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Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system

  • This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.

    Citation: Subramanian Muthaiah, Manigandan Murugesan, Muath Awadalla, Bundit Unyong, Ria H. Egami. Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system[J]. AIMS Mathematics, 2024, 9(6): 16203-16233. doi: 10.3934/math.2024784

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  • This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.



    The following abbreviations are used in this manuscript:

    BVPs Boundary Value Problems
    HHFDEs Hilfer-Hadamard Fractional-order Differential Equations
    HHFIEs Hilfer-Hadamard Fractional-order Integrodifferential Equations
    HFIs Hadamard Fractional Integrals
    HHFDs Hilfer-Hadamard Fractional Derivatives
    CFDs Caputo Fractional Derivatives
    HFDs Hilfer Fractional Derivatives
    HFDEs Hilfer Fractional Differential Equations
    HFDs Hadamard Fractional Derivatives (HFDs)
    CHFDs Caputo-Hadamard Fractional Derivatives (CHFDs)

    This study presents and examines a new nonlinear sequential Hilfer-Hadamard fractional-order integrodifferential equations (HHFIEs) with nonlocal coupled multipoint and Hadamard fractional integrodifferential boundary conditions. The formulation of the problem is as follows:

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ)),(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ)), (1.1)

    and it is enhanced by nonlocal coupled multipoint and Hadamard fractional integrodifferential boundary conditions:

    {S(1)=0,  S(T)=mj=1ηjZ(ξj)+ni=1θiHIϕiZ(ζi)+rk=1λHkDωk1Z(μk),Z(1)=0,  Z(T)=au=1PuS(ψu)+bv=1QvHIδvS(σv)+cw=1MHwDϑw1S(πw). (1.2)

    Here α1,α2(1,2], β1,β2[0,1], λ1,λ2R+, T>1, ηj,θi,λk,Pu,Qv,MwR, ξj,ζi,μk,ψu,σv,πw(1,T), (j=1,2,...,m, i=1,2,...,n, k=1,2,...,r, u=1,2,...,a, v=1,2,...,b, w=1,2,...,c), HHDαi,βj1+ denotes the Hilfer-Hadamard fractional derivative (HHFD) operator of order αi,βj;i=1,2.j=1,2. HIχ is the Hadamard fractional integral (HFI) of order χ{ϕi,δv>0} i=1,2,...,n, v=1,2,...,b, HDψ1 is the Hadamard fractional derivative (HFD) of order ψ{ωk,ϑw} k=1,2,...,r, w=1,2,...,c, F,G:[1,T]×R4R are continuous functions. It's important to highlight and that this study makes a significant contribution to the existing literature by addressing a distinct setup involving sequential HHFIEs along with coupled multipoint and Hadamard fractional integrodifferential boundary conditions. The methodology employed in this study involves using the fixed-point approach to establish both existence and uniqueness results for the problems (1.1) and (1.2). The process involves converting the given problem into an equivalent fixed-point problem, followed by the application of Leray-Schauder alternative and Banach's fixed-point theorem to establish existence and uniqueness results, respectively. Additionally, we investigate the Ulam-Hyers stability of the solution for the proposed system. The findings of this study are innovative and contribute to the existing literature on boundary value problems (BVPs) concerning coupled systems of sequential HHFIEs. In recent decades, fractional calculus has gained considerable attention and become a prominent area of study in mathematical analysis. This growth is largely attributed to the extensive use of fractional calculus techniques in developing innovative mathematical models to represent various phenomena in fields such as economics, mechanics, engineering, science, and others. References [1,2,3,4] offer examples and comprehensive discussions on this subject.

    In the upcoming section, we will provide an overview of relevant scholarly articles pertaining to the discussed problem. Among various fractional derivatives introduced, the Riemann-Liouville and Caputo fractional derivatives (CFDs) have garnered significant attention due to their practical applications. The Hilfer fractional derivative, introduced by Hilfer in [5], incorporates the Riemann-Liouville and CFDs as special cases for certain parameter values. Additional insights into this derivative can be found in [6,7,8,9,10,11,12,13]. References [14,15,16,17,18] offer valuable insights into Hilfer-type initial and BVPs. A recent study [19] explores the Ulam-Hyers stability and existence of a fully coupled system featuring integro-multistrip-multipoint boundary conditions and nonlinear sequential Hilfer fractional differential equations (HFDEs). Furthermore, [20] delves into a hybrid generalized HFDE BVP.

    In 1892, Hadamard introduced the HFD, defined by a logarithmic function with an arbitrary exponent in its kernel [21]. Subsequent studies, such as those in [22,23,24,25,26], have explored variations such as HHFDs and Caputo-Hadamard fractional derivatives (CHFDs). Notably, for specific values of β-β=0 and β=1, respectively—HFDs and CHFDs emerge as particular instances of the HHFD.

    Stability analysis has been a prominent field of study for fractional differential equations in the last several decades and has drawn a lot of interest from scholars. Numerous stability models, including Lyapunov, exponential, and Mittag-Leffler stability, have been thoroughly examined in the literature. We suggest reviewing publications [27,28,29,30,31] for historical perspective on Ulam-Hyers stability and current improvements.

    The problem of existence and Ulam stability of solutions for the following Hilfer-Hadamard fractional differential equations (HHFDEs) [32] is stated as follows:

    {(HDα,β1x)(t)=f(t,u(t)), for tJ:=[1,T],(HI1γ1x)(t)|t=1=φ, (1.3)

    where 0<α<1, 0β1, γ=α+βαβ, T>1, φR, and f:J×RR is a given function. HI1γ1 denotes the left-sided mixed Hadamard integral of order 1γ, and HDα,β1 is the HHFD of order α and type β, introduced by Hilfer. In [33], existence results were established for an HHFDE with nonlocal integro-multipoint boundary conditions:

    {HHDα,β1x(t)=f(t,x(t)),  t[1,T],x(1)=0,  mi=1θix(ξi)=λHIδx(η), (1.4)

    where α(1,2], β[0,1], θi,λR, η,ξi(1,T) (i=1,2,...,m), HIδ is the Hadamard fractional integral (HFI) of order δ>0, and f:[1,T]×RR is a continuous function. Problem (1.4) represents a non-coupled system, in contrast to problems (1.1)-(1.2), which is a coupled system. The systems (1.1)-(1.2) presents nonlocal coupled Hadamard fractional integrodifferential and multipoint boundary conditions, whereas the problem (1.4) involves discrete boundary conditions with HFIs. The authors of [34] established existence results for nonlocal mixed Hilfer-Hadamard fractional BVPs:

    {HHDα,β1x(t)=f(t,x(t)),  t[1,T],x(1)=0,  x(T)=mj=1ηjx(ξj)+ni=1ζiHIϕix(θi)+rk=1λkHDωk1x(μk), (1.5)

    where α(1,2], β[0,1], ηi,ζi,λkR, ξi,θi,μk(1,T), (j=1,2,...,m),(i=1,2,...,n),(k=1,2,...,r), HIϕi is the HFI of order ϕi>0, HDμk1 is the HFD of order μk>0, and f:[1,T]×RR is a continuous function. Equation (1.5) does not represent a coupled system, unlike Eqs (1.1) and (1.2), which does. In the latter, there are nonlocal coupled Hadamard fractional integrodifferential and multipoint boundary conditions. In contrast, Eq (1.5) involves multipoint boundary conditions comprising HFIs and HFDs. Furthermore, in [35], investigations were conducted on coupled HHFDEs within generalized Banach spaces. The authors of the aforementioned study [36] successfully derived existence results for a coupled system of HHFDEs with nonlocal coupled boundary conditions:

    {HHDα,β1u(t)=f(t,u(t),v(t)),  1<α2,  ϖ[1,T],HHDγ,δ1v(t)=g(t,u(t),v(t)),  1<γ2,  ϖ[1,T],u(1)=0,HDς1u(T)=mi=1T1HDϱi1u(s)dHi(s)+ni=1T1HDσi1v(s)dKi(s),v(1)=0,HDϑ1v(T)=pi=1T1HDηi1u(s)dPi(s)+qi=1T1HDθi1v(s)dQi(s), (1.6)

    where α,γ(1,2], β,δ[0,1], T>1, HHDα,β, HHDγ,δ1 denotes the HHFD operator of order α,β,γ,δ. HDχ1+ is the HFD operator of order χ{ς,ϑ,ϱi,ηi,σi,θi}, (i=1,2,...,m),(i=1,2,...,n),(i=1,2,...,p),(i=1,2,...,q), f,g:[1,T]×R×RR are continuous functions. In the boundary conditions, Riemann-Stieltjes integrals with Hi,Ki,Pi,Qi, (i=1,2,...,m),(i=1,2,...,n),(i=1,2,...,p),(i=1,2,...,q), functions of bounded variation. Problem (1.6) involves a coupled system of HHFDEs, while problems (1.1)-(1.2) deals with a coupled system of sequential HHFIEs. In problems (1.1)-(1.2), there is nonlocal coupled multipoint and Hadamard fractional integrodifferential boundary conditions, whereas in problem (1.6), Stieltjes-integral boundary conditions are incorporated, involving HFDs. In problems (1.1)-(1.2), the nonlinearity depends on the unknown function and its fractional integrals at lower orders are included. Conversely, in problem (1.6), the nonlinearity depends on the unknown function, but it does not involve fractional integrals at lower orders. The researchers in [37] performed an examination of a coupled system of HHFDEs with nonlocal coupled HFI boundary conditions:

    {HHDα1,β11+u(t)=ϱ1(t,u(t),v(t)),  1<α12,  ϖE:=[1,T],HHDα2,β21+v(t)=ϱ2(t,u(t),v(t)),  2<α23,  ϖE:=[1,T],u(1)=0,u(T)=λ1HIδ11+v(η1),v(1)=0,v(η2)=0,v(T)=λ2HIδ21+u(η3),    1<η1,η2,η3<T, (1.7)

    where α1(1,2], α2(2,3], β1,β2[0,1], T>1, δ1,δ2>0, λ1,λ2R, HHDαi,βj1+ denotes the HHFD operator of order αi,βj;i=1,2.j=1,2. HIχ1+ is the HFI operator of order χ{δ1,δ2}, and ϱ1,ϱ2:E×R×RR are continuous functions. In Eq (1.7), we encounter a coupled system of HHFDEs, whereas Eqs (1.1) and (1.2) addresses a coupled system of sequential HHFIEs. In the latter, there are nonlocal coupled Hadamard fractional integrodifferential and multipoint boundary conditions, whereas Eq (1.7) involves multipoint and HFI boundary conditions. In Eq (1.7), solutions are obtained for the coupled system of HHFDEs, while in Eqs (1.1) and (1.2), solutions are derived for the coupled system of sequential HHFIEs. In Eqs (1.1) and (1.2), the nonlinearity depends on the unknown function and its fractional integrals at lower orders are included. Conversely, in Eq (1.7), the nonlinearity depends on the unknown function but does not involve fractional integrals at lower orders. Furthermore, in [38], a two-point BVP for a system of nonlinear sequential HHFDEs was investigated:

    {(HHDα1,β11+λ1HHDα11,β11)u(t)=f(t,u(t),v(t)), t[1,e],(HHDα2,β21+λ2HHDα21,β21)v(t)=g(t,u(t),v(t)), t[1,e],u(1)=0,  u(e)=A1,  v(1)=0,  v(e)=A2, (1.8)

    where α1,α2(1,2], β1,β2[0,1], λ1,λ2,A1,A2R+, f,g:[1,e]×R×RR are continuous functions. Equation (1.8) features a two-point boundary condition, while problems (1.1)-(1.2) includes multipoint and Hadamard fractional integrodifferential boundary conditions. In Eqs (1.1) and (1.2), the nonlinearity involves the unknown function and its fractional integrals at lower orders. Conversely, in Eq (1.8), the nonlinearity relies on the unknown function but does not incorporate fractional integrals at lower orders.

    The document is organized as follows in the following sections: The fundamental ideas of fractional calculus relevant to this research are introduced in Section 2. An auxiliary lemma addressing the linear versions of problems (1.1) and (1.2) is provided in Section 3. The primary findings are presented in Section 4 along with illustrative examples. Finally, Section 5 provides a few recommendations.

    Definition 2.1. The HFI of order p>0 for a continuous function F:[a,)R is given by

    HIpa+F(ϖ)=1Γ(p)ϖa(logϖς)p1F(ς)ςdς, (2.1)

    where log()=loge().

    Definition 2.2. The HFD of order p>0 for a function F:[a,)R is defined by

    HDpa+F(ϖ)=δn(HInpa+F)(ϖ),  n=[p]+1, (2.2)

    where δn=ϖndndωn and [p] denotes the integer part of the real number p.

