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An extension of Herstein's theorem on Banach algebra

  • Let A be a (p+q)!-torsion free semiprime ring. We proved that if H,D:AA are two additive mappings fulfilling the algebraic identity 2H(ap+q)=H(ap)aq+apD(aq)+H(aq)ap+aqD(ap) for all aA, then H is a generalized derivation with D as an associated derivation on A. In addition to that, it is also proved in this article that H1 is a generalized left derivation associated with a left derivation δ on A if they fulfilled the algebraic identity 2H1(ap+q)=apH1(aq)+aqδ(ap)+aqH1(ap)+apδ(aq) for all aA. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

    Citation: Abu Zaid Ansari, Suad Alrehaili, Faiza Shujat. An extension of Herstein's theorem on Banach algebra[J]. AIMS Mathematics, 2024, 9(2): 4109-4117. doi: 10.3934/math.2024201

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  • Let A be a (p+q)!-torsion free semiprime ring. We proved that if H,D:AA are two additive mappings fulfilling the algebraic identity 2H(ap+q)=H(ap)aq+apD(aq)+H(aq)ap+aqD(ap) for all aA, then H is a generalized derivation with D as an associated derivation on A. In addition to that, it is also proved in this article that H1 is a generalized left derivation associated with a left derivation δ on A if they fulfilled the algebraic identity 2H1(ap+q)=apH1(aq)+aqδ(ap)+aqH1(ap)+apδ(aq) for all aA. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.



    FDE, fractional differential equation

    LC, Liouville-Caputo

    RL, Riemann-Liouville

    BS, Banach space

    FFIDE, fractional functional integrodifferential equation

    BVP, boundary value problems

    The theory of fractional differential equations (FDEs) generalizes classical differential equations by introducing fractional derivatives, enabling the modeling of complex phenomena exhibiting non-locality, memory, and power-law behavior [1,2]. FDEs have been extensively developed using various fractional derivatives, such as Riemann-Liouville (RL), Caputo, and Grü nwald-Letnikov [3]. These equations describe anomalous diffusion, relaxation, and oscillation processes, making them suitable for modeling real-world problems in physics, engineering, and biology. FDEs have diverse applications across disciplines, including viscoelasticity [4], chaotic dynamics [5], image processing [6], model financial systems [7], population dynamics [8], and electrical circuits [9]. Recent studies have explored fractional-order controllers in robotics [10] and biomedical signal processing[11]. These applications demonstrate the versatility and potential of FDEs in describing complex systems.

    Fractional calculus and fixed point theory have emerged as powerful tools in addressing optimization and inverse problems across various scientific and engineering disciplines. Fractional derivatives and integrals, with their inherent nonlocal properties, are particularly well-suited for modeling systems exhibiting memory effects and long-range dependencies, which are often encountered in optimization problems involving complex systems. Furthermore, fixed point (FP) methods, especially those tailored for non-smooth or set-valued mappings, provide robust frameworks for solving inverse problems, including those arising in image processing and signal reconstruction. The combination of fractional calculus and FP theory offers a synergistic approach, enabling the analysis and solution of challenging optimization and inverse problems that are often intractable using classical techniques. See [12,13,14,15,16] for more information.

    Recent years have witnessed significant interest in boundary value problems (BVPs) of FDEs, encompassing various boundary conditions such as the existence and uniqueness of solutions to fractional boundary value problems [17,18,19,20], the existence and uniqueness of solutions to hybrid fractional systems under multi-point, periodic, and anti-periodic boundary conditions [21,22], and the stability of mixed integral fractional delay dynamic system equations and pantograph differential equations under impulsive effects and nonlocal conditions [23,24]. Integral boundary conditions, in particular, have far-reaching implications in applied fields like heat conduction, electric power networks, elastic stability, telecommunications and electric railway systems. Multi-point BVPs, arising from practical applications, also warrant attention. For example, the existence results for FDEs are established in [25,26,27,28,29]. Alos, the existence of solutions to fractional functional differential equations [30], semilinear fractional differential inclusions [31], Hadamard fractional integro-differential equations [32], systems of multi-point boundary value problems [33], fractional hybrid delay differential equations [34], and nonlinear Atangana-Baleanu-type fractional differential equations [35,36,37,38] have been established. The theory of fractional functional BVPs remains underdeveloped, necessitating further research in mathematical modeling, numerical methods, and computational simulations to address the unresolved aspects.

    Benchohra et al. [26] proved the existence of a solution via Leray-Schauder nonlinear alternative and uniqueness via Banach's FP theorem for fractional functional differential equations with infinite delay. Chauhan et al. [20] explored existence solutions for fractional integro-differential equations with impulses, infinite delay, and integral boundary conditions. Dabas and Gautam [39] examined existence results for impulsive neutral fractional integrodifferential equations featuring state-dependent delays and integral boundary conditions.

    Inspired by the contributions of [20,26,39], first, our study focuses on establishing existence and uniqueness results for a fractional functional integrodifferential equation (FFIDE) featuring infinite delay. It takes the form

    {LCDpϖ(ς)=g(ς,ϖς,ς0Ω(ς,ϑ,ϖϑ)dϑ,σ0Υ(ς,ϑ,ϖϑ)dϑ), p(2,3], ςU=[0,σ]ϖ(ς)=ψ(ς), ς(,0]ϖ(σ)=uj=1bj(Iqj0+ϖ)(λj), 0<λ1<λ2<<λu<σ, (1.1)

    where LCDp is the Liouville-Caputo (LC) fractional derivative with order p. Assume that ={(ς,ϑ):0ϑςσ}, Ξ is a BS, and Θ is a phase space. Then g:U×Θ×ΞΞ, Ω,Υ:×ΘΞ are continuous functions and ψΘ. Furthermore, Iqj0+ refers to the RL fractional integral of order qj>0, and bj represents suitable real constants for j=1,2,,u.

    Supposing that ϖ:(,σ]Θ and ςU, we denote ϖςΘ as an element defined by

    ϖς(ξ)=ϖ(ς+ξ), ξ(,0].

    Throughout this manuscript, we suppose that ϖς(.) is the historical state trajectory from time to ς, and ϖςΘ, where Θ is an abstract phase space.

    The second main result here is to investigate the existence and uniqueness of solutions to the neutral FFIDE with BVPs. It takes the form

    {LCDpς[ϖ(ς)ς0(ςϑ)p1Γ(p)h(ϑ,ϖϑ,ϑ0Ω1(ϑ,μ,ϖμ)dμ,σ0Υ1(ϑ,μ,ϖμ)dμ)]=g(ς,ϖς,ς0Ω2(ς,ϑ,ϖϑ)dϑ,σ0Υ2(ς,ϑ,ϖϑ)dϑ) p(2,3], ς,ϑU=[0,σ]ϖ(ς)=ψ(ς), ς(,0]ϖ(σ)=uj=1bj(Iqj0+ϖ)(λj), 0<λ1<λ2<<λu<σ, (1.2)

    where h,g:U×Θ×ΞΞ, Ω1,Υ1,Ω2, and Υ2:×ΘΞ are continuous functions.

    ● This paper provides a systematic exploration of fractional functional integrodifferential equations.

    ● Section 2 lays the groundwork by establishing the foundational definitions, notation, and preliminary results.

    ● Existence and uniqueness criteria for FFIDEs are developed in Section 3, employing both Krasnoselskii's FP and Banach's FP theorem.

    ● Building upon these findings, Section 4 extends the existence and uniqueness results to neutral FFIDEs with BVPs.

    ● The applicability and practicality of the theoretical framework are demonstrated through illustrative examples provided in Section 5.

    This section presents fundamental definitions, notation, and lemmas essential for the subsequent analysis. Let Ξ denote a BS equipped with the norm .. Furthermore, C(U,Ξ) represents the BS of continuous functions from the interval U to Ξ, endowed with a uniform convergence topology and the norm .C.

    Definition 2.1. [2] For the function gL1(R+)

    (ⅰ) The RL fractional integral of order p>0 is given by

    Ip0+g(ς)=1Γ(p)ς0(ςϑ)p1g(ϑ)dϑ,

    whenever the integral exists.

    (ⅱ) The LC fractional derivative of order p(v1,v] is described as

    LCDpςg(ς)=1Γ(vp)ς0(ςϑ)vp1g(v)(ϑ)dϑ,

    where g has absolutely continuous derivatives up to order (v1).

    Remark 2.2. It should be noted that, if we take v=1 in Definition 2.1 (ⅱ), we have 0<p1 and

    LCDpςg(ς)=1Γ(1p)ς0(ςϑ)pg(ϑ)dϑ,

    where g(ϑ)=dg(ϑ)dϑ.

    Now, for simplicity, we denote LCDpς and Ip0+ by LCDp and Ip, respectively.

    Lemma 2.3. [2] Assume that p,q0, and gL1[b,c]. Then, for all ς[b,c], we have

    (i) IqIpg(ς)=Iq+pg(ς)=IpIqg(ς);

    (ii) LCDpςIqg(ς)=g(ς).

    Theorem 2.4. [40] (Krasnoselskii's theorem) Assume that Λ is a closed and convex subset of a BS Ξ and that , are two operators satisfying

    (i) for ϖ,ϱΛ, ϖ+ϱΛ,

    (ii) is continuous and compact,

    (iii) is a contraction.

    Then, wΛ exists such that w=w+w.

    This paper considers a seminormed linear state space (Θ,.Θ) of functions from (,0] to Ξ satisfying the following hypotheses of Hale and Kato [41]:

    (H1) On the interval (,σ], if ϖ:(,σ]Ξ is continuous and ϖ0Θ, then for ςU, we have the following stipulations:

    (1) ϖςΘ,

    (2) ϖ(ς)ΘκϖςΘ, where κ is a non-negative constant and is independent of ϖ(.),

    (3) There is a continuous function N1:[0,)[0,) and a locally bounded function N2:[0,)[0,) in order that

    ϖςΘN1(ς)sup{ϖ(ϑ):0ϑς}+N2(ς)ϖ(.)Θ,

    where N1 and N2 are independent of ϖ(.).

    (H2) The space Θ is complete.

    (H3) On the interval U, ϖς is a B-valued continuous function, where ϖ(.) is described in (H1).

    Here, we consider N1=supςUN1(ς) and N2=supςUN2(ς).

    This section is devoted to investigating the existence and uniqueness of solution to the considered problem (1.1) by applying Krasnoselskii's and Banach's FP theorems.

    Assume the space

    ˜={ϖ:(,σ]Ξ:ϖ(,0]Θ and ϖU is continuous},

    and select Pϖ(ς)=ς0Ω(ς,ϑ,ϖϑ)dϑ, and Qϖ(ς)=σ0Υ(ς,ϑ,ϖϑ)dϑ.

