi \mathfrak S_1(w_1(t),w_2(t)),\; w_2(t) = \xi \mathfrak S_2(w_1(t),w_2(t)), \end{eqnarray*}
    \begin{eqnarray*} |w_1(t)|& = &\xi|\mathbf{\mathfrak S_1}(w_1(t),w_2(t))| \\ &\le& \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})|ds\\ &&+\dfrac{t}{|\Lambda_1|}\Big(\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}|g(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})|d\tau ds\\ &&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})|ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}|f(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})|d\tau ds\\ &&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}|g(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})|ds\Big)\\ &\leq & \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}\Big[\bar{\alpha_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\alpha_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_\mathfrak{T})\bar{\alpha_3}\Big]ds\\ && +\dfrac{t}{|\Lambda_1|}\Bigg\{\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}\Big[\bar{\varphi_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\varphi_2}\\ &&+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_{\mathfrak{T}})\bar{\varphi_3}\Big]d\tau ds\\ &&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}\Big[\bar{\alpha_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\alpha_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_\mathfrak{T})\bar{\alpha_3}\Big]ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}\Big[\bar{\alpha_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\alpha_2}\\ &&+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_\mathfrak{T})\bar{\alpha_3}\Big]d\tau ds\\ &&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}\Big[\bar{\varphi_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\varphi_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_{\mathfrak{T}})\bar{\varphi_3}\Big]ds\Bigg\}\\ &\leq & \Big[\bar{\alpha_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\alpha_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_\mathfrak{T})\bar{\alpha_3}\Big]\\ &&\times\Big(\frac{1}{\Gamma(\varsigma+1)}+\frac{1}{|\Lambda_1|\Gamma(\varsigma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2\sigma_2^{\varsigma+1}}{2|\Lambda_1|\Gamma(\varsigma+2)}\Big)\\ &&+\Big[\bar{\varphi_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\varphi_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_{\mathfrak{T}})\bar{\varphi_3}\Big]\\ &&\times\Big(\dfrac{|\lambda_1|}{|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}+\dfrac{|\lambda_1|\sigma_1^2}{2|\Lambda_1|\Gamma(\gamma+1)}\Big). \end{eqnarray*}

    Analogously, we can obtain

    \begin{eqnarray*} |w_2(t)| & = & \xi|\mathbf{\mathfrak S_2}(w_1(t),w_2(t))| \\ &\le& \Big[\bar{\alpha_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\alpha_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_\mathfrak{T})\bar{\alpha_3}\Big]\\ &&\times\Big(\dfrac{|\lambda_2|}{|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}+\dfrac{|\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\varsigma+1)}\Big)\\ &&+\Big[\bar{\varphi_1}+(q^*\|w_1\|_{\mathfrak{T'_a}}+p^*\|\eta_1\|_{\mathfrak{T}})\bar{\varphi_2}+(q^*\|w_2\|_{\mathfrak{T'_a}}+p^*\|\eta_2\|_{\mathfrak{T}})\bar{\varphi_3}\Big]\\ &&\times\Big(\frac{1}{\Gamma(\gamma+1)}+\frac{1}{|\Lambda_1|\Gamma(\gamma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}\Big). \end{eqnarray*}

    In consequence, we have

    \begin{eqnarray*} ||w_1||_{\mathfrak{T'_a}} +||w_2||_{\mathfrak{T'_a}} &\leq& \bar{\alpha_1}\Lambda_2+\bar{\varphi_1}\Lambda_3+p^*\|\eta_1\|_{\mathfrak{T}}\Big[\bar{\alpha_2}\Lambda_2+\bar{\varphi_2}\Lambda_3\Big]+p^*\|\eta_2\|_{\mathfrak{T}}\Big[\bar{\alpha_3}\Lambda_2+\bar{\varphi_3}\Lambda_3\Big]\\ &&+q^*\|w_1\|_{\mathfrak{T'_a}}\Big[\bar{\alpha_2}\Lambda_2+\bar{\varphi_2}\Lambda_3\Big]+q^*\|w_2\|_{\mathfrak{T'_a}}\Big[\bar{\alpha_3}\Lambda_2+\bar{\varphi_3}\Lambda_3\Big]. \end{eqnarray*}