    Lemma 2.3. If p,q>0 and 0<a<b< then

    (1)(HIpa+(logϖa)q1)(x)=Γ(q)Γ(q+p)(logxa)q+p1;
    (2)(HDpa+(logϖa)q1)(x)=Γ(q)Γ(qp)(logxa)qp1.

    In particular, for q=1, we have (HDpa+)(1)=1Γ(1p)(logxa)p0,0<p<1.

    Definition 2.4. For n1<p<n and 0q1, the HHFD of order p and q for FL(a,b) is defined as

    (HHDp,qa+)(F(ϖ))=(HIq(np)a+δnHI(np)(1q)a+F)(ϖ)=(HIq(np)a+δnHI(nγ)a+F)(ϖ)=(HIq(np)a+δnHDγa+F)(ϖ), γ=p+nqpq,

    where HI()a+ and HD()a+ are given and defined by (2.1) and (2.2), respectively.

    Theorem 2.5. If FL1(a,b),0<a<b<, and (HInγa+F)(ϖ)ACnδ[a,b], then

    HIpa+(HHDp,qa+F)(ϖ)=HIγa+(HHDγa+F)(ϖ)=F(ϖ)n1j=o(δ(nj1)(HIpa+F))(a)Γ(γj)(logϖa)γj1,

    where p>0,0q1 and γ=p+nqpq,n=[p]+1. Observe that Γ(γj) exists for all j=1,2,,n1 and γ[p,n].

    We'll utilize established fixed point theorems in Banach spaces to demonstrate the existence and uniqueness of solutions for Hilfer-Hadamard fractional differential systems.

    Lemma 2.6. Let H1,H2C([1,T],R) such that

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=H1(ϖ), 1<α12, ϖ[1,T],(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=H2(ϖ),  1<α22, ϖ[1,T], (2.3)

    enhanced by the boundary conditions (1.2) if, and only if,

    S(ϖ)=1Δ[λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς{(logT)γ21}+λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11H1(ς)ςdς, (2.4)

    and

    Z(ϖ)=1Δ[λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς{(logT)γ11}+λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς×{au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11}]λ2ϖ1Z(ς)ςdς+1Γ(α2)ϖ1(logϖς)α21H2(ς)ςdς, (2.5)
    {Δ=A1B2A2B1,A1={(logT)γ11},B2={(logT)γ21},A2={au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11},B1={mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}. (2.6)

    Proof. From the first equation of (2.3), we have

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=H1(ϖ),(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=H2(ϖ). (2.7)

    Taking both sides of the HFI of order α1,α2 (2.7), we obtain

    {HHIα11+(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=HHIα11+H1(ϖ),HHIα21+(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=HHIα21+H2(ϖ). (2.8)

    Equation (2.8) can be written as follows:

    S(ϖ)=c0(logϖ)γ11+c1(logϖ)γ22λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11H1(ς)ςdς, (2.9)

    and

    Z(ϖ)=d0(logϖ)γ21+d1(logϖ)γ22λ2ϖ1Z(ς)ςdς+1Γ(α2)ϖ1(logϖς)α21H2(ς)ςdς, (2.10)

    where c0,d0,c1, and d1 are arbitrary constants. Boundary conditions (1.2) combined with (2.9) and (2.10) produce

    S(1)=c0(log1)γ11+c1(logϖ)2γ1λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11H1(ς)ςdς=0, (2.11)
    Z(1)=d0(log1)γ21+d1(logϖ)2γ2λ2ϖ1Z(ς)ςdς+1Γ(α2)ϖ1(logϖς)α21H2(ς)ςdς=0, (2.12)

    from which we have c1=0 and d1=0. Equations (2.11) and (2.12) can be written as

    S(ϖ)=c0(logϖ)γ11λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11H1(ς)ςdς, (2.13)
    Z(ϖ)=d0(logϖ)γ21λ2ϖ1Z(ς)ςdς+1Γ(α2)ϖ1(logϖς)α21H2(ς)ςdς, (2.14)

    from which we have

    c0=1Δ[λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς{(logT)γ21}+λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}], (2.15)
    d0=1Δ[λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς{(logT)γ11}+λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς{au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11}]. (2.16)

    Substitute the values of c0,c1,d0, and d1 in (2.9) and (2.10), and we get solutions (2.4) and (2.5). The converse follows by direct computation. This completes the proof.

    Let us introduce the Banach space E=T([1,T],R) endowed with the norm defined by ||S||:=supϖ[1,T]|S(ϖ)|. Thus, the product space (E×E,||||E×E) equipped with the norm ||S,Z||E×E=||S||+||Z|| for (S×Z)E×E is also a Banach space.

    In view of Lemma 2.6, we define as operator Υ:E×EE×E by

    Υ(S,Z)(ϖ)=(Υ1(S,Z)(ϖ),Υ2(S,Z)(ϖ)), (3.1)

    where

    Υ1(S,Z)(ϖ)=1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς}{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11H1(ς)ςdς, (3.2)

    and

    Υ2(S,Z)(ϖ)=1Δ[λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς{(logT)γ11}+λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς×{au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11}]λ2ϖ1Z(ς)ςdς+1Γ(α2)ϖ1(logϖς)α21H2(ς)ςdς. (3.3)

    We need the following hypothesis in the sequel:

    Ω1=1Δ[{λ1(logT)+(logT)α1Γ(α1+1)}{(logT)γ21}+{au=1Puλ1(logψu)bv=1QvHIδv1+λ1(logσv)cw=1MwHDϑw1+λ1(logπw)+au=1Pu(logψu)α1Γ(α1+1)+bv=1QvHIδv1+(logσv)α1Γ(α1+1)+cw=1MwHDϑw1+(πw)α1Γ(α1+1)}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1(logT)+(logT)α1Γ(α1+1), (3.4)
    Ω2=1Δ[{λ2mj=1ηj(logξj)λ2ni=1θiHIϕi1+(logζi)λ2rk=1λkHDωk1+(logμk)+mj=1ηj(logξj)α2Γ(α2+1)+ni=1θiHIϕi1+(logζi)α2Γ(α2+1)+rk=1λkHDωk1+(logμk)α2Γ(α2+1)}{(logT)γ21}+{λ2(logT)(logT)α2Γ(α2+1)}{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}], (3.5)
    ˉΩ1=1Δ[{au=1Puλ1(logψu)bv=1QvHIδv1+λ1(logσv)cw=1MwHDϑw1+λ1(logπw)+au=1Pu(logψu)α1Γ(α1+1)+bv=1QvHIδv1+(logσv)α1Γ(α1+1)+cw=1MwHDϑw1+(logπw)α1Γ(α1+1)}{(logT)γ11}+λ1(logT)(logT)α1Γ(α1+1)×{au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11}], (3.6)
    ˉΩ2=1Δ[{λ2(logT)(logT)α2Γ(α2+1)}{(logT)γ11}+λ2mj=1ηj(logξj)λ2ni=1θiHIϕi1+(logζi)λ2rk=1λkHDωk1+(logμk)+mj=1ηj(logξj)α2Γ(α2)+ni=1θiHIϕi1+(logζi)α2Γ(α2+1)+rk=1λkHDωk1+(logμk)α2Γ(α2+1)×{au=1Pu(logψu)γ11bv=1QvHIδv1+(logσv)γ11cw=1MwHDϑw1+(logπw)γ11}]λ2(logT)+(logT)Γ(α2), (3.7)

