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Bifurcation and chaos analysis of the Kadomtsev Petviashvili-modified equal width equation using a novel analytical method: describing ocean waves

  • These authors contributed equally to this work and are co-first authors
  • Received: 13 February 2025 Revised: 08 April 2025 Accepted: 15 April 2025 Published: 25 April 2025
  • MSC : 34G20, 35A20, 35C07, 35C08

  • This work investigated the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation describing ocean waves. Our focus was on the analysis of the KP-mEW equation from various angles, including the study of soliton solutions, bifurcation analysis, multistability, and Lyapunov exponents. First, a transformation was used to transform the partial differential equation (PDE) into an ordinary differential equation (ODE), from which the soliton solutions were obtained by using a new modified (G'/G2)-expansion method. We investigated different types of solutions of the KP-mEW equation with various parameters, including kink, periodic, singular periodic, singular kink, and singular periodic-kink solutions. We also simulated 3D and 2D plots for some solutions to enhance the visualizations. These graphical representations provide important information about the patterns and dynamics of the solutions, leading to a strong understanding of the behavior and applicability of the model. We also observed the chaotic behavior of the system by adding a perturbation term and analyzed the chaotic behavior through bifurcation plots, multistability and time series analysis, and Lyapunov exponents and obtained various dynamic modes such as periodic and quasi-periodic types. A comparison of the obtained solutions with the existing solutions was also presented in the form of Table 1.

    Citation: Amna Mumtaz, Muhammad Shakeel, Abdul Manan, Marouan Kouki, Nehad Ali Shah. Bifurcation and chaos analysis of the Kadomtsev Petviashvili-modified equal width equation using a novel analytical method: describing ocean waves[J]. AIMS Mathematics, 2025, 10(4): 9516-9538. doi: 10.3934/math.2025439

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  • This work investigated the Kadomtsev Petviashvili-modified equal width (KP-mEW) equation describing ocean waves. Our focus was on the analysis of the KP-mEW equation from various angles, including the study of soliton solutions, bifurcation analysis, multistability, and Lyapunov exponents. First, a transformation was used to transform the partial differential equation (PDE) into an ordinary differential equation (ODE), from which the soliton solutions were obtained by using a new modified (G'/G2)-expansion method. We investigated different types of solutions of the KP-mEW equation with various parameters, including kink, periodic, singular periodic, singular kink, and singular periodic-kink solutions. We also simulated 3D and 2D plots for some solutions to enhance the visualizations. These graphical representations provide important information about the patterns and dynamics of the solutions, leading to a strong understanding of the behavior and applicability of the model. We also observed the chaotic behavior of the system by adding a perturbation term and analyzed the chaotic behavior through bifurcation plots, multistability and time series analysis, and Lyapunov exponents and obtained various dynamic modes such as periodic and quasi-periodic types. A comparison of the obtained solutions with the existing solutions was also presented in the form of Table 1.



    The abbreviations used in this manuscript
    BVPs Boundary Value Problems
    HHFDEs Hilfer-Hadamard Fractional-order Differential 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 introduces and investigates a novel nonlinear nonlocal coupled boundary value problem (BVP) encompassing sequential Hilfer-Hadamard fractional-order differential equations (HHFDEs) with varying orders. The problem is formulated as:

    {(HHDψ1,β11++K1HHDψ11,β11+)ϱ(τ)=ρ1(τ,ϱ(τ),φ(τ)),  1<ψ12,  τE:=[1,T],(HHDψ2,β21++K2HHDψ21,β21+)φ(τ)=ρ2(τ,ϱ(τ),φ(τ)),  2<ψ23,  τE:=[1,T], (1.1)

    and it is enhanced by nonlocal coupled Hadamard fractional integral (HFI) boundary conditions:

    {ϱ(1)=0,ϱ(T)=λ1HIδ11+φ(η1),φ(1)=0,φ(η2)=0,φ(T)=λ2HIδ21+ϱ(η3),    1<η1,η2,η3<T. (1.2)

    Here, ψ1(1,2], ψ2(2,3], β1,β2[0,1], K1,K2R+, T>1, δ1,δ2>0, λ1,λ2R, HHDψi,βj1+ denotes the Hilfer-Hadamard fractional derivative (HHFD) operator of order ψi,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. It is noteworthy that this study contributes to the literature by addressing a unique configuration of sequential HHFDEs with distinct orders and coupled HFI boundary conditions. The methodology employed involves the application of the fixed-point approach to establish both existence and uniqueness results for problems (1.1) and (1.2). The conversion of the given problem into an equivalent fixed-point problem is followed by the utilization of the Leray-Schauder alternative and Banach's fixed-point theorem to prove existence and uniqueness results, respectively. The outcomes of this research are novel and enrich the existing body of literature on BVPs involving coupled systems of sequential HHFDEs. Coupled fractional derivatives are essential for modeling systems with non-local interactions and memory effects more accurately than ordinary derivatives. They enable a more precise description of phenomena, such as anomalous diffusion and viscoelasticity, enhancing our understanding of complex physical processes. This improved modeling capability leads to more accurate predictions and insights into real-world phenomena, benefiting various fields ranging from materials science to fluid dynamics and beyond. Over the past few decades, fractional calculus has emerged as a significant and widely explored field within mathematical analysis. The substantial growth observed in this field can be credited to the widespread utilization of fractional calculus methodologies in creating inventive mathematical models to depict diverse phenomena across economics, mechanics, engineering, science, and other domains. References [1,2,3,4] provide examples and detailed discussions on this topic.

