This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo (ABC) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers (UH), generalized UH, Ulam-Hyers-Rassias (UHR) and generalized UHR are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.
Citation: Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function[J]. AIMS Mathematics, 2023, 8(10): 23635-23654. doi: 10.3934/math.20231202
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This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo (ABC) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers (UH), generalized UH, Ulam-Hyers-Rassias (UHR) and generalized UHR are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.
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++K1HHDψ1−1,β11+)ϱ(τ)=ρ1(τ,ϱ(τ),φ(τ)), 1<ψ1≤2, τ∈E:=[1,T],(HHDψ2,β21++K2HHDψ2−1,β21+)φ(τ)=ρ2(τ,ϱ(τ),φ(τ)), 2<ψ2≤3, τ∈E:=[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,K2∈R+, T>1, δ1,δ2>0, λ1,λ2∈R, 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×R→R 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, m∑i=1θix(ξi)=λHIδx(η). | (1.3) |
Here, α∈(1,2], β∈[0,1], θi,λ∈R, η,ξi∈(1,T)\ (i=1,2,...,m), HIδ is the HFI of order δ>0, and f:[1,T]×R→R 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)=m∑j=1ηjx(ξj)+n∑i=1ζiHIϕix(θi)+r∑k=1λkHDωk1x(μk). | (1.4) |
Here, α∈(1,2], β∈[0,1], ηi,ζi,λk∈R, 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]×R→R 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<α≤2, τ∈[1,T],HHDγ,δ1v(t)=g(t,u(t),v(t)), 1<γ≤2, τ∈[1,T],u(1)=0,HDς1u(T)=m∑i=1∫T1HDϱi1u(s)dHi(s)+n∑i=1∫T1HDσi1v(s)dKi(s),v(1)=0,HDϑ1v(T)=p∑i=1∫T1HDηi1u(s)dPi(s)+q∑i=1∫T1HDθi1v(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×R→R 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<α1≤2, τ∈E:=[1,T],HHDα2,β21+v(t)=ϱ2(t,u(t),v(t)), 2<α2≤3, τ∈E:=[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,λ2∈R, 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×R→R 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+λ1HHDα1−1,β11)u(t)=f(t,u(t),v(t)), t∈[1,e],(HHDα2,β21+λ2HHDα2−1,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,A2∈R+, and f,g:[1,e]×R×R→R 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)δ−1φ(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(HIn−δa+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)γ−1)(ϱ)=Γ(γ)Γ(γ+δ)(logϱa)γ+δ−1, |
(2)(HDδa+(logτa)γ−1)(ϱ)=Γ(γ)Γ(γ−δ)(logϱa)γ−δ−1. |
In particular, for γ=1, we have (HDδa+)(1)=1Γ(1−δ)(logϱa)−δ≠0,0<δ<1.
Definition 2.4. For n−1<δ<n and 0≤γ≤1, the HHFD of order δ and γ for φ∈L1(a,b) is defined as
(HHDδ,γa+)=(HIγ(n−δ)a+pnHI(n−δ)(1−γ)a+φ)(τ)=(HIγ(n−δ)a+pnHI(n−q)a+φ)(τ)=(HIγ(n−δ)a+HDqa+φ)(τ), q=δ+nγ−δγ, |
where HI(⋅)a+ and HD(⋅)a+ are given as defined by (2.1) and (2.2), respectively.
Theorem 2.5. If φ∈L1(a,b),0<a<b<∞, and (HIn−qa+φ)(τ)∈ACnp[a,b], then
HIδa+(HHDδ,γa+φ)(τ)=HIqa+(HHDqa+φ)(τ)=φ(τ)−n−1∑j=o(p(n−j−1)(HIδa+φ))(a)Γ(q−j)(logτa)q−j−1, |
where δ>0,0≤γ≤1, and q=δ+nγ−δγ,n=[δ]+1. Observe that Γ(q−j) exists for all j=1,2,⋯,n−1 and q∈[δ,n].