    and

    {Ψ0=(Ω1+¯Ω1)M0+(Ω2+¯Ω2)N0,Ψ1=(Ω1+¯Ω1)(M1+M3Γ(p1+1))+(Ω2+¯Ω2)(N1+N3Γ(q1+1)),Ψ2=(Ω1+¯Ω1)(M2+M4Γ(p2+1))+(Ω2+¯Ω2)(N2+N4Γ(q2+1)). (3.8)

    We give now the assumptions we will use in this section.

    (H1) Assume that there exist real constants Mi,Ni0(i=1,2) and M0>0,N0 such that, for all ϖ[1,T],SiR,i=1,2,3,4,

    |F(ϖ,S1,S2,S3,S4)|M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|,|G(ϖ,S1,S2,S3,S4)|N0+N1|S1|+N2|S2|+N3|S3|+N4|S4|, ϖ[1,T].

    (H2) There exists positive constant L,L1, such that, for all ϖ[1,T],Si,ZiR,i=1,2.

    |F(ϖ,S1,S2,S3,S4)F(ϖ,Z1,Z2,Z3,Z4)|L(|S1Z1|+|S2Z2|+|S3Z3|+|S4Z4|),|G(ϖ,S1,S2,S3,S4)G(ϖ,Z1,Z2,Z3,Z4)|L1(|S1Z1|+|S2Z2|+|S3Z3|+|S4Z4|).

    The first theorem uses the Leray-Schauder alternative to establish the existence of a result.

    Lemma 4.1. [1] Let θ(Ξ)={SE:S=κΞ(S) for some 0<κ<1}, where Ξ:EE is a completely continuous operator. Then, either the set θ(Ξ) is unbounded or there exists at least one fixed for the operator Ξ.

    Theorem 4.2. Suppose condition (H1) is satisfied. Additionally, assume that

    max{Ψ1,Ψ2}<1. (4.1)

    Under these conditions, there exists at least one solution to the problems (1.1) and (1.2) on E.

    Proof. First, let's establish that the operator Υ:E×EE×E, as defined in (3.1), is completely continuous. The continuity of the operator Υ (in terms of Υ1 and Υ2) is evident from the continuity of F and G.

    Next, we aim to demonstrate that the operator Υ is uniformly bounded. To achieve this, consider a bounded set BrE×E. Then, we can find positive constants N1 and N2 satisfying

    {|F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))|N1,|N(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))|N2,(S,Z)Br. (4.2)

    Consequently, we obtain

    ||Υ1(S,Z)(ϖ)||=1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21|N2|(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21|N2|(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21|N2|(ς)ςdς1Γ(α1)T1(logTς)α11|N1|(ς)ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11|N1|(ς)ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11|N1|(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11|N1|(ς)ςdς1Γ(α2)T1(logTς)α21|N2|(ς)ςdς}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11|N1|(ς)ςdς
    N1{1Δ[λ1(logT)+(logT)α1Γ(α1+1)]{(logT)γ21}+{au=1Puλ1(logψu)bv=1QvHIδv1+λ1(logσv)cw=1MwHDϑw1+λ1(logπw)+au=1Pu(logψu)α1Γ(α1+1)+bv=1QvHIδv1+(logσv)α1Γ(α1+1)+cw=1MwHDϑw1+(πw)α1Γ(α1+1)}+{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}λ1(logT)+(logT)α1Γ(α1+1)}+N2{1Δ[{λ2mj=1ηj(logξj)λ2ni=1θiHIϕi1+(logζi)λ2rk=1λkHDωk1+(logμk)+mj=1ηj(logξj)α2Γ(α2+1)+ni=1θiHIϕi1+(logζi)α2Γ(α2+1)+rk=1λkHDωk1+(logμk)α2Γ(α2+1)}{(logT)γ21}+{λ2(logT)(logT)α2Γ(α2+1)}]}. (4.3)

    This observation, in light of the notation (3.4) and (3.5), yields

    ||Υ1(S,Z)||Ω1N1+Ω2N2. (4.4)

    Similarly, employing the notation (3.6) and (3.7), we obtain

    ||Υ2(S,Z)||ˉΩ1N1+ˉΩ2N2. (4.5)

    Then, it follows from (4.4) and (4.5) that

    ||Υ(S,Z)||(Ω1+ˉΩ1)N1+(Ω2+ˉΩ2)N2. (4.6)

    This demonstrates that the operator Υ is uniformly bounded.

    To establish the equicontinuity of Υ, let ϖ1,ϖ2E with ϖ1<ϖ2. Then, we find that

    |Υ1(S,Z)(ϖ2)Υ1(S,Z)(ϖ1)|={1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21H2(ς)ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21H2(ς)ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21H2(ς)ςdς1Γ(α1)T1(logTς)α11H1(ς)ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11H1(ς)ςdς
    +bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11H1(ς)ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11H1(ς)ςdς1Γ(α2)T1(logTς)α21H2(ς)ςdς}{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]+λ1[(logϖ2)γ21(logϖ1)γ11]ϖ11S(ς)ςdς+ϖ2ϖ1S(ς)ςdς+1Γ(α1)ϖ11(logϖ2ς)α11(logϖ1ς)α11H1(ς)ςdς+ϖ2ϖ1(logϖ2ς)α11dςς},0 as ϖ2ϖ1, (4.7)