    Hence, by definition of \Phi and the conditions (3.10), we get

    \begin{equation*} \|(w_1,w_2)\|_{\Pi'} \leq \frac{\bar{\alpha_1}\Lambda_2+\bar{\varphi_1}\Lambda_3+p^*\Big(\|\eta_1\|_{\mathfrak{T}}\Big[\bar{\alpha_2}\Lambda_2+\bar{\varphi_2}\Lambda_3\Big]+\|\eta_2\|_{\mathfrak{T}}\Big[\bar{\alpha_3}\Lambda_2+\bar{\varphi_3}\Lambda_3\Big]\Big)}{\Phi}. \end{equation*}

    This shows that ||(w_1, w_2)||_{\Pi'} is bounded for t \in \Omega , and, as a result, the set \Psi is bounded. Therefore, in view of the conclusion of Lemma 3.1, we deduce that the operator \mathfrak S has at least one fixed point on (-\infty, 1] , and hence there exists at least one solution to problem (1.1) on (-\infty, 1] .

    The next result deals with the uniqueness of the solution for problem (1.1) by utilizing Banach's contraction mapping principle.

    Theorem 3.2. Let f, g:[0, 1]\times \mathfrak{T}\to \mathbb{R} be continuous functions satisfying the condition ( C_2 ). Then there exists a unique solution to problem (1.1) on (-\infty, 1] , if

    \begin{equation} q^*(\ell\Lambda_2+\chi\Lambda_3) < 1, \end{equation} (3.11)

    where \ell = \max\{\ell_1, \ell_2\}, \chi = \max\{\chi_1, \chi_2\} and \Lambda_2, \Lambda_3 are respectively given by (3.8) and (3.9).

    Proof. Let us fix r to satisfy the following:

    \begin{eqnarray*} r > \frac{M_1\Lambda_2+M_2\Lambda_3+p^*(\ell\Lambda_2+\chi\Lambda_3)(\|\eta_1\|_{\mathfrak{T}}+\|\eta_2\|_{\mathfrak{T}})}{1-q^*(\ell\Lambda_2+\chi\Lambda_3)}, \end{eqnarray*}

    where M_1 = \sup_{t\in [0, 1]}|f(t, 0, 0)|, M_2 = \sup_{t\in [0, 1]}|g(t, 0, 0)|, and consider the operator \mathfrak S:{\Pi'} \to {\Pi'}, defined by (3.5). Then we show that \mathfrak S{\mathfrak{P}_r}\subset {\mathfrak{P}_r} , where

    {\mathfrak{P}_r} = \{w\in {\Pi'}:||(w_1,w_2)||_{\Pi'}\leq r\}.