    In the following section, we will present a summary of pertinent scholarly articles related to the discussed problem. The Riemann-Liouville and Caputo fractional derivatives (CFDs), among other fractional derivatives introduced, have drawn a lot of interest due to their applications. The Hilfer fractional derivative (HFD) was introduced by Hilfer in [5]. Its definition includes the Riemann-Liouville and CFDs as special cases for extreme values of the parameter. [6,7] provided further information about this derivative. [8,9,10,11,12] presented noteworthy results on Hilfer-type initial and boundary value problems (BVPs). A new work [13] explores the Ulam-Hyers stability and existence of solutions for a fully coupled system with integro-multistrip-multipoint boundary conditions and nonlinear sequential Hilfer fractional differential equations (HFDEs). Moreover, [14] investigates a hybrid generalized HFDE boundary value problem.

    In 1892, Hadamard proposed the Hadamard fractional derivative (HFD), which is a fractional derivative using a logarithmic function with an arbitrary exponent in its kernel [15]. Later research in [16,17,18,19,20] examined variations such as HHFDs and Caputo-Hadamard fractional derivatives (CHFDs). Importantly, for β values of β=0 and β=1, respectively, HFDs and CHFDs arise as special examples of the HHFD.

    Existence results for an HHFDE with nonlocal integro-multipoint boundary conditions was derived in [21]:

    {HHDα,β1x(t)=f(t,x(t)),  t[1,T],x(1)=0,  mi=1θix(ξi)=λHIδx(η). (1.3)

    Here, α(1,2], β[0,1], θi,λR, η,ξi(1,T)\ (i=1,2,...,m), HIδ is the HFI of order δ>0, and f:[1,T]×RR is a continuous function. Problem (1.3) represents a non-coupled system, in contrast to problems (1.1)–(1.2), which are coupled systems. Problems (1.1)–(1.2) exhibits nonlocal coupled integral and multi-point boundary conditions involving HFIs, while problem (1.3) incorporates discrete boundary conditions with HFIs. Existence results for nonlocal mixed Hilfer-Hadamard fractional BVPs were developed by the authors of [22]:

    {HHDα,β1x(t)=f(t,x(t)),  t[1,T],x(1)=0,  x(T)=mj=1ηjx(ξj)+ni=1ζiHIϕix(θi)+rk=1λkHDωk1x(μk). (1.4)

    Here, 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. Problem (1.4) is not a coupled system, while problems (1.1)–(1.2) are coupled systems. Problems (1.1)–(1.2) features nonlocal coupling with integral and multi-point boundary conditions involving HFIs, whereas problem (1.4) incorporates mixed discrete boundary conditions involving HFIs and derivatives. Additionally, [23] investigated a coupled HHFDEs in generalized Banach spaces. The authors of the aforementioned study [24] 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,  τ[1,T],HHDγ,δ1v(t)=g(t,u(t),v(t)),  1<γ2,  τ[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). (1.5)

    Here, 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), and f,g:[1,T]×R×RR are continuous functions. In the boundary conditions, Riemann-Stieltjes integrals are involved with Hi,Ki,Pi,Qi, (i=1,2,...,m),(i=1,2,...,n),(i=1,2,...,p),(i=1,2,...,q), which are functions of the bounded variation. Problem (1.5) involves a coupled system of HHFDEs, while problems (1.1)–(1.2) deal with coupled systems of sequential HHFDEs. In problems (1.1)–(1.2), there is nonlocal coupling with integral and multi-point boundary conditions involving HFIs, whereas in problem (1.5), Stieltjes-integral boundary conditions are incorporated, involving HFDs. Within problems (1.1)–(1.2), various fractional orders are involved, while problem (1.5) incorporates a uniform fractional order. The authors [25] conducted an analysis on the coupled system of HHFDEs with nonlocal coupled HFI boundary conditions:

    {HHDα1,β11+u(t)=ϱ1(t,u(t),v(t)),  1<α12,  τE:=[1,T],HHDα2,β21+v(t)=ϱ2(t,u(t),v(t)),  2<α23,  τE:=[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. (1.6)

    Here, α1(1,2], α2(2,3], β1,β2[0,1], T>1, δ1,δ2>0, λ1,λ2R, HHDαi,βj1+ denotes the Hilfer-Hadamard Fractional Derivative (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. Problem (1.6) involves a coupled system of HHFDEs, while problems (1.1)–(1.2) deal with coupled systems of sequential HHFDEs. Despite sharing identical boundary conditions in both (1.1)–(1.2) and (1.6), the auxiliary lemma used in problems (1.1)–(1.2) is entirely different from that in problem (1.6). Therefore, problems (1.1)–(1.2) in the manuscript are distinctly separate from problem (1.6). In problem (1.6), solutions are obtained for the coupled system of HHFDEs, whereas in problems (1.1)–(1.2), solutions are derived for the coupled system of sequential HHFDEs. A two-point boundary value problem for a system of nonlinear sequential HHFDEs was investigated in [26]:

    {(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. (1.7)

    Here, α1,α2(1,2], β1,β2[0,1], λ1,λ2,A1,A2R+, and f,g:[1,e]×R×RR are continuous functions. Within problems (1.1)–(1.2), various fractional orders are involved, while problem (1.7) incorporates a uniform fractional order. Problem (1.7) is characterized by a two-point boundary condition, whereas problems (1.1)–(1.2) incorporates multi-point boundary conditions along with HFIs.