Lemma 2.6. Let h1,h2∈C(E,R). Then, the solution to the linear Hilfer-Hadamard coupled BVP is given by:
{(HHDψ1,β11++K1HHDψ1−1,β11+)ϱ(τ)=h1(τ), 1<ψ1≤2,(HHDψ2,β21++K2HHDψ2−1,β21+)φ(τ)=h2(τ), 2<ψ2≤3,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)γ1−2×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1h2(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1h1(ϖ)ϖdϖ](logT)γ2−2log(Tη2)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1h2(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1h1(ϖ)ϖdϖ)](logη1)γ2−2log(η1η2)}−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ1−1h1(ϖ)ϖdϖ, | (2.4) |
and
φ(τ)=(logτ)γ2−2log(τη2)×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1h2(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1h1(ϖ)ϖdϖ](λ2HIδ21+(logη3)γ1−1)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1h2(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1h1(ϖ)ϖdϖ)](logT)γ1−1}+(logτlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫τ1φ(ϖ)ϖdϖ−(logτlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ2−1h2(ϖ)ϖdϖ, | (2.5) |
where
A1=(logT)γ1−1,A2=λ1Γ(γ2−1)(logη1)δ1+γ2−2Γ(δ1+γ2−1){logη2−γ2−1δ1+γ2−1logη1},B1=−λ2Γ(γ1)Γ(δ2+γ1)(logη3)δ2+γ1−1,B2=(logT)γ2−2log(Tη2),Δ=A1B2−A2B1. | (2.6) |
Proof. From the first equation of (2.3), we have
(HHDψ1,β11++K1HHDψ1−1,β11+)ϱ(τ)=h1(τ), | (2.7) |
(HHDψ2,β21++K2HHDψ2−1,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+K1HHDψ1−1,β11+)ϱ(τ)=HIψ11+h1(τ),(HIψ21+HHDψ2,β21++HIψ21+K2HHDψ2−1,β21+)φ(τ)=HIψ11+h2(τ). | (2.9) |
Equation (2.9) can be written as follows,
ϱ(τ)=c0(logτ)γ1−1+c1(logτ)γ1−2−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)∫τ1(logτϖ)ψ1−1h1(ϖ)ϖdϖ. | (2.10) |
φ(τ)=d0(logτ)γ2−1+d1(logτ)γ2−2+d2(logτ)γ2−3−K2∫τ1φ(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ2−1h2(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)γ1−1+c1(logτ)2−γ1−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)∫τ1(logτϖ)ψ1−1h1(ϖ)ϖdϖ=0, | (2.12) |
φ(τ)=d0(logτ)γ2−1+d1(logτ)γ2−2+d2(logτ)3−γ2−K2∫τ1φ(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ2−1h2(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)γ1−1−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)∫τ1(logτϖ)ψ1−1h1(ϖ)ϖdϖ, | (2.14) |
φ(τ)=d0(logτ)γ2−1+d1(logτ)γ2−2−K2∫τ1φ(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ2−1h2(ϖ)ϖdϖ. | (2.15) |
Using the conditions φ(η2)=0 in (2.15), we get
d1=−1(logη2)γ2−2[1Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2ϖdϖ+d0(logη2)γ2−1−K2∫η21φ(ϖ)ϖdϖ], | (2.16) |
and substituting the value of d1 into (2.15), we obtain
φ(τ)=d0(logτ)γ2−2log(τη2)−(logτlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫τ1φ(ϖ)ϖdϖ−(logτlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2ϖdϖ+∫τ1(logη2ϖ)ψ2−1h2ϖdϖ. | (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)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1h2(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1h1(ϖ)ϖdϖ](logT)γ2−2log(Tη2)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1h2(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1h1(ϖ)ϖdϖ)](logη1)γ2−2log(η1η2)}, | (2.19) |