    For (w_1, w_2)\in {\mathfrak{P}_r}, we get

    \begin{eqnarray*} &&|\mathfrak S_1(w_1(t),w_2(t))|\\ &\le& \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}[|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-f(s,0,0)|+M_1]ds\\ &&+\dfrac{t}{|\Lambda_1|}\Big(\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}[|g(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})-g(\tau,0,0)|+M_2]d\tau ds\\&&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}[|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-f(s,0,0)|+M_1]ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}[|f(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})-f(\tau,0,0)|+M_1]d\tau ds\\&&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}[|g(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-g(s,0,0)|+M_2]ds\Big)\\ &\le& \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}[\ell_1||v_{1s}+\bar{w}_{1s}||_{\mathfrak{T}}+\ell_2||v_{2s}+\bar{w}_{2s}||_{\mathfrak{T}}+M_1]ds\\ &&+\dfrac{t}{|\Lambda_1|}\Big(\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}[\chi_1||v_{1\tau}+\bar{w}_{1\tau}||_{\mathfrak{T}}+\chi_2||v_{2\tau}+\bar{w}_{2\tau}||_{\mathfrak{T}}+M_2]d\tau ds\\&&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}[\ell_1||v_{1s}+\bar{w}_{1s}||_{\mathfrak{T}}+\ell_2||v_{2s}+\bar{w}_{2s}||_{\mathfrak{T}}+M_1]ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}[\ell_1||v_{1\tau}+\bar{w}_{1\tau}||_{\mathfrak{T}}+\ell_2||v_{2\tau}+\bar{w}_{2\tau}||_{\mathfrak{T}}+M_1]d\tau ds\\ &&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}[\chi_1||v_{1s}+\bar{w}_{1s}||_{\mathfrak{T}}+\chi_2||v_{2s}+\bar{w}_{2s}||_{\mathfrak{T}}+M_2]ds\Big)\\ &\le& \Big[\ell_1(q^*\|w_1\|_{\mathfrak{T}'_a}+p^*\|\eta_1\|_{\mathfrak{T}})+\ell_2(q^*\|w_2\|_{\mathfrak{T}'_a}+p^*\|\eta_2\|_{\mathfrak{T}})+M_1\Big]\\ &&\times\Big(\frac{1}{\Gamma(\varsigma+1)}+\frac{1}{|\Lambda_1|\Gamma(\varsigma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2\sigma_2^{\varsigma+1}}{2|\Lambda_1|\Gamma(\varsigma+2)}\Big)\\ &&+\Big[\chi_1(q^*\|w_1\|_{\mathfrak{T}'_a}+p^*\|\eta_1\|_{\mathfrak{T}})+\chi_2(q^*\|w_2\|_{\mathfrak{T}'_a}+p^*\|\eta_2\|_{\mathfrak{T}})+M_2\Big]\\ &&\times\Big(\dfrac{|\lambda_1|\sigma_1^{\gamma+1}}{|\Lambda_1|\Gamma(\gamma+2)}+\dfrac{|\lambda_1|\sigma_1^2}{2|\Lambda_1|\Gamma(\gamma+1)}\Big), \end{eqnarray*}

    which yields for t\in \Omega

    \begin{eqnarray*} ||\mathfrak S_1(w_1,w_2)||_{\Pi'} &\le& \Bigg\{\ell\Big(\frac{1}{\Gamma(\varsigma+1)}+\frac{1}{|\Lambda_1|\Gamma(\varsigma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\gamma+1}\Big)\\ &&+\chi\Big(\dfrac{|\lambda_1|}{|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}+\dfrac{|\lambda_1|\sigma_1^2}{2|\Lambda_1|\Gamma(\gamma+1)}\Big)\Bigg\}\\ &&\times\Bigg\{q^*(\|w_1\|_{\mathfrak{T}'_a}+\|w_2\|_{\mathfrak{T}'_a})+p^*(\|\eta_1\|_{\mathfrak{T}}+\|\eta_2\|_{\mathfrak{T}} )\Bigg\}\\ &&+M_1\Big(\frac{1}{\Gamma(\varsigma+1)}+\frac{1}{|\Lambda_1|\Gamma(\varsigma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}\Big)\\ &&+M_2\Big(\dfrac{|\lambda_1|}{|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}+\dfrac{|\lambda_1|\sigma_1^2}{2|\Lambda_1|\Gamma(\gamma+1)}\Big). \end{eqnarray*}

    Similarly, one can find that

    \begin{eqnarray*} ||\mathfrak S_2(w_1,w_2)||_{\Pi'} &\le& \Bigg\{\ell\Big(\dfrac{|\lambda_2|}{|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}+\dfrac{|\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\varsigma+1)}\Big)\\ &&+\chi\Big(\frac{1}{\Gamma(\gamma+1)}+\frac{1}{|\Lambda_1|\Gamma(\gamma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}\Big)\Bigg\}\\ &&\times\Bigg\{q^*(\|w_1\|_{\mathfrak{T}'_a}+\|w_2\|_{\mathfrak{T}'_a})+p^*(\|\eta_1\|_{\mathfrak{T}}+\|\eta_2\|_{\mathfrak{T}} )\Bigg\}\\ &&+M_1\Big(\dfrac{|\lambda_2|}{|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}+\dfrac{|\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\varsigma+1)}\Big)\\ &&+M_2\Big(\frac{1}{\Gamma(\gamma+1)}+\frac{1}{|\Lambda_1|\Gamma(\gamma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}\Big). \end{eqnarray*}