    Definition 3.1. We say that the function ϖ˜ is a solution to the FFIDE (1.1) if it fulfills the problem

    {LCDpϖ(ς)=g(ς,ϖς,Pϖ(ς),Qϖ(ς)),ϖ(ς)=ψ(ς), ς(,0],ϖ(σ)=uj=1bj(Iqjϖ)(λj), 0<λ1<λ2<<λu<σ.

    We initiate our analysis of the nonlinear problem (1.1) by examining its linear counterpart, thereby obtaining a foundational solution.

    Lemma 3.2. Assume that ϖ(ς)C(U,Ξ)  satisfies the following problem:

    {LCDpϖ(ς)=g(ς),p(2,3],ςU,ϖ(ς)=ψ(ς),ς(,0],ϖ(σ)=uj=1bj(Iqjϖ)(λj),0<λ1<λ2<<λu<σ. (3.1)

    Then the unique solution of the fractional BVP (3.1) can be written as

    ϖ(ς)={ψ(ς),ς(,0],Ipg(ς)+ςB(uj=1bjIqj+pg(λj)Ipg(σ))+ψ(0)(1+ςB(uj=1bjλqijΓ(qi+1)1)),ςU,

    where B=σuj=1bjλqi+1jΓ(qi+2)0, provided that uj=1bjλqijΓ(qi+1)>1.

    Proof. Suppose that ρ0,ρ1Ξ are vector constants. Based on [2], the solution of (3.1) takes the form

    ϖ(ς)=Ipg(ς)+ρ0+ρ1ς. (3.2)

    Applying the condition ϖ(ς)=ψ(ς), we get

    ρ0=ψ(0). (3.3)

    Using the condition ϖ(σ)=uj=1bj(Iqjϖ)(λj), we have

    ρ1=1(σuj=1bj λqi+1jΓ(qi+2)){uj=1bjIqj+pg(λj)+ψ(0)(uj=1bj λqijΓ(qi+1)1)Ipg(σ)}. (3.4)

    From (3.3) and (3.4) in (3.2), we can write

    ϖ(ς)=Ipg(ς)+ςB(uj=1bjIqj+pg(λj)Ipg(σ))+ψ(0)(1+ςB(uj=1bj λqijΓ(qi+1)1)).

    After that, we need the following assertions:

    (A1) For all ς,ϑU, ψ1,ψ2Θ and ϖ1,ϖ2,˜ϖ1,˜ϖ2Ξ, g,P,Q exist in order that

    {g(ς,ψ1,ϖ1,˜ϖ1)g(ς,ψ2,ϖ2,˜ϖ2)Ξg(ψ1ψ2Θ+ϖ1ϖ2Ξ+˜ϖ1˜ϖ2Ξ),P(ς,ϑ,ψ1)P(ς,ϑ,ψ2)ΞPψ1ψ2Θ,Q(ς,ϑ,ψ1)Q(ς,ϑ,ψ2)ΞQψ1ψ2Θ.

    (A2) For all (ς,ψ,ϖ1,ϖ2)U×Θ×Ξ×Ξ and (ς,ϑ,ψ)×Θ, VjL1(U,R+) (j=1,2,3,4,5) exists such that

    {g(ς,ψ,ϖ1,ϖ2)ΞV1(ς)ψΘ+V2(ς)ϖ1Ξ+V3(ς)ϖ2Ξ,P(ς,ϑ,ψ)ΞV4(ς)ψΘ,Q(ς,ϑ,ψ)ΞV5(ς)ψΘ.

    (A3) We consider S=gN1{ξ1+ξ2(P+Q)}<1, where

    {ξ1=(1+σ|B|)ν1+σ|B|ν3,ξ2=(1+σ|B|)ν2+σ|B|ν4,ν1=σpΓ(1+p), ν2=σp+1Γ(2+p),ν3=uj=1|bj|λqi+pjΓ(qi+p+1), ν4=uj=1|bj| λqi+p+1jΓ(qi+p+2).

    Now, the first main result in this part is as follows:

    Theorem 3.3. Under Assertions (A1) and (A2), the BVP (1.1) has at least one solution on (,σ], provided that

    =σ|B|gN1{(ν1+ν3)+(ν2+ν4)(P+Q)}<1.

    Proof. The FP technique involves equating a given operator to the problem at hand and seeking a unique FP, which corresponds to the problem's unique solution. Therefore, we convert the BVP (1.1) to an FP problem. Define the operator M:˜˜ as

    (Mϖ)(ς)={ψ(ς), ς(,0],ς0(ςϑ)p1Γ(p)g(ϑ,ϖϑ,Pϖ(ϑ),Qϖ(ϑ))dϑ+ςB(uj=1bjλj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϖϑ,Pϖ(ϑ),Qϖ(ϑ))dϑσ0(σϑ)p1Γ(p)g(ϑ,ϖϑ,Pϖ(ϑ),Qϖ(ϑ))dϑ)+ψ(0)(1+ςB(uj=1bj λqijΓ(qi+1)1)), ςU. (3.5)

    Assume that ϱ(.):(,σ]Ξ is a function described as

    ϱ(ς)={ψ(ς), ς(,0],0, ςU.

    It is clear that ϱ0=0. For every ωC(U,Ξ) with ω(0)=0, we select

    ˜ω(ς)={0, ς(,0],ω(ς), ςU.

    If ϖ(.) fulfills (3.5), then we decompose ϖ(.) as ϖ(ς)=ϱ(ς)+˜ω(ς), which leads to ϖς=ϱς+˜ως for all ςU, and ω(.) satisfies

    ω(ς)=ς0(ςϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ+ςB(uj=1bjλj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑσ0(σϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ)+ψ(0)(1+ςB(uj=1bj λqijΓ(qi+1)1)).

    Put G0={ωC(U,Ξ):ω0=0} and consider .G0 to be the seminorm in G0 given by

    ωG0=supςUω(ς)Ξ+ω0Θ=supςUω(ς)Ξ, ωG0.

    Hence, (G0,.G0) is a BS. Describe the operator Φ:G0G0 as

    (Φω)(ς)=ς0(ςϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ+ςB(uj=1bjλj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑσ0(σϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ)+ψ(0)(1+ςB(uj=1bj λqijΓ(qi+1)1)).

    The existence of an FP for an operator M is equivalent to the operator Φ having an FP. Hence, we focus on establishing the existence of an FP for Φ.

    Consider the set Hs={ωG0:ωG0s}. Hence, Hs is a bounded, closed, and convex subset of G0. Assume that there is a positive constant ε such that ε<s, where

    ε=qL1s[(1+σ|B|)(ν1+ν2)+σ|B|(ν3+ν4)]+ψ(0)(1+ς|B|(uj=1bj λqijΓ(qi+1)1)).

    Now, we decompose Φ as Φ1+Φ2 on Hs, where

    (Φ1ω)(ς)=ς0(ςϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ,

    and

    (Φ2ω)(ς)=ςB(uj=1bjλj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑσ0(σϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ)+ψ(0)(1+ςB(uj=1bj λqijΓ(qi+1)1)).

    Now, if we let ω,ωHs and ςU, we get

    (Φ1ω)(ς)+(Φ2ω)(ς)Ξς0(ςϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑ+ς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑ+σ0(σϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑ)+ψ(0)(1+ς|B|(uj=1|bj| λqijΓ(qi+1)1))ς0(ςϑ)p1Γ(p)[V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ]+ς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)(V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ)dϑ+σ0(σϑ)p1Γ(p)(V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ)dϑ)+ψ(0)(1+ς|B|(uj=1|bj| λqijΓ(qi+1)1))VL1s[(1+σ|B|)(ν1+ν2)+σ|B|(ν3+ν4)]+ψ(0)(1+ς|B|(uj=1bj λqijΓ(qi+1)1))=ε.

    Hence,

    Φ1ω+Φ2ωΞε, (3.6)

    where V(ς)=max{V1(ς),V2(ς),V3(ς),V4(ς)} and

    ϱϑ+˜ωϑΘϱϑΘ+˜ωϑΘN1(ϑ)sup0μϑϱ(μ)+N2(ϑ)ϱ(0)+N1(ϑ)sup0μϑ˜ω(μ)+N2(ϑ)˜ω(0)N1s+N2ψΘs.

    It follows from (3.6) that Φ1ω+Φ2ωHs. Now, we show that Φ2 is a contraction. For this, assume that ω,ωHs, and ςU. We then

    (Φ2ω)(ς)(Φ2ω)(ς)Ξς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ+σ0(σϑ)p1Γ(p)×g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])dϑ)ς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)g((ϱϑ+˜ωϑ)(ϱϑ+˜ωϑ)Θ+P[ϱ(ϑ)+˜ω(ϑ)]P[ϱ(ϑ)+˜ω(ϑ)]Ξ+Q[ϱ(ϑ)+˜ω(ϑ)]Q[ϱ(ϑ)+˜ω(ϑ)]Ξ)dϑ+σ0(σϑ)p1Γ(p)g((ϱϑ+˜ωϑ)(ϱϑ+˜ωϑ)Θ+P[ϱ(ϑ)+˜ω(ϑ)]P[ϱ(ϑ)+˜ω(ϑ)]Ξ+Q[ϱ(ϑ)+˜ω(ϑ)]Q[ϱ(ϑ)+˜ω(ϑ)]Ξ)dϑ)ς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)g(˜ωϑ˜ωϑΘ+P˜ωμ˜ωμΘϑ+Q˜ωμ˜ωμΘϑ)dϑ+σ0(σϑ)p1Γ(p)g(˜ωϑ˜ωϑΘ+P˜ωμ˜ωμΘϑ+Q˜ωμ˜ωμΘϑ)dϑ)ς|B|(uj=1|bj|λj0(λjϑ)qj+p1Γ(qj+p)g(N1supϑ[0,ς]ω(ϑ)ω(ϑ)+PN1supμ[0,ϑ]ω(μ)ω(μ)ϑ+QN1supμ[0,ϑ]ω(μ)ω(μ)ϑ)dϑ+σ0(σϑ)p1Γ(p)g(N1supϑ[0,ς]ω(ϑ)ω(ϑ)+PN1supμ[0,ϑ]ω(μ)ω(μ)ϑ+QN1supμ[0,ϑ]ω(μ)ω(μ)ϑ)dϑ)ς|B|N1g(uj=1|bj|(λj)qj+pΓ(qj+p+1)+(P+Q)uj=1|bj|(λj)qj+p+1Γ(qj+p+2)+σpΓ(p+1)+(P+Q)σp+1Γ(p+2))ωωG0σ|B|gN1{(ν1+ν3)+(ν2+ν4)(P+Q)}ωωG0=ωωG0.