and
d0=1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1h2(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1h1(ϖ)ϖdϖ](λ2HIδ21+(logη3)γ1−1)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1h2(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1h2(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1h1(ϖ)ϖdϖ)](logT)γ1−1}, | (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×X→X×X by
Ω(ϱ,φ)(τ)=(Ω1(ϱ,φ)(τ),Ω2(ϱ,φ)(τ)), | (3.1) |
where
Ω1(ϱ,φ)(τ)=(logτ)γ1−2×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(logT)γ2−2log(Tη2)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logη1)γ2−2log(η1η2)}−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ1)∫τ1(logτϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ, | (3.2) |
and
Ω2(ϱ,φ)(τ)=(logτ)γ2−2log(τη2)×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(λ2HIδ21+(logη3)γ1−1)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫η31(logη3ϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logT)γ1−1}+(logτlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫τ1φ(ϖ)ϖdϖ−(logτlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ. |
We need the following hypotheses in what follows:
(H1) Assume that there exist real constants κi,ˆκi≥0(i=1,2) and κ0>0,ˆκ0>0 such that, for all τ∈[1,T],xi∈R,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,φi∈R,i=1,2,
|ρ1(τ,ϱ1,ϱ2)−ρ1(τ,φ1,φ2)|≤L(|ϱ1−φ1|+|ϱ2−φ2|),|ρ2(τ,ϱ1,ϱ2)−ρ2(τ,φ1,φ2)|≤ˆL(|ϱ1−φ1|+|ϱ2−φ2|). |
Furthermore, we establish the notation:
W1=logTγ1−2Δ[K1(logT)+(logT)ψ1Γ(ψ1+1)](logT)γ2−2log(Tη2)+[λ2K1(logη1)δ2Γ(δ2+2)+λ2(logη3)δ2+ψ1Γ(δ2+ψ1+1)](logη1)γ1−2log(η1η2)+K1(logT)+(logT)ψ1Γ(ψ1+1), | (3.3) |
W2=logTγ1−2Δ[λ1K2(logη1logη2)γ2−2(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ2−2(logη2)ψ2Γ(δ1+ψ2+1)+(logη2)ψ2Γ(δ1+ψ2+1)](logT)γ2−2log(Tη2)+[(logTlogη2)γ2−2K2(logη2)+K2(logT)+(logTlogη2)γ2−2(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logη1)γ1−2log(η1η2), | (3.4) |
^W1=(logT)γ2−2log(Tη2)×(1Δ)[K1(logT)+(logT)ψ1Γ(ψ1+1)]λ2Γ(γ1)(γ1+δ2)(logη3)γ1+δ2−1+[K1λ2(logη3)δ2Γ(δ2+2)+λ2(logη3)ψ1+δ2Γ(ψ1+δ2+1)](logT)γ1−1, | (3.5) |
^W2=(logT)γ2−2log(Tη2)×(1Δ)[λ1K2(logη1logη2)γ2−2(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ2−2(logη2)ψ2+δ1Γ(δ1+ψ2+1)+(logη2)ψ2+δ1Γ(δ1+ψ2+1)]λ2Γ(γ1)(γ1+δ2)(logη3)γ1+δ2−1+[(logTlogη2)γ2−2K2(logη2)+K2(logT)+(logTlogη2)γ2−2(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logT)γ1−1+(logTlogη2)γ2−2K2(logη2)+K2(logT)+(logTlogη2)γ2−2(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)={x∈D:x=kX(x) for some 0<k<1}, where X:D→D 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×X→X×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)γ1−2|×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2|K2∫η21φn(ϖ)−φ(ϖ)ϖdϖ| |
+|K2∫η11φn(ϖ)−φ(ϖ)ϖdϖ|+|(logη1logη2)γ2−2|1Γ(ψ2)|∫η21(logη2ϖ)ψ2−1ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ| |
+|1Γ(ψ2)∫η11(logη1ϖ)ψ2−1ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|} |