    Consequently, for any (w_1, w_2) \in {\mathfrak{P}_r} , we have

    \begin{eqnarray*} ||\mathfrak S(w_1,w_2)||_{\Pi'}& = &||\mathfrak S_1(w_1,w_2)||_{\Pi'}+||\mathfrak S_2(w_1,w_2)||_{\Pi'}\\ &\leq& (\ell\Lambda_2+\chi\Lambda_3)\Big[q^*\|(w_1,w_2)\|_{\Pi'}+p^*(\|\eta_1\|_{\mathfrak{T}},\|\eta_2\|_{\mathfrak{T}})\Big]+M_1\Lambda_2+M_2\Lambda_3\\& < &r. \end{eqnarray*}

    So, we conclude that \mathfrak S maps {\mathfrak{P}_r} into itself.

    Next, to establish the contraction of the operator \mathfrak S , let (w_1, w_2), (w^*_1, w^*_2) \in {\Pi'}, t\in [0, 1] . Then, by (C_2) , we get

    \begin{eqnarray*} &&|\mathfrak S_1(w_1(t),w_2(t))-\mathfrak S_1(w^*_1(t),w^*_2(t))| \\ &\leq& \frac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}\Big[|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-f(s,v_{1s}+\bar{w}^*_{1s},v_{2s}+\bar{w}^*_{2s})|\Big]ds\\ &&+\frac{t}{|\Lambda_1|}\Bigg\{\frac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}\Big[|g(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})-g(\tau,v_{1\tau}+\bar{w}^*_{1\tau},v_{2\tau}+\bar{w}^*_{2\tau})|\Big]d\tau ds\\ &&+\frac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}\Big[|f(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-f(s,v_{1s}+\bar{w}^*_{1s},v_{2s}+\bar{w}^*_{2s})|\Big]ds\\ &&+\frac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}\Big[|f(\tau,v_{1\tau}+\bar{w}_{1\tau},v_{2\tau}+\bar{w}_{2\tau})-f(\tau,v_{1\tau}+\bar{w}^*_{1\tau},v_{2\tau}+\bar{w}^*_{2\tau})|\Big]d\tau ds\\ &&+\frac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}\Big[|g(s,v_{1s}+\bar{w}_{1s},v_{2s}+\bar{w}_{2s})-g(s,v_{1s}+\bar{w}^*_{1s},v_{2s}+\bar{w}^*_{2s})|\Big]ds\Bigg\}\\ &\leq& \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}\Big[\ell_1||{w}_{1s}-{w}^*_{1s}||_{\mathfrak{T}}+\ell_2||{w}_{2s}-{w}^*_{2s}||_{\mathfrak{T}}\Big]ds\\ &&+\dfrac{t}{|\Lambda_1|}\Bigg\{\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1}\Big[\chi_1||{w}_{1\tau}-{w}^*_{1\tau}||_{\mathfrak{T}}+\chi_2||{w}_{2\tau}-{w}^*_{2\tau}||_{\mathfrak{T}}\Big]d\tau ds\\ &&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}\Big[\ell_1||{w}_{1s}-{w}^*_{1s}||_{\mathfrak{T}}+\ell_2||{w}_{2s}-{w}^*_{2s}||_{\mathfrak{T}}\Big]ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1}\Big[\ell_1||{w}_{1\tau}-{w}^*_{1\tau}||_{\mathfrak{T}}+\ell_2||{w}_{2\tau}-{w}^*_{2\tau}||_{\mathfrak{T}}\Big]d\tau ds\\ &&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}\Big[\chi_1||{w}_{1s}-{w}^*_{1s}||_{\mathfrak{T}}+\chi_2||{w}_{2s}-{w}^*_{2s}||_{\mathfrak{T}}\Big]ds\Bigg\}\\ &\leq& \dfrac{1}{\Gamma(\varsigma)}\int_{0}^{t}(t-s)^{\varsigma-1}\; \Big[\ell_1q^*||{w}_{1}-{w}^*_{1}||_{\mathfrak{T}'_a}+\ell_2q^*||{w}_{2}-{w}^*_{2}||_{\mathfrak{T}'_a}\Big]ds\\ &&+\dfrac{t}{|\Lambda_1|}\Bigg\{\dfrac{|\lambda_1|}{\Gamma(\gamma)}\int_{0}^{\sigma_1}\int_{0}^{s}(s-\tau)^{\gamma-1} \; \Big[\chi_1q^*||{w}_{1}-{w}^*_{1}||_{\mathfrak{T}'_a}+\chi_2q^*||{w}_{2}-{w}^*_{2}||_{\mathfrak{T}'_a}\Big]d\tau ds\\ &&+\dfrac{1}{\Gamma(\varsigma)}\int_{0}^{1}(1-s)^{\varsigma-1}\; \Big[\ell_1q^*||{w}_{1}-{w}^*_{1}||_{\mathfrak{T}'_a}+\ell_2q^*||{w}_{2}-{w}^*_{2}||_{\mathfrak{T}'_a}\Big] ds\\ &&+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2\Gamma(\varsigma)}\int_{0}^{\sigma_2}\int_{0}^{s}(s-\tau)^{\varsigma-1} \; \Big[\ell_1q^*||{w}_{1}-{w}^*_{1}||_{\mathfrak{T}'_a}+\ell_2q^*||{w}_{2}-{w}^*_{2}||_{\mathfrak{T}'_a}\Big]d\tau ds\\ &&+\dfrac{|\lambda_1|\sigma_1^2}{2\Gamma(\gamma)}\int_{0}^{1}(1-s)^{\gamma-1}\; \Big[\chi_1q^*||{w}_{1}-{w}^*_{1}||_{\mathfrak{T}'_a}+\chi_2q^*||{w}_{2}-{w}^*_{2}||_{\mathfrak{T}'_a}\Big]ds\Bigg\}\\ &\leq& q^*\Big\{\ell\Big[\frac{1}{\Gamma(\varsigma+1)}+\frac{1}{|\Lambda_1|\Gamma(\varsigma+1)}+\dfrac{|\lambda_1||\lambda_2|\sigma_1^2}{2|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}\Big]\\ &&+\chi\Big[\dfrac{|\lambda_1|}{|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}+\dfrac{|\lambda_1|\sigma_1^2}{2|\Lambda_1|\Gamma(\gamma+1)}\Big]\Big\}(\|w_1-w^*_1\|_{\mathfrak{T}'_a}+\|w_2-w_2^*\|_{\mathfrak{T}'_a}). \end{eqnarray*}