    It follows that

    Φ2ωΦ2ωG0ωωG0.

    Since <1, then, Φ2 is contraction. Because g, P and Q are continuous, and thus Φ1 is continuous. Consider

    (Φ1ω)(ς)Ξς0(ςϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑς0(ςϑ)p1Γ(p)[V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ]dϑVL1s(ν1+ν2).

    This proves that Φ1 is uniformly bounded on Hs. Finally, we prove that Φ1 is compact. Indeed, we claim that Φ1 is equicontinuous. For ς1,ς2U, with ς1<ς2 and ωHs, one can write

    (Φ1ω)(ς2)(Φ1ω)(ς1)Ξς10(ς2ϑ)p1(ς1ϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑ+ς2ς1(ς2ϑ)p1Γ(p)g(ϑ,ϱϑ+˜ωϑ,P[ϱ(ϑ)+˜ω(ϑ)],Q[ϱ(ϑ)+˜ω(ϑ)])Ξdϑς10(ς2ϑ)p1(ς1ϑ)p1Γ(p)[V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ]dϑ+ς2ς1(ς2ϑ)p1Γ(p)[V1(ϑ)ϱϑ+˜ωϑΘ+V2(ϑ)P[ϱ(ϑ)+˜ω(ϑ)]Ξ+V3(ϑ)Q[ϱ(ϑ)+˜ω(ϑ)]Ξ]dϑVL1s(ς10(ς2ϑ)p1(ς1ϑ)p1Γ(p)(1+ϑ)dϑ+ς2ς1(ς2ϑ)p1Γ(p)(1+ϑ)dϑ).

    Clearly, \left\Vert \left(\Phi _{1}\omega \right) \left(\varsigma _{2}\right) -\left(\Phi _{1}\omega \right) \left(\varsigma _{2}\right) \right\Vert _{\Xi }\rightarrow 0 as \varsigma _{1}\rightarrow \varsigma _{2}. Consequently, \Phi _{1} is equicontinuous. Applying the Arzelà -Ascoli theorem, we establish that \Phi _{1} is compact on H_{s} . Consequently, invoking Krasnoselskii's FP theorem, we prove the existence of an FP \omega \in G_{0} , satisfying \Phi \omega = \omega , thereby yielding a solution to the fractional BVP (1.1).

    Now, for the uniqueness, we apply Banach's FP theorem as follows:

    Theorem 3.4. Via Assertions (A _{1} ) and (A _{3} ), the BVP (1.1) owns a unique solution on (-\infty, \sigma ] .

    Proof. Recall the set H_{s} = \left\{ \omega \in G_{0}:\left\Vert \omega \right\Vert _{G_{0}}\leq s\right\} and assume that \omega, \omega ^{\ast }\in G_{0}. For \varsigma \in U, one has

    \begin{eqnarray*} &&\left\Vert \left( \Phi \omega \right) (\varsigma )-\left( \Phi \omega ^{\ast }\right) (\varsigma )\right\Vert _{\Xi } \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P \left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P \left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ &&\left. -\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \right) \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{ \omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q}\left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{\omega } _{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P}\left\Vert \widetilde{ \omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) d\vartheta \right. \\ &&+\left. \int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{ \omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q}\left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) d\vartheta \right) \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \ell _{g}\left( N_{1}^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}+\ell _{P}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta +\ell _{Q}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta \right) \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\ell _{g}\left( N_{1}^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}+\ell _{P}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta +\ell _{Q}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta \right) d\vartheta \right. \\ &&+\left. \int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} \ell _{g}\left( N_{1}^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}+\ell _{P}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta +\ell _{Q}\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}\vartheta \right) d\vartheta \right) \\ &\leq &\ell _{g}N_{1}^{\ast }\left\{ \left[ \frac{\sigma ^{p}}{\Gamma (p+1)}+ \frac{\sigma }{\left\vert B\right\vert }\sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p}}{\Gamma (q_{j}+p+1)}+\frac{ \sigma ^{p+1}}{\left\vert B\right\vert \Gamma (p+1)}\right] \right. \\ &&+\left. \left( \ell _{P}+\ell _{Q}\right) \left[ \frac{\sigma ^{p+1}}{ \Gamma (p+2)}+\frac{\sigma }{\left\vert B\right\vert }\sum\limits_{j = 1}^{u} \left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p+1}}{\Gamma (q_{j}+p+2)}+\frac{\sigma ^{p+2}}{\left\vert B\right\vert \Gamma (p+2)} \right] \right\} \left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}} \\ &\leq &\ell _{g}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P}+\ell _{Q}\right) \right\} \left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}} \\ & = &S\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{eqnarray*}

    Hence,

    \begin{equation*} \left\Vert \Phi \left( \omega \right) -\Phi \left( \omega ^{\ast }\right) \right\Vert _{G_{0}}\leq S\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{equation*}

    By (A _{3} ), S < 1, \Phi is a contraction. By Banach's FP theorem, \Phi possesses a unique FP, which is a unique solution to the problem (1.1) on the interval (-\infty, \sigma ] .

    In this section, we discuss the existence and uniqueness of solution to the considered problem (1.2) by applying Krasnoselskii's and Banach's FP theorems.

    Assume that the space \widetilde{\mho } is defined as in the section above and choose

    \begin{equation*} \left\{ \begin{array}{l} P_{1}\varpi (\varsigma ) = \int_{0}^{\varsigma }\Omega _{1}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) d\vartheta , \\ P_{2}\varpi (\varsigma ) = \int_{0}^{\varsigma }\Omega _{2}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) d\vartheta , \\ Q_{1}\varpi (\varsigma ) = \int_{0}^{\sigma }\Upsilon _{1}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) d\vartheta , \\ Q_{2}\varpi (\varsigma ) = \int_{0}^{\sigma }\Upsilon _{2}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) d\vartheta . \end{array} \right. \end{equation*}

    Definition 4.1. We say that the function \varpi \in \widetilde{\mho } is a solution to the FFIDE (1.1) if it fulfills the problem

    \begin{equation*} \left\{ \begin{array}{l} ^{LC}D^{p}\left[ \varpi (\varsigma )-\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)}h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) \right] = g\left( \varsigma ,\varpi _{\varsigma },P_{2}\varpi (\varsigma ),Q_{2}\varpi (\varsigma )\right) ,\text{ }\varsigma \in U, \\ \varpi (\varsigma ) = \psi \left( \varsigma \right) ,\text{ }\varsigma \in (-\infty ,0], \\ \varpi (\sigma ) = \sum\limits_{j = 1}^{{u} }b_{j}\left( I^{q_{j}}\varpi \right) \left( \lambda _{j}\right) ,\text{ }0 < \lambda _{1} < \lambda _{2} < \dots < \lambda _{u} < \sigma . \end{array} \right. \end{equation*}

    With the aid of Lemma 3.2, the solution of the neutral FFIDE (1.2) takes the form

    \begin{equation*} \varpi (\varsigma ) = \left\{ \begin{array}{l} \psi \left( \varsigma \right) ,\ \varsigma \in (-\infty ,0], \\ \int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varpi _{\vartheta },P_{2}\varpi (\vartheta ),Q_{2}\varpi (\vartheta )\right) +\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1} }{\Gamma (p)}h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) \\ +\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}g\left( \vartheta ,\varpi _{\vartheta },P_{2}\varpi (\vartheta ),Q_{2}\varpi (\vartheta )\right) \right. \\ +\sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}h\left( \vartheta ,\varpi _{\vartheta },P_{2}\varpi (\vartheta ),Q_{2}\varpi (\vartheta )\right) -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varpi _{\vartheta },P_{2}\varpi (\vartheta ),Q_{2}\varpi (\vartheta )\right) \\ \left. -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) \right) \\ +\psi \left( 0\right) \left( 1+\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) }-1\right) \right) ,\text{ }\varsigma \in U, \end{array} \right. \end{equation*}

    where B = \sigma -\sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}+1}}{\Gamma \left(q_{i}+2\right) }\neq 0.

    To accomplish our main task here, we needs the following assertions:

    (A _{4} ) For all \varsigma, \vartheta \in U, \psi _{1}, \psi _{2}\in \Theta and \varpi _{1}, \varpi _{2}, \widetilde{ \varpi }_{1}, \widetilde{\varpi }_{2}\in \Xi, \ell _{g}, \ell _{h}, \ell _{P_{1}}, \ell _{Q_{1}}, \ell _{P_{2}}, \ell _{Q_{2}} exist such that

    \begin{equation*} \left\{ \begin{array}{l} \left\Vert g\left( \varsigma ,\psi _{1},\varpi _{1},\widetilde{\varpi } _{1}\right) -g\left( \varsigma ,\psi _{2},\varpi _{2},\widetilde{\varpi } _{2}\right) \right\Vert _{\Xi }\leq \ell _{g}\left( \left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }+\left\Vert \varpi _{1}-\varpi _{2}\right\Vert _{\Xi }+\left\Vert \widetilde{\varpi }_{1}-\widetilde{\varpi }_{2}\right\Vert _{\Xi }\right) , \\ \left\Vert h\left( \varsigma ,\psi _{1},\varpi _{1},\widetilde{\varpi } _{1}\right) -h\left( \varsigma ,\psi _{2},\varpi _{2},\widetilde{\varpi } _{2}\right) \right\Vert _{\Xi }\leq \ell _{h}\left( \left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }+\left\Vert \varpi _{1}-\varpi _{2}\right\Vert _{\Xi }+\left\Vert \widetilde{\varpi }_{1}-\widetilde{\varpi }_{2}\right\Vert _{\Xi }\right) , \\ \left\Vert P_{1}\left( \varsigma ,\vartheta ,\psi _{1}\right) -P_{1}\left( \varsigma ,\vartheta ,\psi _{2}\right) \right\Vert _{\Xi }\leq \ell _{P_{1}}\left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }, \\ \left\Vert P_{2}\left( \varsigma ,\vartheta ,\psi _{1}\right) -P_{2}\left( \varsigma ,\vartheta ,\psi _{2}\right) \right\Vert _{\Xi }\leq \ell _{P_{2}}\left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }, \\ \left\Vert Q_{1}\left( \varsigma ,\vartheta ,\psi _{1}\right) -Q_{1}\left( \varsigma ,\vartheta ,\psi _{2}\right) \right\Vert _{\Xi }\leq \ell _{Q_{1}}\left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }, \\ \left\Vert Q_{2}\left( \varsigma ,\vartheta ,\psi _{1}\right) -Q_{2}\left( \varsigma ,\vartheta ,\psi _{2}\right) \right\Vert _{\Xi }\leq \ell _{Q_{2}}\left\Vert \psi _{1}-\psi _{2}\right\Vert _{\Theta }. \end{array} \right. \end{equation*}