    independent of (S,Z)Br. Likewise, it can be shown that |Υ1(S,Z)(ϖ2)Υ1(S,Z)(ϖ1)|0 as ϖ2ϖ1 is independent of (S,Z)Br. Consequently, the equicontinuity of Υ1 and Υ2 implies the equicontinuity of the operator Υ. Therefore, by Arzela-Ascoli's theorem, the operator Υ is compact. Finally, we establish the boundedness of the set Θ(Υ)={S,ZE×E:S,Z=κΥ(S,Z);0κ1}. Let (S,Z)Θ(Υ). Then, (S,Z)=κΥ(S,Z), which implies that S(ϖ)=κΥ1(S,Z)(ϖ),Z(ϖ)=κΥ2(S,Z)(ϖ) for any ϖE, and so |S(ϖ)|=|Υ1(S,Z)(ϖ)|,Z(ϖ)=|Υ2(S,Z)(ϖ)|  ϖ[0,1]. Using some inequality proved at the beginning of the proof, we find

    ||Υ1(S,Z)(ϖ)||=1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21N0+N1|S1|+N2|S2|+N3|S3|+N4|S4|ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21N0+N1|S1|+N2|S2|+N3|S3|+N4|S4|ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21N0+N1|S1|+N2|S2|+N3|S3|+N4|S4|ςdς1Γ(α1)T1(logTς)α11M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|ςdς}{(logT)γ21}
    +{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|ςdς1Γ(α2)T1(logTς)α21N0+N1|S1|+N2|S2|+N3|S3|+N4|S4|ςdς}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11M0+M1|S1|+M2|S2|+M3|S3|+M4|S4|ςdςΩ1{M0+M1|S1|+M2|S2|+M3Γ(p1+1)|S1|+M4Γ(p2+1)|S1|}+Ω2{N0+N1|Z1|+N2|Z2|+N3Γ(q1+1)|Z1|+N4Γ(q2+1)|Z1|}, (4.8)

    which implies that

    ||S||=supϖ[1,T]|S(ϖ)|Ω1{M0+M1||S1||+M2||S2||+M3Γ(p1+1)||S1||+M4Γ(p2+1)||S1||}+Ω2{N0+N1||S1||+N2||S2||+N3Γ(q1+1)||S1||+N4Γ(q2+1)||S1||}. (4.9)

    Similarly, one can find that

    ||Z||=supϖ[1,T]|Z(ϖ)|¯Ω1{M0+M1||S1||+M2||S2||+M3Γ(p1+1)||S1||+M4Γ(p2+1)||S1||}+¯Ω2{N0+N1||S1||+N2||S2||+N3Γ(q1+1)||S1||+N4Γ(q2+1)||S1||}. (4.10)

    From (4.9) and (4.10), we obtained

    ||S||+||Z||(Ω1+¯Ω1)M0+(Ω2+¯Ω2)N0+||S||{(Ω1+¯Ω1)(M1+M3Γ(p1+1))+(Ω2+¯Ω2)(N1+N3Γ(q2+1))}+||Z||{(Ω1+¯Ω1)(M2+M4Γ(p1+1))+(Ω2+¯Ω2)(N2+N4Γ(q2+1))},=Ψ0+Ψ1||S||+Ψ2||Z||Ψ0+max{Ψ1,Ψ2}||(S,Z)||E, (4.11)

    where Ψi i=0,1,2 are given by (3.8). By (4.1), we deduce that

    ||(S,Z)||E=Ψ01max{Ψ1,Ψ2}. (4.12)

    As a result, Θ(Υ) is bounded. Consequently, the conclusion of Lemma 2.6 applies, implying that the operator Υ has at least one fixed point, which indeed serves as a solution to the problems (1.1) and (1.2).

    Now, we introduce the constants

    F0=supϖ[0,1]|F(ϖ,0,0,0,0)|,  G0=supϖ[0,1]|G(ϖ,0,0,0,0)|, (4.13)
    {ρ1=max{1+1Γ(p1+1),1+1Γ(p2+1)},ρ2=max{1+1Γ(q1+1),1+1Γ(q2+1)},D1=z0ρ1Ω1+k0ρ2Ω2,D2=z0ρ1¯Ω1+k0ρ2¯Ω2,G1=F0Ω1+G0Ω2,G2=F0¯Ω1+G0¯Ω2, (4.14)

    where Ω1,Ω2,¯Ω1,¯Ω2 are given by (3.4)–(3.7).

    The subsequent result will establish the existence of a unique solution to the problems (1.1) and (1.2) through the application of a fixed point theorem attributed to Banach.

    Theorem 4.3. If the assumption (H2) is satisfied and that

    D1+D2<1, (4.15)

    where Ωi and ˉΩi, (i = 1, 2) are given in (3.4)–(3.7), then the problems (1.1) and (1.2) have a unique solution on E.

    Proof. By using condition (4.15), we define the positive number

    RG1+G21(D1+D2), (4.16)

    where G1,G2,D1,D2 are given by (4.12). We will prove that A(BR)BR, where BR={(S,Z)E×E:||(S,Z)R}. For (S,Z)BR and ϖ[0,1], we obtain

    |F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))||F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))F(ϖ,0,0,0,0)|+|F(ϖ,0,0,0,0)|z0(|S(ϖ)|+|Z(ϖ)|+|Ip1S(ϖ)|+|Ip2Z(ϖ)|)+F0z0(||S||+||Z||+1Γ(p1+1)||S||+1Γ(p2+1)||Z||)+F0z0max{1+1Γ(p1+1),1+1Γ(p2+1)}||S,Z||E+F0z0max{1+1Γ(p1+1),1+1Γ(p2+1)}R+F0z0ρ1R+F0,

    and

    |G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))||G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))G(ϖ,0,0,0,0)|+|G(ϖ,0,0,0,0)|k0(|S(ϖ)|+|Z(ϖ)|+|Iq1S(ϖ)|+|Iq2Z(ϖ)|)+G0k0(||S||+||Z||+1Γ(q1+1)||S||+1Γ(q2+1)||Z||)+G0k0max{1+1Γ(q1+1),1+1Γ(q2+1)}||S,Z||E+G0k0max{1+1Γ(q1+1),1+1Γ(q2+1)}R+G0k0ρ2R+G0.