    The sections of this document are organized as follows: The fundamental ideas of fractional calculus relating 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. For a continuous function φ:[a,)R, the HFI of order δ>0 is given by

    HIδa+φ(τ)=1Γ(δ)τa(logτϖ)δ1φ(ϖ)ϖdϖ, (2.1)

    where log()=loge().

    Definition 2.2. For a continuous function φ:[a,)R, the HFD of order δ>0 is given by

    HDδa+φ(τ)=pn(HInδa+φ)(τ),  n=[δ]+1, (2.2)

    where pn=τndndtn, and [δ] represents the integer parts of the real number δ.

    Lemma 2.3. If δ,γ>0 and 0<a<b<, then

    (1)(HIδa+(logτa)γ1)(ϱ)=Γ(γ)Γ(γ+δ)(logϱa)γ+δ1,
    (2)(HDδa+(logτa)γ1)(ϱ)=Γ(γ)Γ(γδ)(logϱa)γδ1.

    In particular, for γ=1, we have (HDδa+)(1)=1Γ(1δ)(logϱa)δ0,0<δ<1.

    Definition 2.4. For n1<δ<n and 0γ1, the HHFD of order δ and γ for φL1(a,b) is defined as

    (HHDδ,γa+)=(HIγ(nδ)a+pnHI(nδ)(1γ)a+φ)(τ)=(HIγ(nδ)a+pnHI(nq)a+φ)(τ)=(HIγ(nδ)a+HDqa+φ)(τ), q=δ+nγδγ,

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

    Theorem 2.5. If φL1(a,b),0<a<b<, and (HInqa+φ)(τ)ACnp[a,b], then

    HIδa+(HHDδ,γa+φ)(τ)=HIqa+(HHDqa+φ)(τ)=φ(τ)n1j=o(p(nj1)(HIδa+φ))(a)Γ(qj)(logτa)qj1,

    where δ>0,0γ1, and q=δ+nγδγ,n=[δ]+1. Observe that Γ(qj) exists for all j=1,2,,n1 and q[δ,n].

    Lemma 2.6. Let h1,h2C(E,R). Then, the solution to the linear Hilfer-Hadamard coupled BVP is given by:

    {(HHDψ1,β11++K1HHDψ11,β11+)ϱ(τ)=h1(τ),  1<ψ12,(HHDψ2,β21++K2HHDψ21,β21+)φ(τ)=h2(τ),  2<ψ23,ϱ(1)=0,ϱ(T)=λ1HIδ11+φ(η1),φ(1)=0,φ(η2)=0,φ(T)=λ2HIδ21+ϱ(η3),    1<η1,η2,η3<T, (2.3)
    ϱ(τ)=(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21h2(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11h1(ϖ)ϖdϖ](logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21h2(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11h1(ϖ)ϖdϖ)](logη1)γ22log(η1η2)}K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ11h1(ϖ)ϖdϖ, (2.4)

    and

    φ(τ)=(logτ)γ22log(τη2)×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21h2(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11h1(ϖ)ϖdϖ](λ2HIδ21+(logη3)γ11)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21h2(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11h1(ϖ)ϖdϖ)](logT)γ11}+(logτlogη2)γ22K2η21φ(ϖ)ϖdϖK2τ1φ(ϖ)ϖdϖ(logτlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ21h2(ϖ)ϖdϖ, (2.5)

    where

    A1=(logT)γ11,A2=λ1Γ(γ21)(logη1)δ1+γ22Γ(δ1+γ21){logη2γ21δ1+γ21logη1},B1=λ2Γ(γ1)Γ(δ2+γ1)(logη3)δ2+γ11,B2=(logT)γ22log(Tη2),Δ=A1B2A2B1. (2.6)

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

    (HHDψ1,β11++K1HHDψ11,β11+)ϱ(τ)=h1(τ), (2.7)
    (HHDψ2,β21++K2HHDψ21,β21+)φ(τ)=h2(τ). (2.8)

    Taking the Hadamard fractional integral of order ψ1 and ψ2 on both sides of (2.7) and (2.8), we get

    (HIψ11+HHDψ1,β11++HIψ11+K1HHDψ11,β11+)ϱ(τ)=HIψ11+h1(τ),(HIψ21+HHDψ2,β21++HIψ21+K2HHDψ21,β21+)φ(τ)=HIψ11+h2(τ). (2.9)

    Equation (2.9) can be written as follows,

    ϱ(τ)=c0(logτ)γ11+c1(logτ)γ12K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)τ1(logτϖ)ψ11h1(ϖ)ϖdϖ. (2.10)
    φ(τ)=d0(logτ)γ21+d1(logτ)γ22+d2(logτ)γ23K2τ1φ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ21h2(ϖ)ϖdϖ. (2.11)

    Here, c0,c1,d0,d1, and d2 are arbitrary constants. Now, using boundary conditions (1.2) together with (2.10) and (2.11), one can get

    ϱ(τ)=c0(logτ)γ11+c1(logτ)2γ1K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)τ1(logτϖ)ψ11h1(ϖ)ϖdϖ=0, (2.12)
    φ(τ)=d0(logτ)γ21+d1(logτ)γ22+d2(logτ)3γ2K2τ1φ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ21h2(ϖ)ϖdϖ=0, (2.13)

    from which we have c1=0 and d2=0. Equations (2.12) and (2.13) can be written as

    ϱ(τ)=c0(logτ)γ11K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)τ1(logτϖ)ψ11h1(ϖ)ϖdϖ, (2.14)
    φ(τ)=d0(logτ)γ21+d1(logτ)γ22K2τ1φ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ21h2(ϖ)ϖdϖ. (2.15)