    (A _{5} ) For all \left(\varsigma, \psi, \varpi _{1}, \varpi _{2}\right) \in U\times \Theta \times \Xi \times \Xi and \left(\varsigma, \vartheta, \psi \right) \in \mho \times \Theta, V_{j}\in L^{1}(U, \mathbb{R} _{+}) \left(j = 1, 2, 3, 4, 5\right) exists in order that

    \begin{equation*} \left\{ \begin{array}{l} \left\Vert g\left( \varsigma ,\psi ,\varpi _{1},\varpi _{2}\right) \right\Vert _{\Xi }\leq V_{1}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }+V_{2}(\varsigma )\left\Vert \varpi _{1}\right\Vert _{\Xi }+V_{3}(\varsigma )\left\Vert \varpi _{2}\right\Vert _{\Xi }, \\ \left\Vert h\left( \varsigma ,\psi ,\varpi _{1},\varpi _{2}\right) \right\Vert _{\Xi }\leq V_{4}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }+V_{5}(\varsigma )\left\Vert \varpi _{1}\right\Vert _{\Xi }+V_{6}(\varsigma )\left\Vert \varpi _{2}\right\Vert _{\Xi }, \\ \left\Vert P_{1}\left( \varsigma ,\vartheta ,\psi \right) \right\Vert _{\Xi }\leq V_{7}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }, \\ \left\Vert P_{2}\left( \varsigma ,\vartheta ,\psi \right) \right\Vert _{\Xi }\leq V_{8}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }, \\ \left\Vert Q_{1}\left( \varsigma ,\vartheta ,\psi \right) \right\Vert _{\Xi }\leq V_{9}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }, \\ \left\Vert Q_{2}\left( \varsigma ,\vartheta ,\psi \right) \right\Vert _{\Xi }\leq V_{10}(\varsigma )\left\Vert \psi \right\Vert _{\Theta }. \end{array} \right. \end{equation*}

    (A _{6} ) Assume that S^{\ast } = \ell _{g}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left(\ell _{P_{2}}+\ell _{Q_{2}}\right) \right\} +\ell _{h}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left(\ell _{P_{1}}+\ell _{Q_{1}}\right) \right\} < 1, where

    \begin{equation*} \left\{ \begin{array}{l} \xi _{1} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{1}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{3}, \\ \xi _{2} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{2}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{4}, \\ \nu _{1} = \frac{\sigma ^{p}}{\Gamma (1+p)},\text{ }\nu _{2} = \frac{\sigma ^{p+1}}{\Gamma (2+p)}, \\ \nu _{3} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\lambda _{j}^{q_{i}+p}}{\Gamma \left( q_{i}+p+1\right) },\text{ }\nu _{4} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\text{ }\lambda _{j}^{q_{i}+p+1}}{\Gamma \left( q_{i}+p+2\right) }. \end{array} \right. \end{equation*}

    Theorem 4.2. Under Assertions (A _{4} ) and (A _{5} ), the neutral BVP (1.2) has at least one solution on (-\infty, \sigma ] , provided that

    \begin{equation*} \ell ^{\ast } = \frac{\sigma N_{1}^{\ast }}{\left\vert B\right\vert }\left( \ell _{g}\left[ \left( \nu _{1}+\nu _{3}\right) +\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left( \nu _{2}+\nu _{4}\right) \right] +\ell _{h}\left[ \left( \nu _{1}+\nu _{3}\right) +\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left( \nu _{2}+\nu _{4}\right) \right] \right) < 1. \end{equation*}

    Proof. Define the operator \Game :\widetilde{\mho }\rightarrow \widetilde{\mho } as

    \begin{equation} \left( \Game \varpi \right) (\varsigma ) = \left\{ \begin{array}{l} \psi \left( \varsigma \right) ,\ \varsigma \in (-\infty ,0], \\ \int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \varsigma ,\varpi _{\varsigma },P_{2}\varpi (\varsigma ),Q_{2}\varpi (\varsigma )\right) d\varsigma \\ +\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) d\varsigma \\ +\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}g\left( \varsigma ,\varpi _{\varsigma },P_{2}\varpi (\varsigma ),Q_{2}\varpi (\varsigma )\right) d\varsigma \right. \\ +\sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) d\varsigma \\ -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \varsigma ,\varpi _{\varsigma },P_{2}\varpi (\varsigma ),Q_{2}\varpi (\varsigma )\right) d\varsigma \\ \left. -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varpi _{\vartheta },P_{1}\varpi (\vartheta ),Q_{1}\varpi (\vartheta )\right) d\varsigma \right) \\ +\psi \left( 0\right) \left( 1+\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) }-1\right) \right) ,\text{ }\varsigma \in U. \end{array} \right. \end{equation} (4.1)

    Analogous to Theorem 3.3, define the operator \Re :G_{0}\rightarrow G_{0} as

    \begin{equation*} \left( \Re \varpi \right) (\varsigma ) = \left\{ \begin{array}{l} \int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ +\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ +\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ +\sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ \left. -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right) \\ +\psi \left( 0\right) \left( 1+\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) }-1\right) \right) ,\text{ }\varsigma \in U. \end{array} \right. \end{equation*}

    Describe the set H_{\widehat{s}} as H_{\widehat{s}} = \left\{ \omega \in G_{0}:\left\Vert \omega \right\Vert _{G_{0}}\leq \widehat{s}\right\}. Let there be a positive constant \varepsilon ^{\ast } such that \varepsilon ^{\ast } < \widehat{s}, where

    \begin{equation*} \varepsilon ^{\ast } = 2\left\Vert V^{\ast }\right\Vert _{L^{1}}\widehat{s} ^{\ast }\left[ \xi _{1}+\xi _{2}\right] +\left\Vert \psi \left( 0\right) \right\Vert \left( 1+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) }-1\right) \right), \end{equation*}

    where

    \begin{equation*} V^{\ast }(\varsigma ) = \max \left\{ V_{1}(\varsigma ),V_{2}(\varsigma ),V_{3}(\varsigma ),V_{4}(\varsigma ),V_{5}(\varsigma ),V_{6}(\varsigma ),V_{7}(\varsigma ),V_{8}(\varsigma ),V_{9}(\varsigma ),V_{10}(\varsigma )\right\} . \end{equation*}

    Now, we decompose \Re as \Re _{1}+\Re _{2} on H_{\widehat{s}} , where

    \begin{equation*} \left( \Re _{1}\varpi \right) (\varsigma ) = \left\{ \begin{array}{l} \int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ +\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) , \end{array} \right. \end{equation*}

    and

    \begin{equation*} \left( \Re _{2}\varpi \right) (\varsigma ) = \left\{ \begin{array}{l} \frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ +\sum\limits_{j = 1}^{u}b_{j}\int_{0}^{\lambda _{j}}\frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \\ \left. -\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right) \\ +\psi \left( 0\right) \left( 1+\frac{\varsigma }{B}\left( \sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) }-1\right) \right) ,\text{ }\varsigma \in U. \end{array} \right. \end{equation*}

    Now, for \omega, \omega ^{\ast }\in H_{\widehat{s}} and \varsigma \in U , we have

    \begin{eqnarray*} &&\left\Vert \left( \Re _{1}\omega \right) (\varsigma )+\left( \Re _{2}\omega ^{\ast }\right) (\varsigma )\right\Vert _{\Xi } \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \right. \\ &&+\sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&\left. +\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \right) \\ &&+\left\Vert \psi \left( 0\right) \right\Vert \left( 1+\frac{\varsigma }{ \left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\frac{\left\vert b_{j}\right\vert \text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) } -1\right) \right) \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left[ V_{1}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }\right\Vert _{\Theta }+V_{2}(\vartheta )\left\Vert P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{3}(\vartheta )\left\Vert Q_{2}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }\right. \\ &&\left. +V_{4}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{ \omega }_{\vartheta }\right\Vert _{\Theta }+V_{5}(\vartheta )\left\Vert P_{1} \left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{6}(\vartheta )\left\Vert Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi } \right] d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left[ V_{1}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }\right\Vert _{\Theta }+V_{2}(\vartheta )\left\Vert P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }\right. \right. \\ &&+V_{3}(\vartheta )\left\Vert Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{4}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega }_{\vartheta }\right\Vert _{\Theta }+V_{5}(\vartheta )\left\Vert P_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }(\vartheta )\right] \right\Vert _{\Xi } \\ &&\left. +V_{6}(\vartheta )\left\Vert Q_{1}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }\right] d\theta +\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left[ V_{1}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }\right\Vert _{\Theta }\right. \\ &&+q_{2}(\vartheta )\left\Vert P_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }(\vartheta )\right] \right\Vert _{\Xi }+q_{3}(\vartheta )\left\Vert Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{4}(\vartheta )\left\Vert \varrho _{\vartheta }+ \widetilde{\omega }_{\vartheta }\right\Vert _{\Theta } \\ &&+\left. V_{5}(\vartheta )\left\Vert P_{1}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{6}(\vartheta )\left\Vert Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right\Vert _{\Xi }\right] d\theta \\ &&+\left\Vert \psi \left( 0\right) \right\Vert \left( 1+\frac{\varsigma }{ \left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\frac{\left\vert b_{j}\right\vert \text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) } -1\right) \right) \\ &\leq &2\left\Vert V^{\ast }\right\Vert _{L^{1}}\widehat{s}^{\ast }\left[ \xi _{1}+\xi _{2}\right] +\left\Vert \psi \left( 0\right) \right\Vert \left( 1+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u} \frac{b_{j}\text{ }\lambda _{j}^{q_{i}}}{\Gamma \left( q_{i}+1\right) } -1\right) \right) \\ & = &\varepsilon ^{\ast }, \end{eqnarray*}

    where V^{\ast }(\varsigma) = \max \left\{ V_{1}(\varsigma), V_{2}(\varsigma), V_{3}(\varsigma), V_{4}(\varsigma), V_{5}(\varsigma), V_{6}(\varsigma)\right\} and

    \begin{equation*} \left\Vert \varrho _{\vartheta }+\widetilde{\omega }_{\vartheta }\right\Vert _{\Theta }\leq N_{1}^{\ast }\widehat{s}+N_{2}^{\ast }\widehat{s}\left\Vert \psi \left( 0\right) \right\Vert _{\Theta }\leq \widehat{s}^{\ast }. \end{equation*}

    Hence,

    \begin{equation*} \left\Vert \Re _{1}\omega +\Re _{2}\omega ^{\ast }\right\Vert _{G_{0}}\leq \varepsilon ^{\ast }. \end{equation*}