    Then we deduce that

    |Υ1(S,Z)(ϖ)|(z0ρ1R+F0)Ω1+(k0ρ2R+G0)Ω2=(z0ρ1Ω1+k1ρ2Ω2)R+F0Ω1+G0Ω2=D1R+G1, (4.17)

    and

    |Υ2(S,Z)(ϖ)|(k0ρ1R+F0)¯Ω1+(k0ρ2R+G0)¯Ω2=(z0ρ1¯Ω1+k1ρ2¯Ω2)R+F0¯Ω1+G0¯Ω2=D2R+G2. (4.18)

    Therefore, by (4.17), (4.18), and the definition of R, we conclude that

    ||Υ(S,Z)||E=||Υ1(S,Z)||+||Υ2(S,Z)||(D1+D2)R+G1+G2=R,(S,Z)BR, (4.19)

    which gives us Υ(BR)BR.

    We will prove next that Υ is a contraction operator. By using (H2), for (Si,Zi)BR,i=1,2, and for any ϖ[0,1], we find:

    Letting K1=supϖ[1,T]|F(ϖ,0,0)|< and K2=supϖ[1,T]|G(ϖ,0,0)|<, it follows by the assumption (H1) that

    |F(ϖ,S,Z)|L1(||S||+||Z||)+K1L1(||S||+||Z||)+K1,

    and

    |G(ϖ,S,Z)|L2(||S||+||Z||)+K2.

    To begin, we show that ΥBρBρ, where Bρ={(S,Z)E×E:||(S,Z)ρ}, with

    ρ(Ω1+ˉΩ1)K1+(Ω2+ˉΩ2)K21((Ω1+ˉΩ1)L1+(Ω2+ˉΩ2)L2). (4.20)

    For (S,Z)Bρ, we have

    ||Υ1(S,Z)||=supϖ[1,T]|Υ1(S,Z)(ϖ)|1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21|G(ς,S(ς),Z(ς))|ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21|G(ς,S(ς),Z(ς))|ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21|G(ς,S(ς),Z(ς))|ςdς1Γ(α1)T1(logTς)α11|F(ς,S(ς),Z(ς))|ςdς}{(logT)γ21}
    +{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11|F(ς,S(ς),Z(ς))|ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11|F(ς,S(ς),Z(ς))|ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11|F(ς,S(ς),Z(ς))|ςdς1Γ(α2)T1(logTς)α21|G(ς,S(ς),Z(ς))|ςdς}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11|F(ς,S(ς),Z(ς))|ςdς,

    which yields

    ||Υ1(S,Z)||(L1ρ+K1){1Δ[λ1(logT)+(logT)α1Γ(α1+1)]{(logT)γ21}+{au=1Puλ1(logψu)bv=1QvHIδv1+λ1(logσv)cw=1MwHDϑw1+λ1(logπw)+au=1Pu(logψu)α1Γ(α1+1)+bv=1QvHIδv1+(logσv)α1Γ(α1+1)+cw=1MwHDϑw1+(πw)α1Γ(α1+1)}+{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}λ1(logT)+(logT)α1Γ(α1+1)}+{1Δ[{λ2mj=1ηj(logξj)λ2ni=1θiHIϕi1+(logζi)λ2rk=1λkHDωk1+(logμk)+mj=1ηj(logξj)α2Γ(α2+1)+ni=1θiHIϕi1+(logζi)α2Γ(α2+1)+rk=1λkHDωk1+(logμk)α2Γ(α2+1)}{(logT)γ21}+{λ2(logT)(logT)α2Γ(α2+1)}]}.

    Using the notation (3.4)-(3.5), we get

    ||Υ1(S,Z)||(L1Ω1+L2Ω2)+Ω1K1+Ω2K2. (4.21)

    Likewise, we can find that

    ||Υ2(S,Z)||(L1ˉΩ1+L2ˉΩ2)+ˉΩ1K1+ˉΩ2K2. (4.22)

    Then, it follows from (4.21)-(4.22) that

    ||Υ(S,Z)||=||Υ1(S,Z)||+||Υ2(S,Z)||ρ.

    Therefore, ΥBρBρ as (S,Z)Bρ is an arbitrary element.

    In order to verify that the operator Υ is a contraction, let Si,ZiBρ,i=1,2. Then, we get

    ||Υ1(S1,Z1)Υ1(S1,Z1)||1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21|G(ς,S1(ς),Z1(ς))G(ς,S2(ς),Z2(ς))|ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21|G(ς,S1(ς),Z1(ς))G(ς,S2(ς),Z2(ς))|ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21|G(ς,S1(ς),Z1(ς))G(ς,S2(ς),Z2(ς))|ςdς1Γ(α1)T1(logTς)α11|F(ς,S1(ς),Z1(ς))F(ς,S2(ς),Z2(ς))|ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdς
    cw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11|F(ς,S1(ς),Z1(ς))F(ς,S2(ς),Z2(ς))|ςdς+bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11|F(ς,S1(ς),Z1(ς))F(ς,S2(ς),Z2(ς))|ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11|F(ς,S1(ς),Z1(ς))F(ς,S2(ς),Z2(ς))|ςdς1Γ(α2)T1(logTς)α21|G(ς,S1(ς),Z1(ς))G(ς,S2(ς),Z2(ς))|ςdς}{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11|F(ς,S1(ς),Z1(ς))F(ς,S2(ς),Z2(ς))|ςdς,

    which, by (H2), yields

    ||Υ1(S1,Z1)Υ1(S2,Z2)||(Ω1L1+Ω2L2)[||S1S2||+||Z1Z2||]. (4.23)

    Similarly, we can observe that

    ||Υ2(S1,Z1)Υ2(S2,Z2)||(ˉΩ1L1+ˉΩ2L2)[||S1S2||+||Z1Z2||]. (4.24)

    Consequently, it follows from (4.23) and (4.24) that

    ||Υ(S1,Z1)Υ(S2,Z2)||=||Υ1(S1,Z2)Υ1(S1,Z2)||+||Υ2(S1,Z2)Υ2(S1,Z2)||[(Ω1+ˉΩ1)L1+(Ω2+ˉΩ2)L2][||S1S2||+||Z1Z2||], (4.25)

    and by condition (4.15), it follows that Υ is a contraction. Consequently, the operator Υ possesses a unique fixed point as a direct application of the Banach fixed point theorem. Thus, there exists a unique solution for the problems (1.1) and (1.2) on E.

    This section is devoted to the investigation of Hyers-Ulam stability for our proposed system. Consider the following inequality:

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))ε1,  ϖ[1,T],(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))ε2,  ϖ[1,T], (5.1)

    where ε1,ε2 are given two positive real numbers.