+K1|∫T1ϱn(ϖ)−ϱ(ϖ)ϖdϖ|+1Γ(ψ1)|∫T1(logTϖ)ψ1−1ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|]×(logT)γ2−2log(Tη2)+[(logTlogη2)γ2−2K2|∫η21φn(ϖ)−φ(ϖ)ϖdϖ|+K2|∫T1φn(ϖ)−φ(ϖ)ϖdϖ| |
+(logTlogη2)γ2−21Γ(ψ2)|∫η21(logη2ϖ)ψ2−1ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|+1Γ(ψ2)|∫T1(logTϖ)ψ2−1ρ2(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)ψ1−1ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|)]×(logη1)γ2−2log(η1η2)}+K1|∫τ1ϱn(ϖ)−ϱ(ϖ)ϖdϖ| |
+1Γ(ψ2)|∫τ1(logτϖ)ψ1−1ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ|≤(logτ)γ1−2×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21|φn(ϖ)−φ(ϖ)|ϖdϖ+K2∫η11|φn(ϖ)−φ(ϖ)|ϖdϖ+(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ |
+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ}+K1∫T1|ϱn(ϖ)−ϱ(ϖ)|ϖdϖ+1Γ(ψ1)∫T1(logTϖ)ψ1−1|ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ]×(logT)γ2−2log(Tη2) |
+[(logTlogη2)γ2−2K2∫η21|φn(ϖ)−φ(ϖ)|ϖdϖ+K2∫T1|φn(ϖ)−φ(ϖ)|ϖdϖ+(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ+1Γ(ψ2)∫T1(logTϖ)ψ2−1|ρ2(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ+λ2Iδ21+(K1∫η31|ϱn(ϖ)−ϱ(ϖ)|ϖdϖ |
+1Γ(ψ2)∫η31(logη3ϖ)ψ1−1|ρ1(s,ϱn(ϖ),φn(ϖ))(ϖ)−ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)|ϖdϖ)]×(logη1)γ2−2log(η1η2)}+K1∫τ1|ϱn(ϖ)−ϱ(ϖ)|ϖdϖ |
+1Γ(ψ2)∫τ1(logτϖ)ψ1−1|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(ϱn−φn)−Ω1(ϱ−φ)‖→0 as (ϱn,φn)→(ϱ,φ). | (3.10) |
In the same way, we obtain
‖Ω2(ϱn−φn)−Ω2(ϱ−φ)‖→0 as (ϱn,φn)→(ϱ,φ). | (3.11) |
It follows from (3.10) and (3.11) that
‖Ω(ϱn−φn)−Ω(ϱ−φ)‖→0 as (ϱn,φn)→(ϱ,φ). | (3.12) |
Hence, Ω is continuous. Let us initially establish the complete continuity of the operator Ω:X×X→X×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 M⊂X×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)γ1−2×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ]×(logT)γ2−2log(Tη2)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1ρ2(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ)](logη1)γ2−2log(η1η2)}−K1∫τ1ϱ(ϖ)ϖdϖ+1Γ(ψ2)∫τ1(logτϖ)ψ1−1ρ1(s,ϱ(ϖ),φ(ϖ))(ϖ)ϖdϖ,≤N1{logTγ1−2Δ[K1(logT)+(logT)ψ1Γ(ψ1+1)](logT)γ2−2log(Tη2)+[λ2K1(logη1)δ2Γ(δ2+2)+λ2(logη3)δ2+ψ1Γ(δ1+ψ1+1)](logη1)γ1−2log(η1η2)+K1(logT)+(logT)ψ1Γ(ψ1+1)}+N2{logTγ1−2Δ[λ1K2(logη1logη2)γ2−2(logη2)δ1Γ(δ1+2)+λ1K2(logη2)δ1Γ(δ1+2)+λ1(logη1logη2)γ2−2(logη2)ψ2Γ(δ1+ψ2+1)+(logη2)ψ2Γ(δ1+ψ2+1)](logT)γ2−2log(Tη2)+[(logTlogη2)γ2−2K2(logη2)+K2(logT)+(logTlogη2)γ2−2(logη2)ψ2Γ(ψ2+1)+(logT)ψ2Γ(ψ2+1)]×(logη1)γ1−2log(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)γ1−2−(logτ1)γ1−2×1Δ{[λ1HIδ11+{(logη1logη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫η11φ(ϖ)ϖdϖ−(logη1logη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1dϖϖ+1Γ(ψ2)∫η11(logη1ϖ)ψ2−1dϖϖ}+K1∫T1ϱ(ϖ)ϖdϖ−1Γ(ψ1)∫T1(logTϖ)ψ1−1dϖϖ](logT)γ2−2log(Tη2)−[(logTlogη2)γ2−2K2∫η21φ(ϖ)ϖdϖ−K2∫T1φ(ϖ)ϖdϖ−(logTlogη2)γ2−21Γ(ψ2)∫η21(logη2ϖ)ψ2−1dϖϖ−1Γ(ψ2)∫T1(logTϖ)ψ2−1dϖϖ+λ2Iδ21+(K1∫η31ϱ(ϖ)ϖdϖ−1Γ(ψ2)∫η31(logη3ϖ)ψ1−1dϖϖ)](logη1)γ2−2log(η1η2)}−K1∫τ1τ2ϱ(ϖ)ϖdϖ+1Γ(ψ2)∫τ11|(logτ2ϖ)ψ1−1−(logτ1ϖ)ψ1−1|dϖϖ+1Γ(ψ2)∫τ1τ2(logτ2ϖ)ψ1−1dϖϖ, →0 as τ2→τ1, | (3.16) |