    Thus, \Re _{1}\omega +\Re _{2}\omega ^{\ast }\in H_{\widehat{s}}. Now, we prove that \Re _{2} is a contraction. Let \omega, \omega ^{\ast }\in H_{ \widehat{s}} and \varsigma \in U . We then

    \begin{eqnarray*} &&\left\Vert \left( \Re _{2}\omega \right) (\varsigma )-\left( \Re _{2}\omega ^{\ast }\right) (\varsigma )\right\Vert _{\Xi } \\ &\leq &\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left[ \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \right. \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi } \\ &&+\left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \\ &&-\left. \left. h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }\right] d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left[ \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi } \\ &&+\left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \\ &&-\left. \left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }\right] d\vartheta \\ &\leq &\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left[ \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{\omega } _{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P_{2}}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q_{2}}\left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right. \right. \\ &&+\left. \ell _{h}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P_{1}}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q_{1}}\left\Vert \widetilde{ \omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right] d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left[ \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{ \omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P_{2}}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q_{2}}\left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right. \\ &&+\left. \left. \ell _{h}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\ell _{P_{1}}\left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta +\ell _{Q_{1}}\left\Vert \widetilde{ \omega }_{\mu }-\widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right] d\vartheta \right) \\ &\leq &\frac{\sigma N_{1}^{\ast }}{\left\vert B\right\vert }\left[ \begin{array}{c} \ell _{g}\left\{ \left( \nu _{1}+\nu _{3}\right) +\left( \nu _{2}+\nu _{4}\right) \left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \right\} \\ +\ell _{h}\left\{ \left( \nu _{1}+\nu _{3}\right) +\left( \nu _{2}+\nu _{4}\right) \left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \right\} \end{array} \right] \left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}} \\ & = &\ell ^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{eqnarray*}

    It follows that

    \begin{equation*} \left\Vert \Re _{2}\omega -\Re _{2}\omega ^{\ast }\right\Vert _{G_{0}}\leq \ell ^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{equation*}

    Since \ell ^{\ast } < 1, then, \Re _{2} is contraction. Since g , P_{1}, P_{2}, Q_{1} and Q_{2} are continuous, then \Re _{1} is continuous. Furthermore,

    \begin{eqnarray*} &&\left\Vert \left( \Re _{1}\omega \right) \left( \varsigma \right) \right\Vert _{\Xi } \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right\Vert _{\Xi }d\vartheta \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left[ V_{1}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }\right\Vert _{\Theta }+V_{2}(\vartheta )\left\Vert P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{3}(\vartheta )\left\Vert Q_{2}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }\right] d\vartheta \\ &&+\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left[ V_{4}(\vartheta )\left\Vert \varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }\right\Vert _{\Theta }+V_{5}(\vartheta )\left\Vert P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }+V_{6}(\vartheta )\left\Vert Q_{1}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right\Vert _{\Xi }\right] d\vartheta \\ &\leq &2\left\Vert q^{\ast }\right\Vert _{L^{1}}\widehat{s}^{\ast }\left( \nu _{1}+\nu _{2}\right) . \end{eqnarray*}

    Hence, \Re _{1} is uniformly bounded on H_{s}. Ultimately, we claim that \Re _{1} is compact. Indeed, we prove that \Re _{1} is equicontinuous. For \varsigma _{1}, \varsigma _{2}\in U, with \varsigma _{1} < \varsigma _{2} and \omega \in H_{\widehat{s}}, one has

    \begin{eqnarray*} &&\left\Vert \left( \Re _{1}\omega \right) \left( \varsigma _{2}\right) -\left( \Re _{1}\omega \right) \left( \varsigma _{1}\right) \right\Vert _{\Xi } \\ &\leq &\int_{0}^{\varsigma _{1}}\frac{(\varsigma _{2}-\vartheta )^{p-1}-(\varsigma _{1}-\vartheta )^{p-1}}{\Gamma (p)}\left[ \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right\Vert _{\Xi }\right. \\ &&+\left. \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{ \omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } (\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } (\vartheta )\right] \right) \right\Vert _{\Xi }\right] d\vartheta \\ &&+\int_{\varsigma _{1}}^{\varsigma _{2}}\frac{(\varsigma _{2}-\vartheta )^{p-1}}{\Gamma (p)}\left[ \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+ \widetilde{\omega }(\vartheta )\right] \right) \right\Vert _{\Xi }\right. \\ &&+\left. \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{ \omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } (\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } (\vartheta )\right] \right) \right\Vert _{\Xi }\right] d\vartheta \\ &\leq &\int_{0}^{\varsigma _{1}}\frac{(\varsigma _{2}-\vartheta )^{p-1}-(\varsigma _{1}-\vartheta )^{p-1}}{\Gamma (p)}\left[ V_{1}(\vartheta )\widehat{s}^{\ast }+V_{2}(\vartheta )V_{8}(\vartheta )\widehat{s}^{\ast }\vartheta +V_{3}(\vartheta )V_{10}(\vartheta )\widehat{s}^{\ast }\vartheta \right. \\ &&+\left. V_{4}(\vartheta )\widehat{s}^{\ast }+V_{5}(\vartheta )V_{7}(\vartheta )\widehat{s}^{\ast }\vartheta +V_{6}(\vartheta )V_{9}(\vartheta )\widehat{s}^{\ast }\vartheta \right] d\vartheta \\ &&+\int_{\varsigma _{1}}^{\varsigma _{2}}\frac{(\varsigma _{2}-\vartheta )^{p-1}}{\Gamma (p)}\left[ V_{1}(\vartheta )\widehat{s}^{\ast }+V_{2}(\vartheta )V_{8}(\vartheta )\widehat{s}^{\ast }\vartheta +V_{3}(\vartheta )V_{10}(\vartheta )\widehat{s}^{\ast }\vartheta \right. \\ &&+\left. V_{4}(\vartheta )\widehat{s}^{\ast }+V_{5}(\vartheta )q_{7}(\vartheta )\widehat{s}^{\ast }\vartheta +V_{6}(\vartheta )V_{9}(\vartheta )\widehat{s}^{\ast }\vartheta \right] d\vartheta \\ &\leq &2\left\Vert V^{\ast }\right\Vert _{L^{1}}\widehat{s}^{\ast }\left( \int_{0}^{\varsigma _{1}}\frac{(\varsigma _{2}-\vartheta )^{p-1}-(\varsigma _{1}-\vartheta )^{p-1}}{\Gamma (p)}(1+\vartheta )d\vartheta +\int_{\varsigma _{1}}^{\varsigma _{2}}\frac{(\varsigma _{2}-\vartheta )^{p-1}}{\Gamma (p)} (1+\vartheta )d\vartheta \right) . \end{eqnarray*}

    Therefore, \left\Vert \left(\Re _{1}\omega \right) \left(\varsigma _{2}\right) -\left(\Re _{1}\omega \right) \left(\varsigma _{2}\right) \right\Vert _{\Xi }\rightarrow 0 as \varsigma _{1}\rightarrow \varsigma _{2}. Hence, \Phi _{1} is equicontinuous. By the Arzelà-Ascoli theorem, we establish that \Re _{1} is compact on H_{\widehat{s}} . Consequently, invoking Krasnoselskii's FP theorem, \omega \in G_{0} exists such that \Re \omega = \omega , which is a solution the neutral BVP (1.2).

    For the uniqueness, we have the following theorem:

    Theorem 4.3. Via Assertions (A _{4} ) and (A _{6} ), the neutral BVP (1.2) has a unique solution on (-\infty, \sigma ] .

    Proof. Define the set H_{\widehat{s}} = \left\{ \omega \in G_{0}:\left\Vert \omega \right\Vert _{G_{0}}\leq s\right\} and assume that \omega, \omega ^{\ast }\in G_{0}. For \varsigma \in U, we get

    \begin{eqnarray*} &&\left\Vert \left( \Re \omega \right) (\varsigma )-\left( \Re \omega ^{\ast }\right) (\varsigma )\right\Vert _{\Xi } \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta ) \right] \right) \right. \\ &&-\left. h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ &&-\left. h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left\Vert g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ &&-\left. g\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{2}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{2}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left\Vert h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega }_{\vartheta },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{\omega }(\vartheta )\right] \right) \right. \\ &&-\left. \left. h\left( \vartheta ,\varrho _{\vartheta }+\widetilde{\omega } _{\vartheta }^{\ast },P_{1}\left[ \varrho (\vartheta )+\widetilde{\omega } ^{\ast }(\vartheta )\right] ,Q_{1}\left[ \varrho (\vartheta )+\widetilde{ \omega }^{\ast }(\vartheta )\right] \right) \right\Vert _{\Xi }d\vartheta \right) \\ &\leq &\int_{0}^{\varsigma }\frac{(\varsigma -\vartheta )^{p-1}}{\Gamma (p)} \left[ \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right. \\ &&+\left. \ell _{h}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right] d\vartheta \\ &&+\frac{\varsigma }{\left\vert B\right\vert }\left( \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \int_{0}^{\lambda _{j}} \frac{(\lambda _{j}-\vartheta )^{q_{j}+p-1}}{\Gamma (q_{j}+p)}\left[ \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{\omega } _{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left\Vert \widetilde{\omega }_{\mu }-\widetilde{\omega } _{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right. \right. \\ &&+\left. \ell _{h}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right] d\vartheta \\ &&+\int_{0}^{\sigma }\frac{(\sigma -\vartheta )^{p-1}}{\Gamma (p)}\left[ \ell _{g}\left( \left\Vert \widetilde{\omega }_{\vartheta }-\widetilde{ \omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right. \\ &&\left. \left. +\ell _{h}\left( \left\Vert \widetilde{\omega }_{\vartheta }- \widetilde{\omega }_{\vartheta }^{\ast }\right\Vert _{\Theta }+\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left\Vert \widetilde{\omega }_{\mu }- \widetilde{\omega }_{\mu }^{\ast }\right\Vert _{\Theta }\vartheta \right) \right] d\vartheta \right) \\ &\leq &\left\{ \ell _{g}N_{1}^{\ast }\left[ \frac{\sigma ^{p}}{\Gamma (p+1)}+ \frac{\sigma }{\left\vert B\right\vert }\sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p}}{\Gamma (q_{j}+p+1)}+\frac{ \sigma ^{p+1}}{\left\vert B\right\vert \Gamma (p+1)}\right. \right. \\ &&+\left. \left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left( \frac{\sigma ^{p+1}}{\Gamma (p+2)}+\frac{\sigma }{\left\vert B\right\vert } \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p+1}}{\Gamma (q_{j}+p+2)}+\frac{\sigma ^{p+2}}{\left\vert B\right\vert \Gamma (p+2)}\right) \right] \\ &&+\ell _{h}N_{1}^{\ast }\left[ \frac{\sigma ^{p}}{\Gamma (p+1)}+\frac{ \sigma }{\left\vert B\right\vert }\sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p}}{\Gamma (q_{j}+p+1)}+\frac{ \sigma ^{p+1}}{\left\vert B\right\vert \Gamma (p+1)}\right. \\ &&+\left. \left. \left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left( \frac{ \sigma ^{p+1}}{\Gamma (p+2)}+\frac{\sigma }{\left\vert B\right\vert } \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{(\lambda _{j})^{q_{j}+p+1}}{\Gamma (q_{j}+p+2)}+\frac{\sigma ^{p+2}}{\left\vert B\right\vert \Gamma (p+2)}\right) \right] \right\} \left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}} \\ &\leq &\ell _{g}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \right\} +\ell _{h}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \right\} \left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}} \\ & = &S^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{eqnarray*}

    Hence,

    \begin{equation*} \left\Vert \Re \left( \omega \right) -\Re \left( \omega ^{\ast }\right) \right\Vert _{G_{0}}\leq S^{\ast }\left\Vert \omega -\omega ^{\ast }\right\Vert _{G_{0}}. \end{equation*}

    By (A _{6} ), S^{\ast } < 1. Thus, \Re is a contraction. By Banach's FP theorem, \Re has a unique FP, which is a unique solution to the problem (1.2) on (-\infty, \sigma ] .