    Using the conditions φ(η2)=0 in (2.15), we get

    d1=1(logη2)γ22[1Γ(ψ2)η21(logη2ϖ)ψ21h2ϖdϖ+d0(logη2)γ21K2η21φ(ϖ)ϖdϖ], (2.16)

    and substituting the value of d1 into (2.15), we obtain

    φ(τ)=d0(logτ)γ22log(τη2)(logτlogη2)γ22K2η21φ(ϖ)ϖdϖK2τ1φ(ϖ)ϖdϖ(logτlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2ϖdϖ+τ1(logη2ϖ)ψ21h2ϖdϖ. (2.17)

    Now, using (2.14) and (2.17) in the conditions:

    ϱ(T)=λ1HIδ11+φ(η1),φ(T)=λ2HIδ21+ϱ(η3),

    we find that

    {c0A1+d1A2=I1,c0B1+d1B2=I2. (2.18)

    Thus, we get,

    c0=1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21h2(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11h1(ϖ)ϖdϖ](logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21h2(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11h1(ϖ)ϖdϖ)](logη1)γ22log(η1η2)}, (2.19)

    and

    d0=1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21h2(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11h1(ϖ)ϖdϖ](λ2HIδ21+(logη3)γ11)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21h2(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21h2(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11h1(ϖ)ϖdϖ)](logT)γ11}, (2.20)

    where Δ is defined in (2.6). By substituting the value of c0 obtained from (2.19) into (2.14), and substituting the values of d0 and d1 obtained from (2.20) and (2.16) into (2.15), the resulting solution is given by (2.4) and (2.5).

    Denote by X={ϱ(τ)|ϱ(τ)C([1,T],R) as the Banach space of all functions (continuous) from [1,T] into R equipped with the norm ϱ=supτ[1,T]|ϱ(τ)|. Obviously, (X,) is a Banach space and, as a result, the product space (X×X,) is a Banach space with the norm (ϱ,φ)=ϱ+φ for (ϱ,φ)(X×X). In view of Lemma 2.4, we define an operator Ω:X×XX×X by

    Ω(ϱ,φ)(τ)=(Ω1(ϱ,φ)(τ),Ω2(ϱ,φ)(τ)), (3.1)

    where

    Ω1(ϱ,φ)(τ)=(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logη1)γ22log(η1η2)}K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)τ1(logτϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ, (3.2)

    and

    Ω2(ϱ,φ)(τ)=(logτ)γ22log(τη2)×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(λ2HIδ21+(logη3)γ11)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ1)η31(logη3ϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logT)γ11}+(logτlogη2)γ22K2η21φ(ϖ)ϖdϖK2τ1φ(ϖ)ϖdϖ(logτlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ.

    We need the following hypotheses in what follows:

    (H1) Assume that there exist real constants κi,ˆκi0(i=1,2) and κ0>0,ˆκ0>0 such that, for all τ[1,T],xiR,i=1,2,

    |ρ1(τ,ϱ,φ)|κ0+κ1|ϱ|+κ2|φ|,|ρ2(τ,ϱ,φ)|ˆκ0+ˆκ1|ϱ|+ˆκ2|φ|.

    (H2) There exist positive constants L,ˆL, such that, for all τ[1,T],ϱi,φiR,i=1,2,

    |ρ1(τ,ϱ1,ϱ2)ρ1(τ,φ1,φ2)|L(|ϱ1φ1|+|ϱ2φ2|),|ρ2(τ,ϱ1,ϱ2)ρ2(τ,φ1,φ2)|ˆL(|ϱ1φ1|+|ϱ2φ2|).

    Furthermore, we establish the notation:

    W1=logTγ12Δ[K1(logT)+(logT)ψ1Γ(ψ1+1)](logT)γ22log(Tη2)+[λ2K1(logη1)δ2Γ(δ2+2)+λ2(logη3)δ2+ψ1Γ(δ2+ψ1+1)](logη1)γ12log(η1η2)+K1(logT)+(logT)ψ1Γ(ψ1+1), (3.3)
    W2=logTγ12Δ[λ1K2(logη1logη2)γ22(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ22(logη2)ψ2Γ(δ1+ψ2+1)+(logη2)ψ2Γ(δ1+ψ2+1)](logT)γ22log(Tη2)+[(logTlogη2)γ22K2(logη2)+K2(logT)+(logTlogη2)γ22(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logη1)γ12log(η1η2), (3.4)
    ^W1=(logT)γ22log(Tη2)×(1Δ)[K1(logT)+(logT)ψ1Γ(ψ1+1)]λ2Γ(γ1)(γ1+δ2)(logη3)γ1+δ21+[K1λ2(logη3)δ2Γ(δ2+2)+λ2(logη3)ψ1+δ2Γ(ψ1+δ2+1)](logT)γ11, (3.5)
    ^W2=(logT)γ22log(Tη2)×(1Δ)[λ1K2(logη1logη2)γ22(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ22(logη2)ψ2+δ1Γ(δ1+ψ2+1)+(logη2)ψ2+δ1Γ(δ1+ψ2+1)]λ2Γ(γ1)(γ1+δ2)(logη3)γ1+δ21+[(logTlogη2)γ22K2(logη2)+K2(logT)+(logTlogη2)γ22(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logT)γ11+(logTlogη2)γ22K2(logη2)+K2(logT)+(logTlogη2)γ22(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1), (3.6)
    Φ=min{1[(W1+^W1)κ1+(W2+^W2)^κ1],1[(W1+^W1)κ2+(W2+^W2)^κ2]}. (3.7)

    To demonstrate the existence of solutions for problems (1.1) and (1.2), we employ the following established result.