    Definition 5.1. Problem (1.1) is Hyers-Ulam stable if there exist Ωi>0,i=1,2,3,4 such that for a given ε1,ε2>0 and for each solution (S,Z)C([1,T],×R2) of inequality (5.1), there exists a solution (S,Z)C([1,T],×R2) of problem (1.1) with

    {|S(ϖ)S(ϖ)|Ω1ε1+Ω2ε2,  ϖ[1,T],|Z(ϖ)Z(ϖ)|¯Ω1ε1+¯Ω2ε2,  ϖ[1,T]. (5.2)

    Remark 5.1. (S,Z) is a solution of inequality (5.1) if there exist functions QiC([1,T],R),i=1,2, which depend upon S,Z, respectively, such that

    i)|Q1(ϖ)|ε1,  ii)|Q2(ϖ)|ε2,   ϖ[1,T]. (5.3)
    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))+Q1(ϖ),  ϖ[1,T],(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))+Q2(ϖ),  ϖ[1,T], (5.4)

    Remark 5.2. If (S,Z), respectively, is a solution of inequality (5.1), then (S,Z) is a solution of the following inequality:

    {|S(ϖ)S(ϖ)|Ω1ε1+Ω2ε2,  ϖ[1,T],|Z(ϖ)Z(ϖ)|¯Ω1ε1+¯Ω2ε2,  ϖ[1,T]. (5.5)

    As from Remark 5.1, we have

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ))+Q1(ϖ),  ϖ[1,T],(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ))+Q2(ϖ),  ϖ[1,T]. (5.6)

    With the help of Definition 5.1 and Remark 5.1, we verified Remark 5.2 in the following lines:

    |S(ϖ)S(ϖ)||1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21|G(ς,S(ς),Z(ς))|ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21|G(ς,S(ς),Z(ς))|ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21|G(ς,S(ς),Z(ς))|ςdς1Γ(α1)T1(logTς)α11|F(ς,S(ς),Z(ς))|ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11|F(ς,S(ς),Z(ς))|ςdς
    +bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11|F(ς,S(ς),Z(ς))|ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11|F(ς,S(ς),Z(ς))|ςdς1Γ(α2)T1(logTς)α21|G(ς,S(ς),Z(ς))|ςdς}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11|F(ς,S(ς),Z(ς))|ςdς||1Δ[{λ1T1S(ς)ςdςλ2mj=1ηjξj1(Z)ςςdςλ2ni=1θiHIϕi1+ζi1Z(ς)ςdςλ2rk=1λkHDωk1+μk1Z(ς)ςdς+mj=1ηj1Γ(α2)ξj1(logξjς)α21|Q2(ς)|ςdς+ni=1θiHIϕi1+1Γ(α2)ζi1(logζiς)α21|Q2(ς)|ςdς+rk=1λkHDωk1+1Γ(α2)μk1(logμkς)α21|Q2(ς)|ςdς1Γ(α1)T1(logTς)α11|Q1(ς)|ςdς}{(logT)γ21}+{λ2T1Z(ς)ςdςau=1Puλ1ψu1(S)ςςdςbv=1QvHIδv1+λ1σv1(S)ςςdςcw=1MwHDϑw1+λ1πw1(S)ςςdς+au=1Pu1Γ(α1)ψu1(logψuς)α11|Q1(ς)|ςdς
    +bv=1QvHIδv1+1Γ(α1)σv1(logσvς)α11|Q1(ς)|ςdς+cw=1MwHDϑw1+1Γ(α1)πw1(logπwς)α11|Q1(ς)|ςdς1Γ(α2)T1(logTς)α21|Q2(ς)|ςdς}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]λ1ϖ1S(ς)ςdς+1Γ(α1)ϖ1(logϖς)α11|Q1(ς)|ςdς|ε1{1Δ[{λ1(logT)+(logT)α1Γ(α1+1)}{(logT)γ21}+{au=1Puλ1(logψu)bv=1QvHIδv1+λ1(logσv)cw=1MwHDϑw1+λ1(logπw)+au=1Pu(logψu)α1Γ(α1+1)+bv=1QvHIδv1+(logσv)α1Γ(α1+1)+cw=1MwHDϑw1+(πw)α1Γ(α1+1)}×{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]
    λ1(logT)+(logT)α1Γ(α1+1)}+ε2{1Δ[{λ2mj=1ηj(logξj)λ2ni=1θiHIϕi1+(logζi)λ2rk=1λkHDωk1+(logμk)+mj=1ηj(logξj)α2Γ(α2+1)+ni=1θiHIϕi1+(logζi)α2Γ(α2+1)+rk=1λkHDωk1+(logμk)α2Γ(α2+1)}{(logT)γ21}+{λ2(logT)(logT)α2Γ(α2+1)}{mj=1ηi(logξi)γ21+ni=1θiHIϕi1+(logζi)γ21+rk=1λkHDωk1+(logμk)γ21}]},|S(ϖ)S(ϖ)|Ω1ε1+Ω2ε2. (5.7)

    By the same method, we can obtain that

    |Z(ϖ)Z(ϖ)|¯Ω1ε1+¯Ω2ε2, (5.8)

    where Ω1,Ω2,¯Ω1,¯Ω2 are given by (3.4)–(3.7). Hence, Remark 5.2 is verified, with the help of (5.7) and (5.8). Thus, the nonlinear sequential coupled system of HHFDEs is Hyers-Ulam stable and, consequently, the system (1.1) is Hyers-Ulam stable.

    Consider the following Hilfer-Hadamard fractional BVP:

    {(HHDα1,β11++λ1HHDα11,β11+)S(ϖ)=F(ϖ,S(ϖ),Z(ϖ),Ip1S(ϖ),Ip2Z(ϖ)),(HHDα2,β21++λ2HHDα21,β21+)Z(ϖ)=G(ϖ,S(ϖ),Z(ϖ),Iq1S(ϖ),Iq2Z(ϖ)),S(1)=0,  S(T)=mj=1ηjZ(ξj)+ni=1θiHIϕi1Z(ζi)+rk=1λHkDωk1Z(μk),Z(1)=0,  Z(T)=au=1PuS(ψu)+bv=1QvHIδv1+S(σv)+cw=1MHwDϑw1+S(πw), (6.1)

    with α1=5/4, α2=3/2, β1=1/2, β2=1/4, m=2, n=2, r=2, a=2, b=2, c=2, η1=1/5, η2=1/10, ξ1=4/3, ξ2=3/2, ϕ1=5/3, ϕ2=7/3, θ1=1/2, θ2=1/2, ζ1=7/3, ζ2=5/2, λ1=1, λ2=1, ω1=1/4, ω2=2/3, μ1=4, μ2=4/3, P1=1/18, P2=1/9, ψ1=4/3, ψ2=5/2, Q1=1/4, Q2=1/7, δ1=1/4, δ2=3/5, σ1=5/3, σ2=3/2, M1=2/3, M2=2/5, ϑ1=2/3, ϑ2=3/5, π1=5/2, π2=3/2. Using the given data, it is found that γ1=13/8,γ2=13/8, Δ=0.639100490745,A1=0.799441,A2=0.799441,B1=0.1184655,B2=0.1251315,Ω1=1.3283929,Ω2=0.718823345,¯Ω1=1.028734,¯Ω2=0.97432874,T=2,p1=11/5,p2=25/6,q1=11/5,q2=22/7.