    This section is devoted to testing the conditions of the proposed systems and their effectiveness, which leads to supporting and enhancing the theoretical results we obtained.

    Example 5.1. Assume the following FFIDE:

    \begin{equation} \left\{ \begin{array}{l} ^{LC}D^{\frac{5}{2}}\varpi (\varsigma ) = \frac{1}{\left( \varsigma +7\right) ^{2}}e^{-\nu \varsigma }\frac{\left\vert \varpi _{\varsigma }\right\vert }{ 1+\left\vert \varpi _{\varsigma }\right\vert }+\frac{1}{32} \int_{0}^{\varsigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{ 16(1+\vartheta )}\frac{\cos (\varpi _{\vartheta })}{1+\cos (\varpi _{\vartheta })}d\vartheta +\frac{1}{64}\int_{0}^{\sigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{20(1+\vartheta )}\frac{\sin (\varpi _{\vartheta })}{1+\sin (\varpi _{\vartheta })}d\vartheta ,\text{ }\varsigma \in \lbrack 0,1], \\ \varpi (\varsigma ) = \psi \left( \varsigma \right) ,\text{ }\varsigma \in (-\infty ,0], \\ \varpi (1) = \sum\limits_{j = 1}^{3}b_{j}\left( I_{0^{+}}^{q_{j}}\varpi \right) \left( \lambda _{j}\right) ,\text{ }0 < \lambda _{1} < \lambda _{2} < \lambda _{3} < 1, \end{array} \right. \end{equation} (5.1)

    where \nu > 0 is a real constant and define the set H_{\nu } as

    \begin{equation*} H_{\nu } = \left\{ \omega \in C\left( (-\infty ,0], \mathbb{R} \right) :\lim\limits_{\phi \rightarrow -\infty }e^{\nu \phi }\omega (\phi )\text{ exists in } \mathbb{R} \right\} , \end{equation*}

    under the norm

    \begin{equation*} \left\Vert \omega \right\Vert _{\nu } = \sup\limits_{\phi \in (-\infty ,0]}e^{\nu \phi }\left\vert \omega (\phi )\right\vert . \end{equation*}

    Assume that \varpi :(-\infty, \sigma ]\rightarrow \Xi in order that \varpi _{0} = \psi \in H_{\nu }. Then

    \begin{equation*} \lim\limits_{\phi \rightarrow -\infty }e^{\nu \phi }\omega _{\varsigma }(\phi ) = \lim\limits_{\phi \rightarrow -\infty }e^{\nu \phi }\omega (\varsigma +\phi ) = \lim\limits_{\phi \rightarrow -\infty }e^{\nu (\phi -\varsigma )}\omega (\phi ) = e^{-\nu \varsigma }\lim\limits_{\phi \rightarrow -\infty }e^{\nu \phi }\omega _{0}(\phi ) < \infty . \end{equation*}

    Therefore, \omega _{\varsigma }\in H_{\nu }. Select N_{1} = N_{2} = \kappa = 1. Hence, we show the condition

    \begin{equation*} \left\Vert \omega _{\varsigma }\right\Vert _{\nu }\leq N_{1}(\varsigma )\sup \left\{ \left\vert \varpi (\vartheta )\right\vert :0\leq \vartheta \leq \varsigma \right\} +N_{2}(\varsigma )\left\Vert \varpi _{0}\right\Vert _{\nu }. \end{equation*}

    Clearly, \left\vert \omega _{\varsigma }(\phi)\right\vert = \left\vert \omega (\varsigma +\phi)\right\vert. If \varsigma +\phi \leq 0, we get

    \begin{equation*} \left\vert \omega _{\varsigma }(\phi )\right\vert \leq \sup \left\{ \left\vert \varpi (\vartheta )\right\vert :-\infty < \vartheta \leq 0\right\} . \end{equation*}

    In the case of \varsigma +\phi \geq 0, we have

    \begin{equation*} \left\vert \omega _{\varsigma }(\phi )\right\vert \leq \sup \left\{ \left\vert \varpi (\vartheta )\right\vert :0 < \vartheta \leq \varsigma \right\} . \end{equation*}

    Hence, if \varsigma +\phi \in \lbrack 0, 1], we can write

    \begin{equation*} \left\vert \omega _{\varsigma }(\phi )\right\vert \leq \sup \left\{ \left\vert \varpi (\vartheta )\right\vert :-\infty < \vartheta \leq 0\right\} +\sup \left\{ \left\vert \varpi (\vartheta )\right\vert :0\leq \vartheta \leq \varsigma \right\} , \end{equation*}

    which implies that

    \begin{equation*} \left\Vert \omega _{\varsigma }\right\Vert _{\nu }\leq \sup \left\{ \left\vert \varpi (\vartheta )\right\vert :0\leq \vartheta \leq \varsigma \right\} +\left\Vert \varpi _{0}\right\Vert _{\nu }. \end{equation*}

    Furthermore, the pair (H_{\nu }, \left\Vert \omega \right\Vert) is a BS and H_{\nu } is a phase space. Here, p = \frac{5}{2}, u = 3, and we choose

    \begin{equation*} \begin{array}{ccc} b_{1} = \frac{1}{6}, & b_{2} = \frac{1}{8}, & b_{3} = 4, \\ \lambda _{1} = \frac{1}{9}, & \lambda _{2} = \frac{1}{4}, & \lambda _{3} = \frac{7 }{11}, \\ q_{1} = \frac{1}{3}, & q_{2} = \frac{1}{2}, & q_{3} = \frac{6}{5}. \end{array} \end{equation*}

    By simple calculation, we have

    \begin{equation*} \left\{ \begin{array}{l} B = \sigma -\sum\limits_{j = 1}^{u}\frac{b_{j}\text{ }\lambda _{j}^{q_{i}+1}}{ \Gamma \left( q_{i}+2\right) } = 1-\sum\limits_{j = 1}^{3}\frac{b_{j}\text{ } \lambda _{j}^{q_{i}+1}}{\Gamma \left( q_{i}+2\right) }\approx 0.3729\neq 0, \\ \nu _{1} = \frac{\sigma ^{p}}{\Gamma (1+p)}\approx 0.3009,\text{ }\nu _{2} = \frac{\sigma ^{p+1}}{\Gamma (2+p)}\approx 0.0859, \\ \nu _{3} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\lambda _{j}^{q_{i}+p}}{\Gamma \left( q_{i}+p+1\right) }\approx 0.0088,\text{ }\nu _{4} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\lambda _{j}^{q_{i}+p+1}}{\Gamma \left( q_{i}+p+2\right) }\approx 0.0008, \\ \xi _{1} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{1}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{3}\approx 1.1314, \\ \xi _{2} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{2}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{4}\approx 0.3184. \end{array} \right. \end{equation*}

    From (5.1), we have

    \begin{equation*} g\left( \varsigma ,\varpi _{\varsigma },P(\varsigma ),Q(\varsigma )\right) = \frac{1}{\left( \varsigma +7\right) ^{2}}e^{-\nu \varsigma }\frac{\left\vert \varpi _{\varsigma }\right\vert }{1+\left\vert \varpi _{\varsigma }\right\vert }+\frac{1}{32}P\varpi (\varsigma )+\frac{1}{64}Q\varpi (\varsigma ), \end{equation*}

    where

    \begin{eqnarray*} P\varpi (\varsigma ) & = &\int_{0}^{\varsigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{16(1+\vartheta )}\frac{\cos (\varpi _{\vartheta })}{ 1+\cos (\varpi _{\vartheta })}d\vartheta , \\ Q\varpi (\varsigma ) & = &\int_{0}^{\sigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{20(1+\vartheta )}\frac{\sin (\varpi _{\vartheta })}{1+\sin (\varpi _{\vartheta })}d\vartheta . \end{eqnarray*}