    Lemma 3.1. The Leray-Schauder alternative. Let F(X)={xD:x=kX(x) for some 0<k<1}, where X:DD is a completely continuous operator. Then, either the set F(X) is unbounded or there exists at least one fixed point for operator X.

    We establish an existence result in this section using the Leray-Schauder alternative.

    Theorem 3.2. Presume (H1) is true. Furthermore, it is presumed that

    (W1+W2)κ1+(^W1+^W2)^κ1<1, (3.8)

    and

    (W1+W2)κ2+(^W1+^W2)^κ2<1. (3.9)

    Then, systems (1.1) and (1.2) have at least one solution on [1,T].

    Proof. To demonstrate that Ω, defined by (3.1), has a fixed point, we shall employ the Leray-Schauder alternative. The proof is split into two parts. Step 1, we show that Ω:X×XX×X, defined by (3.1), is completely continuous (C.C).

    First we show that Ω is continuous. Let {(ϱn,φn)} be a sequence such that (ϱn,φn)(ϱ,φ) in X×X. Then, for each τ[1,T], we have

    |Ω1(ϱn,φn)Ω1(ϱ,φ)||(logτ)γ12|×1Δ{[λ1HIδ11+{(logη1logη2)γ22|K2η21φn(ϖ)φ(ϖ)ϖdϖ|
    +|K2η11φn(ϖ)φ(ϖ)ϖdϖ|+|(logη1logη2)γ22|1Γ(ψ2)|η21(logη2ϖ)ψ21ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|
    +|1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|}
    +K1|T1ϱn(ϖ)ϱ(ϖ)ϖdϖ|+1Γ(ψ1)|T1(logTϖ)ψ11ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|]×(logT)γ22log(Tη2)+[(logTlogη2)γ22K2|η21φn(ϖ)φ(ϖ)ϖdϖ|+K2|T1φn(ϖ)φ(ϖ)ϖdϖ|
    +(logTlogη2)γ221Γ(ψ2)|η21(logη2ϖ)ψ21ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|+1Γ(ψ2)|T1(logTϖ)ψ21ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|+λ2Iδ21+(K1|η31ϱn(ϖ)ϱ(ϖ)ϖdϖ|
    +1Γ(ψ2)|η31(logη3ϖ)ψ11ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|)]×(logη1)γ22log(η1η2)}+K1|τ1ϱn(ϖ)ϱ(ϖ)ϖdϖ|
    +1Γ(ψ2)|τ1(logτϖ)ψ11ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21|φn(ϖ)φ(ϖ)|ϖdϖ+K2η11|φn(ϖ)φ(ϖ)|ϖdϖ+(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ
    +1Γ(ψ2)η11(logη1ϖ)ψ21|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ}+K1T1|ϱn(ϖ)ϱ(ϖ)|ϖdϖ+1Γ(ψ1)T1(logTϖ)ψ11|ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ]×(logT)γ22log(Tη2)
    +[(logTlogη2)γ22K2η21|φn(ϖ)φ(ϖ)|ϖdϖ+K2T1|φn(ϖ)φ(ϖ)|ϖdϖ+(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ+1Γ(ψ2)T1(logTϖ)ψ21|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ+λ2Iδ21+(K1η31|ϱn(ϖ)ϱ(ϖ)|ϖdϖ
    +1Γ(ψ2)η31(logη3ϖ)ψ11|ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ)]×(logη1)γ22log(η1η2)}+K1τ1|ϱn(ϖ)ϱ(ϖ)|ϖdϖ
    +1Γ(ψ2)τ1(logτϖ)ψ11|ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ.

    Since ρ1 is continuous, we get

    |ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|0   as (ϱn,φn)(ϱ,φ),

    and

    |ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|0   as (ϱn,φn)(ϱ,φ).

    Then,

    Ω1(ϱnφn)Ω1(ϱφ)0  as  (ϱn,φn)(ϱ,φ). (3.10)

    In the same way, we obtain

    Ω2(ϱnφn)Ω2(ϱφ)0  as  (ϱn,φn)(ϱ,φ). (3.11)

    It follows from (3.10) and (3.11) that

    Ω(ϱnφn)Ω(ϱφ)0  as  (ϱn,φn)(ϱ,φ). (3.12)

    Hence, Ω is continuous. Let us initially establish the complete continuity of the operator Ω:X×XX×X as defined in (3.1). Evidently, the continuity of the operator Ω in terms of Ω1 and Ω2 is a consequence of the continuity of ρ1 and ρ2. Subsequently, we proceed to demonstrate that the operator Ω is uniformly bounded.