    In a similar manner, we get

    \begin{eqnarray*} &&| \mathfrak S_2(w_1(t),w_2(t))-\mathfrak S_2(w^*_1(t),w^*_2(t))|\\ &\leq& q^*\Big\{\ell\Big[\dfrac{|\lambda_2|}{|\Lambda_1|\Gamma(\varsigma+2)}\sigma_2^{\varsigma+1}+\dfrac{|\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\varsigma+1)}\Big]\\ &&+\chi\Big[\frac{1}{\Gamma(\gamma+1)}+\frac{1}{|\Lambda_1|\Gamma(\gamma+1)}+\dfrac{|\lambda_1|\lambda_2|\sigma_2^2}{2|\Lambda_1|\Gamma(\gamma+2)}\sigma_1^{\gamma+1}\Big]\Big\}(\|w_1-w^*_1\|_{\mathfrak{T}'_a}+\|w_2-w_2^*\|_{\mathfrak{T}'_a}). \end{eqnarray*}

    Consequently, it follows from the foregoing inequalities that

    \begin{eqnarray*} ||\mathfrak S(w_1,w_2)-\mathfrak S(w^*_1,w^*_2)||_{\Pi'}& = & ||\mathfrak S_1(w_1,w_2)-\mathfrak S_1(w^*_1,w^*_2)||_{\Pi'} +||\mathfrak S_2(w_1,w_2)-\mathfrak S_2(w^*_1,w^*_2)||_{\Pi'}\\ &\leq& q^*( \ell\Lambda_2+\chi\Lambda_3)||({w}_{1},{w}_{2})-({w}^*_{1},{w}^*_{2})||_{\Pi'}, \end{eqnarray*}

    which, together with the condition (3.11), implies that \mathfrak S is a contraction mapping. Therefore, we deduce from the conclusion of the Banach fixed-point theorem that \mathfrak S has a unique fixed point. This ensures the existence of a unique solution to problem (1.1) on (-\infty, 1] .