    Now, for \varpi _{\varsigma }, \varrho _{\varsigma }\in H_{\nu }, we have

    \begin{eqnarray} \left\vert P\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -P\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert \frac{\varsigma \vartheta e^{-\nu \varsigma }}{16(1+\vartheta ) }\frac{\cos (\varpi _{\vartheta })}{1+\cos (\varpi _{\vartheta })}-\frac{ \varsigma \vartheta e^{-\nu \varsigma }}{16(1+\vartheta )}\frac{\cos (\varrho _{\vartheta })}{1+\cos (\varrho _{\vartheta })}\right\vert \\ &\leq &\frac{1}{16}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.2)
    \begin{eqnarray} \left\vert Q\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -Q\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert \frac{\varsigma \vartheta e^{-\nu \varsigma }}{20(1+\vartheta ) }\frac{\sin (\varpi _{\vartheta })}{1+\sin (\varpi _{\vartheta })}-\frac{ \varsigma \vartheta e^{-\nu \varsigma }}{20(1+\vartheta )}\frac{\sin (\varrho _{\vartheta })}{1+\sin (\varrho _{\vartheta })}\right\vert \\ &\leq &\frac{1}{20}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.3)
    \begin{eqnarray} &&\left\vert g\left( \varsigma ,\varpi _{\varsigma },P\varpi (\varsigma ),Q\varpi (\varsigma )\right) -g\left( \varsigma ,\varrho _{\varsigma },P\varrho (\varsigma ),Q\varrho (\varsigma )\right) \right\vert \\ &\leq &\frac{1}{\left( \varsigma +7\right) ^{2}}e^{-\nu \varsigma }\frac{ \left\vert \varpi _{\varsigma }-\varrho _{\varsigma }\right\vert }{\left( 1+\left\vert \varpi _{\varsigma }\right\vert \right) \left( 1+\left\vert \varrho _{\varsigma }\right\vert \right) }+\frac{1}{32}\left\vert P\varpi (\varsigma )-P\varrho (\varsigma )\right\vert +\frac{1}{64}\left\vert Q\varpi (\varsigma )-P\varrho (\varsigma )\right\vert \\ &\leq &\frac{1}{64}\left( \left\Vert \varpi -\varrho \right\Vert _{\nu }+ \frac{1}{8}\left\Vert \varpi -\varrho \right\Vert _{\nu }+\frac{1}{20} \left\Vert \varpi -\varrho \right\Vert _{\nu }\right) , \end{eqnarray} (5.4)
    \begin{eqnarray} &&\left\vert g\left( \varsigma ,\psi ,\varpi ,\varrho \right) \right\vert \\ & = &\left\vert \frac{1}{\left( \varsigma +7\right) ^{2}}e^{-\nu \varsigma } \frac{\left\vert \psi _{\varsigma }\right\vert }{1+\left\vert \psi _{\varsigma }\right\vert }+\frac{1}{32}\int_{0}^{\varsigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{16(1+\vartheta )}\frac{\cos (\varpi _{\vartheta })}{1+\cos (\varpi _{\vartheta })}d\vartheta +\frac{1}{64} \int_{0}^{\sigma }\frac{\varsigma \vartheta e^{-\nu \varsigma }}{ 20(1+\vartheta )}\frac{\sin (\varrho _{\vartheta })}{1+\sin (\varrho _{\vartheta })}d\vartheta \right\vert \\ &\leq &\frac{1}{64}\left\vert \psi \right\vert +\frac{1}{32}\left\vert \varpi \right\vert +\frac{1}{64}\left\vert \varrho \right\vert , \end{eqnarray} (5.5)
    \begin{equation} \left\vert P\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert \frac{\varsigma \vartheta e^{-\nu \varsigma }}{16(1+\vartheta )} \frac{\cos (\varpi _{\vartheta })}{1+\cos (\varpi _{\vartheta })}\right\vert \leq \frac{1}{32}\left\vert \varpi \right\vert , \end{equation} (5.6)

    and

    \begin{equation} \left\vert Q\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert \frac{\varsigma \vartheta e^{-\nu \varsigma }}{20(1+\vartheta )} \frac{\sin (\varpi _{\vartheta })}{1+\sin (\varpi _{\vartheta })}\right\vert \leq \frac{1}{40}\left\vert \varpi \right\vert . \end{equation} (5.7)

    It follows from (5.2)–(5.7) that \ell _{g} = \frac{1}{64}, \ell _{P} = \frac{1}{16}, \ell _{Q} = \frac{1}{20}, V_{1}(\varsigma) = \frac{1 }{64}, V_{2}(\varsigma) = \frac{1}{32}, V_{3}(\varsigma) = \frac{1}{64}, V_{4}(\varsigma) = \frac{1}{32}, V_{5}(\varsigma) = \frac{1}{40}, and N_{1}^{\ast } = 1. Hence

    \begin{equation*} \ell = \frac{\sigma }{\left\vert B\right\vert }\ell _{g}N_{1}^{\ast }\left\{ \left( \nu _{1}+\nu _{3}\right) +\left( \nu _{2}+\nu _{4}\right) \left( \ell _{P}+\ell _{Q}\right) \right\} \approx 0.0138 < 1, \end{equation*}

    and

    \begin{equation*} S = \ell _{g}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P}+\ell _{Q}\right) \right\} \approx 0.0182 < 1. \end{equation*}

    Therefore, all the requirements of Theorems 3.3 and 3.4 are satisfied. Then, the considered problem (5.1) has a unique solution on (-\infty, \sigma ].

    Example 5.2. Assume the following neutral FFIDE:

    \begin{equation} \left\{ \begin{array}{l} ^{LC}D_{\varsigma }^{\frac{5}{2}}\left[ \varpi (\varsigma )-\int_{0}^{\varsigma }\sqrt{\frac{(\varsigma -\vartheta )^{p-1}}{\pi }} \left( \frac{e^{-\nu \varsigma }}{20}\frac{\varpi _{\varsigma }^{2}}{ 1+\varpi _{\varsigma }^{2}}+\frac{1}{20}\int_{0}^{\varsigma }\frac{e^{-\nu \varsigma }}{6}\ln \left( 1+\varpi _{\varsigma }\right) d\vartheta +\frac{1}{ 25}\int_{0}^{\sigma }\frac{e^{-\nu \varsigma }}{4}\frac{\tan ^{-1}\left( \varpi _{\varsigma }\right) }{1+\tan ^{-1}\left( \varpi _{\varsigma }\right) }d\vartheta \right) \right] \\ = \frac{\left( 1+e^{-\varsigma }\right) e^{-\nu \varsigma }}{\left( 34+e^{\varsigma }\right) }\frac{\left\vert \varpi _{\varsigma }\right\vert }{ 1+\left\vert \varpi _{\varsigma }\right\vert }+\frac{1}{15} \int_{0}^{\varsigma }e^{-\nu \varsigma }\cos (\frac{\varpi _{\vartheta }}{5} )d\vartheta +\frac{1}{35}\int_{0}^{\sigma }e^{-\nu \varsigma }\sin (\frac{ \varpi _{\vartheta }}{6})d\vartheta ,\text{ }\varsigma \in \lbrack 0,1], \\ \varpi (\varsigma ) = \psi \left( \varsigma \right) ,\text{ }\varsigma \in (-\infty ,0], \\ \varpi (1) = \sum\limits_{j = 1}^{3}b_{j}\left( I_{0^{+}}^{q_{j}}\varpi \right) \left( \lambda _{j}\right) ,\text{ }0 < \lambda _{1} < \lambda _{2} < \lambda _{3} < 1. \end{array} \right. \end{equation} (5.8)

    Assume that H_{\nu } is the phase space, which is defined in Example 5.1, where p = \frac{5}{2}, u = 3 , and

    \begin{equation*} \begin{array}{ccc} b_{1} = \frac{1}{5}, & b_{2} = \frac{1}{4}, & b_{3} = 5, \\ \lambda _{1} = \frac{1}{7}, & \lambda _{2} = \frac{1}{2}, & \lambda _{3} = \frac{7 }{12}, \\ q_{1} = \frac{1}{2}, & q_{2} = \frac{1}{3}, & q_{3} = \frac{7}{2}. \end{array} \end{equation*}

    By simple calculation, we have

    \begin{equation*} \left\{ \begin{array}{l} B\approx 0.5231\neq 0, \\ \nu _{1} = \frac{\sigma ^{p}}{\Gamma (1+p)}\approx 0.4187,\text{ }\nu _{2} = \frac{\sigma ^{p+1}}{\Gamma (2+p)}\approx 0.3979, \\ \nu _{3} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\lambda _{j}^{q_{i}+p}}{\Gamma \left( q_{i}+p+1\right) }\approx 0.0132,\text{ }\nu _{4} = \sum\limits_{j = 1}^{u}\left\vert b_{j}\right\vert \frac{\lambda _{j}^{q_{i}+p+1}}{\Gamma \left( q_{i}+p+2\right) }\approx 0.0037, \\ \xi _{1} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{1}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{3}\approx 1.5841, \\ \xi _{2} = \left( 1+\frac{\sigma }{\left\vert B\right\vert }\right) \nu _{2}+ \frac{\sigma }{\left\vert B\right\vert }\nu _{4}\approx 0.7239. \end{array} \right. \end{equation*}

    From (5.8), one can write

    \begin{eqnarray*} g\left( \varsigma ,\varpi _{\varsigma },P_{2}(\varsigma ),Q_{2}(\varsigma )\right) & = &\frac{\left( 1+e^{-\varsigma }\right) e^{-\nu \varsigma }}{ \left( 34+e^{\varsigma }\right) }\frac{\left\vert \varpi _{\varsigma }\right\vert }{1+\left\vert \varpi _{\varsigma }\right\vert }+\frac{1}{15} P_{2}\varpi (\varsigma )+\frac{1}{35}Q_{2}\varpi (\varsigma ), \\ h\left( \varsigma ,\varpi _{\varsigma },P_{1}(\varsigma ),Q_{1}(\varsigma )\right) & = &\frac{e^{-\nu \varsigma }}{20}\frac{\varpi _{\varsigma }^{2}}{ 1+\varpi _{\varsigma }^{2}}+\frac{1}{20}P_{1}\varpi (\varsigma )+\frac{1}{25} Q_{1}\varpi (\varsigma ), \end{eqnarray*}

    where

    \begin{eqnarray*} P_{2}\varpi (\varsigma ) & = &\int_{0}^{\varsigma }\frac{e^{-\nu \varsigma }}{6 }\ln \left( 1+\varpi _{\varsigma }\right) d\vartheta , \\ Q_{2}\varpi (\varsigma ) & = &\int_{0}^{\sigma }\frac{e^{-\nu \varsigma }}{4} \frac{\tan ^{-1}\left( \varpi _{\varsigma }\right) }{1+\tan ^{-1}\left( \varpi _{\varsigma }\right) }d\vartheta , \\ P_{1}\varpi (\varsigma ) & = &\int_{0}^{\varsigma }e^{-\nu \varsigma }\cos ( \frac{\varpi _{\vartheta }}{5})d\vartheta , \\ Q_{1}\varpi (\varsigma ) & = &\int_{0}^{\sigma }e^{-\nu \varsigma }\sin (\frac{ \varpi _{\vartheta }}{6})d\vartheta . \end{eqnarray*}