    To achieve this, let MX×X be a bounded set. Consequently, we can identify positive constants N1 and N2 satisfying ρ1|(τ,ϱ(τ),φ(τ))|N1 and ρ2|(τ,ϱ(τ),φ(τ))|N2, (ϱ,φ)M. Consequently, we obtain

    ||Ω1(ϱ,φ)||=sup|Ω1(ϱ,φ)(τ)|(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logη1)γ22log(η1η2)}K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ,N1{logTγ12Δ[K1(logT)+(logT)ψ1Γ(ψ1+1)](logT)γ22log(Tη2)+[λ2K1(logη1)δ2Γ(δ2+2)+λ2(logη3)δ2+ψ1Γ(δ1+ψ1+1)](logη1)γ12log(η1η2)+K1(logT)+(logT)ψ1Γ(ψ1+1)}+N2{logTγ12Δ[λ1K2(logη1logη2)γ22(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ22(logη2)ψ2Γ(δ1+ψ2+1)+(logη2)ψ2Γ(δ1+ψ2+1)](logT)γ22log(Tη2)+[(logTlogη2)γ22K2(logη2)+K2(logT)+(logTlogη2)γ22(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logη1)γ12log(η1η2)}.

    This, considering the notation in (3.3) and (3.4), results in:

    ||Ω1(ϱ,φ)||W1N1+W2N2. (3.13)

    Likewise, using the notation of (3.5) and (3.6), we have

    ||Ω2(ϱ,φ)||^W1N1+^W2N2. (3.14)

    Then, it follows from (3.13) and (3.14) that

    ||Ω(ϱ,φ)||(W1+^W1)N1+(W2+^W2)N2. (3.15)

    This demonstrates that the operator Ω is uniformly bounded.

    To establish the equicontinuity of Ω, we consider τ1,τ2[1,T] with τ1<τ2. Then, we find that

    |Ω1(ϱ,φ)(τ2)Ω1(ϱ,φ)(τ1)|(logτ2)γ12(logτ1)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21dϖϖ+1Γ(ψ2)η11(logη1ϖ)ψ21dϖϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11dϖϖ](logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21dϖϖ1Γ(ψ2)T1(logTϖ)ψ21dϖϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11dϖϖ)](logη1)γ22log(η1η2)}K1τ1τ2ϱ(ϖ)ϖdϖ+1Γ(ψ2)τ11|(logτ2ϖ)ψ11(logτ1ϖ)ψ11|dϖϖ+1Γ(ψ2)τ1τ2(logτ2ϖ)ψ11dϖϖ,  0 as τ2τ1, (3.16)

    independent of (ϱ,φ)M. Likewise, it can be shown that |Ω2(ϱ,φ)(τ2)Ω2(ϱ,φ)(τ1)|0 as τ2τ1 independent of (ϱ,φ)M. Thus, the equicontinuity of Ω1 and Ω2 implies that the operator Ω is equicontinuous. Hence, the operator Ω is equicontinuous. Therefore, the operator Ω satisfies the conditions for compactness according to Arzela-Ascoli's theorem. Lastly, we confirm the boundedness of the set: Θ(Ω)={(ϱ,φ)X×X:(ϱ,φ)=κΩ(ϱ,φ);0κ1}. Let (ϱ,φ)Θ(Ω). Then (ϱ,φ)=κΩ(ϱ,φ), which implies that

    ϱ(τ)=κΩ1(ϱ,φ)(τ),φ(τ)=κΩ2(ϱ,φ)(τ),

    for any τ[1,T].

    Based on the assumption (H1), we obtain:

    |ϱ(τ)|(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖ+K2η11φ(ϖ)ϖdϖ+(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21[^κ0+^κ1|ϱ|+^κ2|φ|]ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21[^κ0+^κ1|ϱ|+^κ2|φ|]ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11[κ0+κ1|ϱ|+κ2|φ|]ϖdϖ](logT)γ22log(Tη2)+[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖ+K2T1φ(ϖ)ϖdϖ+(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21[^κ0+^κ1|ϱ|+^κ2|φ|]ϖdϖ+1Γ(ψ2)T1(logTϖ)ψ21[^κ0+^κ1|ϱ|+^κ2|φ|]ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ+1Γ(ψ2)η31(logη3ϖ)ψ11[κ0+κ1|ϱ|+κ2|φ|]ϖdϖ)]×(logη1)γ22log(η1η2)}+K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ11[κ0+κ1|ϱ|+κ2|φ|]ϖdϖW1[κ0+κ1|ϱ|+κ2|φ|]+W2[^κ0+^κ1|ϱ|+^κ2|φ|], (3.17)

    which implies that

    ||ϱ||=supτ[1,T]|ϱ(τ)|W1κ0+W2^κ0+(W1κ1+W2^κ1)||ϱ||+(W1κ2+W2^κ2)||φ||. (3.18)

    Similarly, one can find that

    ||φ||^W1κ0+^W2^κ0+(^W1κ1+^W2^κ1)||ϱ||+(^W1κ2+^W2^κ2)||φ||. (3.19)

    From (3.18) and (3.19), we obtain

    ||ϱ||+||φ||(W1+^W1)κ0+(W2+^W2)^κ0+(W1+^W1)κ1+(W2+^W2)^κ1||ϱ||+(W1+^W1)κ2+(W2+^W2)^κ2||φ||.

    Which, by ||(ϱ,φ)||=||ϱ||+||φ||, yields

    ||(ϱ,φ)||1Φ[(W1+^W1)κ0+(W2+^W2)^κ0].

    As a result, Θ(Ω) is constrained within bounds. Consequently, the conclusion of Lemma 3.1 is applicable, implying that the operator Ω possesses at least one fixed point. This fixed point indeed corresponds to a solution of problems (1.1) and (1.2).

    In the forthcoming findings, the application of Banach's fixed-point theorem will be utilized to demonstrate the existence of a unique solution for the problems (1.1) and (1.2).