    Consider the following coupled system:

    \begin{equation} \left\{\begin{array}{ll} {}^CD_{0^+}^{3/2}u_1(t) = f(t,u_{1t},u_{2t}), \; \; t\in \Omega: = [0,1],\\ u_1(t) = \eta_1(t),\; \; t\in (-\infty,0],\\ {}^CD_{0^+}^{5/4}u_2(t) = g(t,u_{1t},u_{2t}), \; \; t\in \Omega: = [0,1],\\ u_2 = \eta_2(t),\; \; t\in (-\infty,0],\\ u_1(1) = 1/2\int_0^{2/5} u_2(s) ds,\; \; \; u_2(1) = \int_0^{1/3} u_1(s) ds, \end{array} \right. \end{equation} (3.12)

    where \varsigma = 3/2, \, \gamma = 5/4, \, \lambda_1 = 1/2, \, \lambda_2 = 1, \, \sigma_1 = 2/5, \, \sigma_2 = 1/3, \, and f(t, u_{1t}, u_{2t}), \, g(t, u_{1t}, u_{2t}), \, \eta_1(t) , and \eta_2(t) will be fixed later.

    We find by using the data given in (3.12) that \Lambda_1 = 0.9977778, \, \Lambda_2 = 1.568185488 , \Lambda_3 = 1.828971108, where \Lambda_{1} , \Lambda_2 and \Lambda_3 are respectively given by (2.2), (3.8), and (3.9).

    Let \delta > 0 and set {\mathfrak{T}}_{\delta} = \{u\in C((-\infty, 0], \mathbb{R}): \lim\limits_{\tau \to -\infty} e^{\delta \tau}u(\tau) exists in \mathbb{R}\}, with the norm \|u\|_{\delta} = \sup \limits_{-\infty < \tau \leq 0} e^{\delta\tau}|u(\tau)|. It is clear that the space {\mathfrak{T}}_{\delta} satisfies the axioms of phase space, and p(t) = q(t) = L = 1 , see [24]. Now, let us take the space \Pi_\delta = \mathfrak{T}_\delta \times \mathfrak{T}_\delta with the norm

    \|(u_1,u_2)\|_{{\Pi}_\delta} = \|u_{1}\|_{\mathfrak{T}_\delta}+\|u_{2}\|_{\mathfrak{T}_\delta},\; \; \mbox{for all} \; \; (u_1,u_2)\in {{\Pi}_\delta}.

    One can take \eta_1(t) = \dfrac{e^{t}-1}{2} and \eta_2(t) = e^{3 t}-1 , which are continuous functions such that \eta_1(0) = \eta_2(0) = 0 and \lim\limits_{t \to -\infty} e^{\delta t}\eta_1(t) < \infty, \lim\limits_{t \to -\infty} e^{\delta t}\eta_2(t) < \infty . Thus, \eta_1, \eta_2 \in {\mathfrak{T}}_{\delta}. Obviously, (\eta_1, \eta_2)\in \Pi_{\delta} and (\eta_1(0), \eta_2(0)) = (0, 0) .

    In order to illustrate Theorem 3.2, we chose

    \begin{equation} f(t,u_{1t},u_{2t}) = \frac{4t^2}{8+t^3}+\frac{e^{-\delta t}\sin u_{1t}}{\sqrt{64+t}}+\frac{e^{-\delta t}u_{2t}|u_{1t}|}{\sqrt{t^2+900}\; (1+|u_{1t}|)} , \end{equation} (3.13)
    \begin{equation} g(t,u_{1t},u_{2t}) = \frac{2t^4}{5}+\frac{e^{-\delta t} \tan^{-1}u_{1t}}{16(t^2+1)}+\frac{e^{-\delta t}u_{2t}}{\sqrt{t+12}} . \end{equation} (3.14)