    Now, for \varpi _{\varsigma }, \varrho _{\varsigma }\in H_{\nu }, we have

    \begin{eqnarray} \left\vert P_{2}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -P_{2}\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert \frac{e^{-\nu \varsigma }}{6}\ln \left( 1+\varpi _{\varsigma }\right) -\frac{e^{-\nu \varsigma }}{6}\ln \left( 1+\varrho _{\varsigma }\right) \right\vert \\ &\leq &\frac{1}{6}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.9)
    \begin{eqnarray} \left\vert Q_{2}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -Q_{2}\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert \frac{e^{-\nu \varsigma }}{4}\frac{\tan ^{-1}\left( \varpi _{\varsigma }\right) }{1+\tan ^{-1}\left( \varpi _{\varsigma }\right) } d\vartheta -\frac{e^{-\nu \varsigma }}{4}\frac{\tan ^{-1}\left( \varpi _{\varsigma }\right) }{1+\tan ^{-1}\left( \varpi _{\varsigma }\right) } d\vartheta \right\vert \\ &\leq &\frac{1}{4}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.10)
    \begin{eqnarray} \left\vert P_{1}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -P_{1}\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert e^{-\nu \varsigma }\cos (\frac{\varpi _{\vartheta }}{5} )-e^{-\nu \varsigma }\cos (\frac{\varrho _{\vartheta }}{5})\right\vert \\ &\leq &\frac{1}{5}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.11)
    \begin{eqnarray} \left\vert Q_{1}\left( \varsigma ,\vartheta ,\varpi _{\vartheta }\right) -Q_{1}\left( \varsigma ,\vartheta ,\varrho _{\vartheta }\right) \right\vert & = &\left\vert e^{-\nu \varsigma }\sin (\frac{\varpi _{\vartheta }}{6} )-e^{-\nu \varsigma }\sin (\frac{\varrho _{\vartheta }}{6})\right\vert \\ &\leq &\frac{1}{6}\left\Vert \varpi -\varrho \right\Vert _{\nu }, \end{eqnarray} (5.12)
    \begin{eqnarray} &&\left\vert g\left( \varsigma ,\varpi _{\varsigma },P_{2}\varpi (\varsigma ),Q_{2}\varpi (\varsigma )\right) -g\left( \varsigma ,\varrho _{\varsigma },P_{2}\varrho (\varsigma ),Q_{2}\varrho (\varsigma )\right) \right\vert \\ &\leq &\frac{\left( 1+e^{-\varsigma }\right) e^{-\nu \varsigma }}{\left( 34+e^{\varsigma }\right) }\frac{\left\vert \varpi _{\varsigma }-\varrho _{\varsigma }\right\vert }{\left( 1+\left\vert \varpi _{\varsigma }\right\vert \right) \left( 1+\left\vert \varrho _{\varsigma }\right\vert \right) }+\frac{1}{15}\left\vert P_{2}\varpi (\varsigma )-P_{2}\varrho (\varsigma )\right\vert +\frac{1}{35}\left\vert Q_{2}\varpi (\varsigma )-P_{2}\varrho (\varsigma )\right\vert \\ &\leq &\frac{1}{5}\left( \left\Vert \varpi -\varrho \right\Vert _{\nu }+ \frac{1}{3}\left\Vert \varpi -\varrho \right\Vert _{\nu }+\frac{1}{7} \left\Vert \varpi -\varrho \right\Vert _{\nu }\right) , \end{eqnarray} (5.13)
    \begin{eqnarray} &&\left\vert h\left( \varsigma ,\varpi _{\varsigma },P_{1}\varpi (\varsigma ),Q_{1}\varpi (\varsigma )\right) -h\left( \varsigma ,\varrho _{\varsigma },P_{1}\varrho (\varsigma ),Q_{1}\varrho (\varsigma )\right) \right\vert \\ &\leq &\frac{e^{-\nu \varsigma }}{20}\frac{\varpi _{\varsigma }^{2}}{ 1+\varpi _{\varsigma }^{2}}+\frac{1}{20}\left\vert P_{2}\varpi (\varsigma )-P_{2}\varrho (\varsigma )\right\vert +\frac{1}{25}\left\vert Q_{2}\varpi (\varsigma )-P_{2}\varrho (\varsigma )\right\vert \\ &\leq &\frac{1}{4}\left( \left\Vert \varpi -\varrho \right\Vert _{\nu }+ \frac{1}{5}\left\Vert \varpi -\varrho \right\Vert _{\nu }+\frac{1}{5} \left\Vert \varpi -\varrho \right\Vert _{\nu }\right) , \end{eqnarray} (5.14)
    \begin{eqnarray} &&\left\vert g\left( \varsigma ,\psi ,\varpi ,\varrho \right) \right\vert \\ & = &\left\vert \frac{\left( 1+e^{-\varsigma }\right) e^{-\nu \varsigma }}{ \left( 34+e^{\varsigma }\right) }\frac{\left\vert \psi _{\varsigma }\right\vert }{1+\left\vert \psi _{\varsigma }\right\vert }+\frac{1}{15} \int_{0}^{\varsigma }e^{-\nu \varsigma }\cos (\frac{\varpi _{\vartheta }}{5} )d\vartheta +\frac{1}{35}\int_{0}^{\sigma }e^{-\nu \varsigma }\sin (\frac{ \varrho _{\vartheta }}{6})d\vartheta \right\vert \\ &\leq &\frac{1}{35}\left\vert \psi \right\vert +\frac{1}{15}\left\vert \varpi \right\vert +\frac{1}{35}\left\vert \varrho \right\vert , \end{eqnarray} (5.15)
    \begin{eqnarray} &&\left\vert h\left( \varsigma ,\psi ,\varpi ,\varrho \right) \right\vert \\ & = &\left\vert \frac{e^{-\nu \varsigma }}{20}\frac{\psi _{\varsigma }^{2}}{ 1+\psi _{\varsigma }^{2}}+\frac{1}{20}\int_{0}^{\varsigma }\frac{e^{-\nu \varsigma }}{6}\ln \left( 1+\varpi _{\varsigma }\right) d\vartheta +\frac{1}{ 25}\int_{0}^{\sigma }\frac{e^{-\nu \varsigma }}{4}\frac{\tan ^{-1}\left( \varrho _{\varsigma }\right) }{1+\tan ^{-1}\left( \varrho _{\varsigma }\right) }d\vartheta \right\vert \\ &\leq &\frac{1}{20}\left\vert \psi \right\vert +\frac{1}{160}\left\vert \varpi \right\vert +\frac{1}{100}\left\vert \varrho \right\vert , \end{eqnarray} (5.16)
    \begin{equation} \left\vert P_{2}\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert \frac{e^{-\nu \varsigma }}{6}\ln \left( 1+\varpi _{\varsigma }\right) \right\vert \leq \frac{1}{6}\left\vert \varpi \right\vert , \end{equation} (5.17)
    \begin{equation} \left\vert Q_{2}\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert \frac{e^{-\nu \varsigma }}{4}\frac{\tan ^{-1}\left( \varpi _{\varsigma }\right) }{1+\tan ^{-1}\left( \varpi _{\varsigma }\right) } \right\vert \leq \frac{1}{4}\left\vert \varpi \right\vert , \end{equation} (5.18)
    \begin{equation} \left\vert P_{1}\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert e^{-\nu \varsigma }\cos (\frac{\varpi _{\vartheta }}{5} )\right\vert \leq \frac{1}{5}\left\vert \varpi \right\vert , \end{equation} (5.19)

    and

    \begin{equation} \left\vert Q_{1}\left( \varsigma ,\vartheta ,\varpi \right) \right\vert = \left\vert e^{-\nu \varsigma }\sin (\frac{\varpi _{\vartheta }}{6} )\right\vert \leq \frac{1}{60}\left\vert \varpi \right\vert . \end{equation} (5.20)

    From (5.9)–(5.20), we have \ell _{g} = \ell _{P_{1}} = V_{7}(\varsigma) = \frac{1}{5}, \ell _{h} = \ell _{Q_{2}} = V_{10}(\varsigma) = \frac{1}{4}, \ell _{P_{2}} = \ell _{Q_{1}} = V_{8}(\varsigma) = \frac{1}{6}, V_{1}(\varsigma) = V_{3}(\varsigma) = \frac{1}{35}, V_{2}(\varsigma) = \frac{1}{15}, V_{4}(\varsigma) = \frac{1 }{20}, V_{5}(\varsigma) = \frac{1}{160}, V_{6}(\varsigma) = \frac{1}{100}, , V_{9}(\varsigma) = \frac{1}{60}, N_{1}^{\ast } = 1. Thus, we can write

    \begin{equation*} S^{\ast } = \ell _{g}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \right\} +\ell _{h}N_{1}^{\ast }\left\{ \xi _{1}+\xi _{2}\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \right\} \approx 0.8395 < 1, \end{equation*}

    and

    \begin{equation*} \ell ^{\ast } = \frac{\sigma N_{1}^{\ast }}{\left\vert B\right\vert }\left( \ell _{g}\left[ \left( \nu _{1}+\nu _{3}\right) +\left( \ell _{P_{2}}+\ell _{Q_{2}}\right) \left( \nu _{2}+\nu _{4}\right) \right] +\ell _{h}\left[ \left( \nu _{1}+\nu _{3}\right) +\left( \ell _{P_{1}}+\ell _{Q_{1}}\right) \left( \nu _{2}+\nu _{4}\right) \right] \right) \approx 0.5059 < 1. \end{equation*}

    Hence, all the assertions of Theorems 3.4 and 4.2 are fulfilled. Therefore, the supposed problem (5.8) has a unique solution on (-\infty, \sigma ].

    The study of FFIDEs presents a formidable challenge due to the inherent complexities arising from the interplay of fractional-order derivatives, functional arguments, and integral operators. Traditional methods often fall short in addressing these equations due to the non-local nature of fractional derivatives and the intricate dependence on past states introduced by functional arguments. Overcoming these difficulties requires the development and application of sophisticated mathematical tools, including specialized FP theorems tailored for fractional settings, careful treatment of infinite delay, and the construction of appropriate function spaces that accommodate the combined effects of these operators. Furthermore, the presence of multi-term fractional integral boundary conditions adds another layer of complexity, demanding innovative techniques for handling the non-local and distributed nature of the boundary constraints. Successfully navigating these hurdles necessitates a deep understanding of fractional calculus, functional analysis, and operator theory, ultimately paving the way for a more comprehensive understanding of the dynamics governed by FFIDEs. This paper investigates the existence and uniqueness of solutions for a class of hybrid fractional-order functional and neutral functional integrodifferential equations, featuring infinite delay and multi-term fractional integral boundary conditions. A rigorous mathematical framework is developed, leveraging FP theorems, to analyze these complex equations. The LC definition of fractional derivatives is employed, facilitating a comprehensive study of nonlocal dynamics. Illustrative examples are provided to demonstrate the applicability and practical relevance of the theoretical results. Future work includes exploring more complex equations (e.g., variable-order, generalized functional arguments), investigating stability, controllability, and numerical methods, and applying these equations to real-world problems. Developing new fixed point theorems tailored for fractional functional integrodifferential equations and studying associated inverse problems are also promising research avenues. Finally, we also look forward to extending the study period outside the proposed period [2,3].

    Manal Elzain Mohamed Abdalla: Writing–review-editing, formal analysis, funding acquisition; Hasanen A. Hammad: Writing–original draft, conceptualization, investigation, methodology. All authors have read and approved the final version of the manuscript for publication.

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

    Data sharing is not applicable to the article as no data sets were generated or analyzed during the current study.

    The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through the Large Research Project under grant number R.G.P. 2/217/45.

    All authors confirm that they have no conflict of interest.



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