    Theorem 3.3. If condition (H2) is met, and the inequality

    (W1+^W1)L1+(W2+^W2)L2<1, (3.20)

    holds, where Wi and ^Wi are defined in (3.3)–(3.6), then problems (1.1) and (1.2) possess unique solutions over the interval [1,T].

    Proof. Denoting K1={supτ[1,T]|ρ1(τ,0,0)|<} and K2={supτ[1,T]|ρ2(τ,0,0)|<}, it can be inferred from assumption (H1) that

    |ρ1(τ,ϱ,φ)|L1(||ϱ||+||φ||)+K1L1||(ϱ,φ)||+K1,

    and

    |ρ2(τ,ϱ,φ)|L2||(ϱ,φ)||+K2.

    First, we show that ΩBrBr, where Br={(ϱ,φ)X×X:||(ϱ,φ)||r}, with

    r(W1+^(W1))K1+(W2+^(W2))K21(W1+ˆ(W1))L1+(W2+^(W2))ˆL. (3.21)

    For (ϱ,φ)Br, we have

    ||Ω1(ϱ,φ)||=supτ[1,T]|Ω1(ϱ,φ)(τ)|(logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21φ(ϖ)ϖdϖK2η11φ(ϖ)ϖdϖ(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1T1ϱ(ϖ)ϖdϖ1Γ(ψ1)T1(logTϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(logT)γ22log(Tη2)[(logTlogη2)γ22K2η21φ(ϖ)ϖdϖK2T1φ(ϖ)ϖdϖ(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ1Γ(ψ2)T1(logTϖ)ψ21ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1η31ϱ(ϖ)ϖdϖ1Γ(ψ2)η31(logη3ϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logη1)γ22log(η1η2)}K1τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ11ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ (3.22)

    and

    ||Ω(ϱ,φ)||(L1r+K1){logTγ12Δ[K1(logT)+(logT)ψ1Γ(ψ1+1)](logT)γ22log(Tη2)+[λ2K1(logη1)δ2Γ(δ2+2)+λ2(logη3)δ2+ψ1Γ(δ1+ψ1+1)](logη1)γ12log(η1η2)+K1(logT)+(logT)ψ1Γ(ψ1+1)+logTγ12Δ[λ1K2(logη1logη2)γ22(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ22(logη2)ψ2Γ(δ1+ψ2+1)+(logη2)ψ2Γ(δ1+ψ2+1)](logT)γ22log(Tη2)+[(logTlogη2)γ22K2(logη2)+K2(logT)+(logTlogη2)γ22(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logη1)γ12log(η1η2)}.

    Making use of the notation of (3.3)–(3.6), we get

    ||Ω1(ϱ,φ)||(L1W1+L2W2)r+W1K1+W2K2. (3.23)

    Likewise, we can find that

    ||Ω2(ϱ,φ)||(L1^W1+L2^W2)r+W1^K1+^W2K2. (3.24)

    Then, it follows from (3.23)–(3.24) that

    ||Ω(ϱ,φ)||||Ω1(ϱ,φ)||+||Ω2(ϱ,φ)||r.

    Therefore, ΩBrBr as (ϱ,φ)Br is an arbitrary element.

    To confirm the contraction property of the operator Ω, consider (ϱi,φj)Br for i=1,2. Subsequently, we obtain

    Ω1(ϱ1,φ1)Ω1(ϱ,φ)|(logτ)γ12|×1Δ{[λ1HIδ11+{(logη1logη2)γ22|K2η21φ1(ϖ)φ(ϖ)ϖdϖ|
    +|K2η11φ1(ϖ)φ(ϖ)ϖdϖ|+|(logη1logη2)γ22|1Γ(ψ2)|η21(logη2ϖ)ψ21ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|
    +|1Γ(ψ2)η11(logη1ϖ)ψ21ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|}+K1|T1ϱ1(ϖ)ϱ(ϖ)ϖdϖ|+1Γ(ψ1)|T1(logTϖ)ψ11ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|]
    ×(logT)γ22log(Tη2)+[(logTlogη2)γ22K2|η21φ1(ϖ)φ(ϖ)ϖdϖ|+K2|T1φ1(ϖ)φ(ϖ)ϖdϖ|+(logTlogη2)γ221Γ(ψ2)|η21(logη2ϖ)ψ21ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|
    +1Γ(ψ2)|T1(logTϖ)ψ21ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|+λ2Iδ21+(K1|η31ϱ1(ϖ)ϱ(ϖ)ϖdϖ|+1Γ(ψ2)|η31(logη3ϖ)ψ11ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|)]
    ×(logη1)γ22log(η1η2)}+K1|τ1ϱ1(ϖ)ϱ(ϖ)ϖdϖ|+1Γ(ψ2)|τ1(logτϖ)ψ11ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|,
    (logτ)γ12×1Δ{[λ1HIδ11+{(logη1logη2)γ22K2η21|φ1(ϖ)φ2(ϖ)|ϖdϖ+K2η11|φ1(ϖ)φ2(ϖ)|ϖdϖ
    +(logη1logη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21|ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ+1Γ(ψ2)η11(logη1ϖ)ψ21|ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ}+K1T1|ϱ1(ϖ)ϱ2(ϖ)|ϖdϖ
    +1Γ(ψ1)T1(logTϖ)ψ11|ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ]×(logT)γ22log(Tη2)+[(logTlogη2)γ22K2η21|φ1(ϖ)φ2(ϖ)|ϖdϖ+K2T1|φ1(ϖ)φ2(ϖ)|ϖdϖ
    +(logTlogη2)γ221Γ(ψ2)η21(logη2ϖ)ψ21|ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ+1Γ(ψ2)T1(logTϖ)ψ21|ρ2(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ2(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ+λ2Iδ21+(K1η31|ϱ1(ϖ)ϱ2(ϖ)|ϖdϖ
    +1Γ(ψ2)η31(logη3ϖ)ψ11|ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ)]×(logη1)γ22log(η1η2)}+K1τ1|ϱ1(ϖ)ϱ2(ϖ)|ϖdϖ+1Γ(ψ2)τ1(logτϖ)ψ11|ρ1(s,ϱ1(ϖ),φ1(ϖ))(ϖ)ρ1(s,ϱ2(ϖ),φ2(ϖ))(ϖ)|ϖdϖ.