    Clearly,

    \begin{equation*} |f(t,u_{1t},u_{2t})| = \frac{4t^2}{8+t^3}+\frac{1} {\sqrt{64+t}}||u_{1t}||_{{\mathfrak{T}}_{\delta}}+\frac{1}{\sqrt{t^2+900}}||u_{2t}||_{{\mathfrak{T}}_{\delta}} , \end{equation*}
    \begin{equation*} |g(t,u_{1t},u_{2t})| = \frac{2t^4}{5}+\frac{1 }{16(t^2+1)}||u_{1t}||_{{\mathfrak{T}}_{\delta}}+\frac{1}{\sqrt{t+12}}||u_{2t}||_{{\mathfrak{T}}_{\delta}}, \end{equation*}

    and note that the assumption (C_1) is satisfied with \alpha_1(t) = \frac{4t^2}{8+t^3}, \alpha_2(t) = \frac{1}{\sqrt{64+t}}, \alpha_3 = \frac{1}{\sqrt{t^2+900}} and \varphi_1 = \frac{2t^4}{5}, \varphi_2 = \frac{1}{16(t^2+1)}, \varphi_3 = \frac{1}{\sqrt{t+12}} . Moreover, we find

    \begin{eqnarray*} q^*(\bar{\alpha_2}\Lambda_2+\bar{\varphi_2}\Lambda_3) \approx 0.3103338802 < 1, \\ q^*(\bar{\alpha_3}\Lambda_2+\bar{\varphi_3}\Lambda_3) \approx 0.5802513305 < 1. \end{eqnarray*}

    Hence all the conditions of Theorem 3.1 are satisfied, and as a consequence, the problem (3.12) with f(t, u_{1t}, u_{2t}) and g(t, u_{1t}, u_{2t}) given by (3.13) and (3.14), respectively, has at least one solution on (-\infty, 1] .

    Next, to demonstrate the applicability of Theorem 3.2, let us assume

    \begin{equation} f(t,u_{1t},u_{2t}) = \frac{e^{-\delta t}\sin u_{1t}}{25+t}+\frac{e^{-\delta t} |u_{2t}|}{12+|u_{2t}|}+\ln7, \end{equation} (3.15)
    \begin{equation} g(t,u_{1t},u_{2t}) = \frac{t e^{-\delta t}u_{1t}}{24}+\frac{e^{-\delta t}\sin u_{2t}}{\sqrt{t^2+36}}. \end{equation} (3.16)

    Clearly, f and g satisfy condition (C_2) with \ell_1 = 1/25, \ell_2 = 1/12, \chi_1 = 1/24, \chi_2 = 1/6, and so \ell = 1/12, \chi = 1/6. Also,

    \begin{equation*} q^*(\ell\Lambda_2+\chi\Lambda_3)\approx 0.4355106420 < 1. \end{equation*}

    So, all the assumptions of Theorem 3.1 hold true, and, according to its conclusion, problem (3.12) with f(t, u_{1t}, u_{2t}) and g(t, u_{1t}, u_{2t}) given by (3.15) and (3.16), respectively, has a unique solution on (-\infty, 1] .

    As coupled systems have gained intensive interest due to their important applications in real-world phenomena, we have considered in this paper a new class of coupled systems involving nonlinear fractional differential equations that are affected by infinite delay and complemented with nonlocal integral boundary conditions. We have investigated the existence of solutions for problem (1.1) by applying Schaefer's fixed point theorem, while for the uniqueness result, the contraction mapping principle has been employed. To deal with the differential equations with infinite delay, we needed to select an appropriate phase space that satisfies the axioms given in [22]. To guarantee the applicability of our results, illustrative examples have been constructed. The results presented in this paper take on importance as a new contribution to the study of nonlinear coupled systems with infinite delay that extends the literature on this subject. For further studies of the coupled systems with infinite delay, by following the papers [14,16,30], we can extend our work in this article by discussing the stability and simulation results for the solutions of the obtained system. Also, the differential equations in the problem at hand can be replaced by implicit differential equations of different types of fractional derivatives with the p -Laplacian operator, and a variety of fixed-point theorems can be applied based on our previous works [12,15].

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

    The authors gratefully acknowledge the referees for their useful comments on their paper.

    The authors declare that they have no conflict interests.



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