    Example 6.1. For illustrating Theorem 4.2, we take

    {|F(ϖ,S1,S2,S3,S4)|ϖϖ2+1(cosϖ+18sin(S1+S2))19(ϖ+1)S3+19arctanS4,|G(ϖ,S1,S2,S3,S4)|1(ϖ+2)2[7eϖ+13S1+4S2]ϖ+35sin(S3+S4), (6.2)

    for all ϖ[0,1],SiR,i=1,2,3,4.

    We obtained the inequalities

    {|F(ϖ,S1,S2,S3,S4)|=12+116|S1|+116|S2|+118|S3|+19|S4|,|G(ϖ,S1,S2,S3,S4)|=79+127|S1|+427|S2|+45|S3|+45|S4|, (6.3)

    for all ϖ[0,1] and SiR,i=1,2,3,4. We also have M0=12,M1=116,M2=116,M3=118, M4=19,N0=79,N1=127, N2=427,N3=45,N4=45,

    We find here Ψ10.8228588 and Ψ20.5948088559. We deduce that the condition max{Ψ1,Ψ2}=Ψ1<1 is satisfied. Then, by Theorem 4.2, we conclude that the problem (6.1) with the nonlinearities (6.2) has at least one solution ϖ[0,1].

    Example 6.2. For illustrating Theorem 4.3, we take

    {|F(ϖ,S1,S2,S3,S4)|ϖ+13+19(ϖ+2)(S1+|S2|1+|S2|)1(1+ϖ)2cosS3+ϖ4arctanS4,|G(ϖ,S1,S2,S3,S4)|ϖ2+2ϖ3+217S1+18sinS2+15(ϖ+3)sinS3e2ϖ|S4|8(1+|S4|), (6.4)

    for all ϖ[0,1],SiR,i=1,2,3,4.

    We obtain here the following inequalities

    {|F(ϖ,S1,S2,S3,S4)F(ϖ,Z1,Z2,Z3,Z4)|(1/27|S1Z1|+1/27|S2Z2|+1/4|S3Z3|+1/4|S4Z4|),|G(ϖ,S1,S2,S3,S4)G(ϖ,Z1,Z2,Z3,Z4)|(1/7|S1Z1|+1/8|S2Z2|+1/20|S3Z3|+1/8|S4Z4|), (6.5)

    for all ϖ[0,1]. So, we have c0=1/4 and d0=1/20. In addition, we find ρ11.4125480,ρ21.13889158, D10.510067622,D20.43933889. Then, D1+D20.9410020109<1, that is, the condition (4.15) is satisfied. Therefore, by Theorem 4.3, we conclude that problem (6.1) with the nonlinearities (6.4) has a unique solution ϖ[0,1].

    We have presented criteria for the existence, uniqueness, and Ulam-Hyers stability of solutions to a coupled system of nonlinear sequential HHFIEs and nonlocal coupled Hadamard fractional integrodifferential and multipoint boundary conditions. We derive the expected results using a methodology that uses modern analytical tools. It is imperative to emphasize that the results offered in this specific context are novel and contribute to the corpus of existing literature on the topic. Furthermore, our results encompass cases where the system reduces to the boundary conditions of the form:

    When ηj=Pu=0, then we get

    {S(1)=0,  S(T)=ni=1θiHIϕiZ(ζi)+rk=1λHkDωk1Z(μk),Z(1)=0,  Z(T)=bv=1QvHIδvS(σv)+cw=1MHwDϑw1S(πw).

    If θi=Qv=0, we get:

    {S(1)=0,  S(T)=mj=1ηjZ(ξj)+rk=1λHkDωk1Z(μk),Z(1)=0,  Z(T)=au=1PuS(ψu)+cw=1MHwDϑw1S(πw).

    When λk=Mw=0, the outcome is:

    {S(1)=0,  S(T)=mj=1ηjZ(ξj)+ni=1θiHIϕiZ(ζi),Z(1)=0,  Z(T)=au=1PuS(ψu)+bv=1QvHIδvS(σv).

    In addition, if ηj=Pu=λk=Mw=0, we obtain:

    {S(1)=0,  S(T)=ni=1θiHIϕiZ(ζi),Z(1)=0,  Z(T)=bv=1QvHIδvS(σv).

    When ηj=Pu=θi=Qv=0, the boundary condition is:

    {S(1)=0,  S(T)=rk=1λHkDωk1Z(μk),Z(1)=0,  Z(T)=cw=1MHwDϑw1S(πw).

    If λk=Mw=θi=Qv=0, we obtain:

    {S(1)=0,  S(T)=mj=1ηjZ(ξj),Z(1)=0,  Z(T)=au=1PuS(ψu).

    These cases represent new findings. Looking ahead, our future plans include extending this work to a coupled system of nonlinear sequential HHFIEs enhanced by the nonlocal coupled mixed integro-differential and discrete type boundary conditions. We also intend to investigate the multivalued analogue of the problem studied in this paper.

    Subramanian Muthaiah: Developed the conceptualization and proposed the method, wrote–original draft, reviewed and edited the paper; Manigandan Murugesan: Developed the conceptualization and proposed the method, wrote–original draft, investigated, processed and provided examples; Muath Awadalla: Investigated, processed and provided examples, reviewed and edited the paper; Bundit Unyong: Developed the conceptualization and proposed the method, reviewed and edited the paper; Ria H Egami: Investigated, processed and provided examples, reviewed and edited the paper. All authors have read and agreed to the published version of the manuscript.

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

    This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [GrantA113]. This study is supported via funding from Prince Sattam bin Abdulaziz University, project number (PSAU/2024/R/1445). M.Manigandan gratefully acknowledges the Center for Computational Modeling, Chennai Institute of Technology, India, vide funding number CIT/CCM/2023/RP-018.

    The authors declare no conflicts of interest.



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