    Which, by (H2), yields

    ||Ω1(ϱ1,φ1)Ω1(ϱ2,φ2)||(W1L1+W2L2)[||ϱ1ϱ2+||φ1φ2||]. (3.25)

    Similarly, we can discover that

    ||Ω2(ϱ1,φ1)Ω1(ϱ2,φ2)||(^W1L1+^W2L2)[||ϱ1ϱ2+||φ1φ2||]. (3.26)

    Consequently, it follows from (3.25) and (3.26) that

    ||Ω(ϱ1,φ1)Ω(ϱ1,φ1)||=||Ω1(ϱ1,φ1)Ω1(ϱ1,φ1)||+||Ω2(ϱ1,φ1)Ω1(ϱ1,φ1)||([W1+^W1]L1+[W2+^W2]L2)[||ϱ1ϱ2+||φ1φ2||]. (3.27)

    This, in line with condition (3.20), implies that Ω acts as a contraction. As a result, the operator Ω has a unique fixed point, following the application of the Banach fixed-point theorem. Consequently, there exists a unique solution for problems (1.1) and (1.2) over the interval [1,T].

    The sequential fractional differential system under consideration, involving the coupled Hilfer-Hadamard operators, is expressed as:

    {(HHDψ1,β11++K1HHDψ11,β11+)ϱ(τ)=ρ1(τ,ϱ(τ),φ(τ)),  1<ψ12,  τE:=[1,T],(HHDψ2,β21++K2HHDψ21,β21+)φ(τ)=ρ2(τ,ϱ(τ),φ(τ)),  2<ψ23,  τE:=[1,T], (4.1)

    supplemented with nonlocal coupled Hadamard integral boundary conditions:

    {ϱ(1)=0,ϱ(T)=λ1HIδ11+φ(η1),φ(1)=0,φ(η2)=0,φ(T)=λ2HIδ21+ϱ(η3),    1<η1,η2,η3<T. (4.2)

    Here, ψ1=54,ψ1=32,β1=12,β2=12,T=10,δ1=13,δ=34,η1=6,η2=43,η3=5,λ1=3,λ2=2,γ1=1116,γ2=1116,K1=17,K2=19,Δ=0.114465 with the given data, and it is found that W1=2.79137199,W2=1.574688,ˆW1=6.799260,ˆW2=0.91745564.

    In order to demonstrate Theorem 3.2, we use

    ρ1(τ,ϱ(τ),φ(τ))=2τ+1+|u(τ)|25(1+|ϱ(τ)|)+cosφ(τ)5τ+10,ρ2(τ,ϱ(τ),φ(τ))=e2τ+tan1ϱ(τ)30τ)+145sinφ(τ). (4.3)

    It is evident that condition (H1) is fulfilled with parameter values: κ0=3, κ1=125, κ2=115, ^κ0=1e2, ^κ1=130, and ^κ2=145. Moreover, we have

    (W1+W2)125+(^W1+^W2)1300.4318658333<1, (4.4)

    and

    (W1+W2)115+(^W1+^W2)1450.462553055<1. (4.5)

    Hence, the assumptions of Theorem 3.2 are satisfied. Consequently, the outcome of Theorem 3.2 is applicable, and therefore, problems (1.1) and (1.2), with ρ1 and ρ2 specified in (4.3), possess at least one solution over the interval [1,10].

    To demonstrate Theorem 3.3, we take into account

    ρ1(τ,ϱ(τ),φ(τ))=1τ2+4+1102τ+7(sinϱ(τ)+|φ(τ)|),ρ2(τ,ϱ(τ),φ(τ))=e2τ+15(τ+4)(tan1ϱ(τ)+cosφ(τ)). (4.6)

    Put simply, we discover that L=130 and ˆL=125, and

    (W1+^W1)130+(W2+^W2)1250.41937332<1.

    Since the conditions of Theorem 3.3 are satisfied, it can be concluded, according to its findings, that problems (1.1) and (1.2), with ρ1 and ρ2 defined in (4.6), possess unique solutions over the interval [1,10].

    We have presented criteria for the existence of solutions to a coupled system of nonlinear sequential HHFDEs with distinct orders, coupled with nonlocal HFI 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 one with boundary conditions of the following form: When λ1=λ2=0, we get

    {ϱ(1)=0,ϱ(T)=0,φ(1)=0,φ(η2)=0,φ(T)=0,    1<η1,η2<T.

    These cases represent new findings. Looking ahead, our future plans include extending this work to a tripled system of nonlinear sequential HHFDEs with varying orders and integro-multipoint boundary conditions. We also intend to investigate the multivalued analogue of the problem studied in this paper.

    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 (grant no. 5979). 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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