independent of (\varrho, \varphi) \in \mathcal{M}. Likewise, it can be shown that |\Omega_{2}(\varrho, \varphi)(\tau_{2})- \Omega_{2}(\varrho, \varphi)(\tau_{1})|\rightarrow 0 as \tau_{2} \rightarrow \tau_{1} independent of (\varrho, \varphi) \in \mathcal{M} . Thus, the equicontinuity of \Omega_{1} and \Omega_{2} implies that the operator \Omega is equicontinuous. Hence, the operator \Omega is equicontinuous. Therefore, the operator \Omega satisfies the conditions for compactness according to Arzela-Ascoli's theorem. Lastly, we confirm the boundedness of the set: \Theta(\Omega) = \{(\varrho, \varphi) \in \mathcal{X} \times \mathcal{X}: (\varrho, \varphi) = \kappa \Omega (\varrho, \varphi); 0 \leq \kappa \leq 1\} . Let (\varrho, \varphi) \in \Theta (\Omega) . Then (\varrho, \varphi) = \kappa \Omega(\varrho, \varphi) , which implies that
\begin{align*} & \varrho(\tau) = \kappa\Omega_{1}(\varrho, \varphi)(\tau), \\& \varphi(\tau) = \kappa\Omega_{2}(\varrho, \varphi)(\tau), \end{align*} |
for any \tau \in [1, \mathfrak{T}] .
Based on the assumption (\mathcal{H}_{1}) , we obtain:
\begin{align} |\varrho(\tau)|& \leq (\log \tau)^{\gamma_{1}-2} \times \frac{1}{\Delta} \Biggl\{ \Bigg[\lambda_{1} {}^{\mathcal{H}}\mathcal{I}_{1+}^{\delta_{1}}\Biggl\{ \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{\varphi({\varpi})}{\varpi}d\varpi + \mathcal{K}_{2} \int_{1}^{\eta_{1}} \frac{\varphi({\varpi})}{\varpi}d\varpi \\& \quad + \Bigg(\frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{[\hat{\kappa_{0}} +\hat{\kappa_{1}}|\varrho| +\hat{ \kappa_{2}}|\varphi|]}{\varpi}d\varpi \\& \quad + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{1}} \Bigg( \log \frac{\eta_{1}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{[\hat{\kappa_{0}} +\hat{\kappa_{1}}|\varrho| +\hat{ \kappa_{2}}|\varphi|]}{\varpi}d\varpi\Biggr\} \\& \quad + \mathcal{K}_{1} \int_{1}^{\mathfrak{T}}\frac{\varrho({\varpi})}{\varpi}d\varpi- \frac{1}{\varGamma(\psi_{1})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{[\kappa_{0} +\kappa_{1}|\varrho| + \kappa_{2}|\varphi|]}{\varpi}d\varpi\Bigg](\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\& \quad + \Bigg[\Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{\varphi({\varpi})}{\varpi}d\varpi + \mathcal{K}_{2} \int_{1}^{\mathfrak{T}} \frac{\varphi({\varpi})}{\varpi}d\varpi \\& \quad + \Bigg(\frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{[\hat{\kappa_{0}} +\hat{\kappa_{1}}|\varrho| +\hat{ \kappa_{2}}|\varphi|]}{\varpi}d\varpi \\& \quad + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{[\hat{\kappa_{0}} +\hat{\kappa_{1}}|\varrho| +\hat{ \kappa_{2}}|\varphi|]}{\varpi}d\varpi\\& \quad + \lambda_{2} \mathcal{I}_{1+}^{\delta_{2}} \Bigg(\mathcal{K}_{1} \int_{1}^{\eta_{3}} \frac{\varrho({\varpi})}{\varpi}d\varpi + \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{3}} \Bigg( \log \frac{\eta_{3}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{[\kappa_{0} +\kappa_{1}|\varrho| + \kappa_{2}|\varphi|]}{\varpi}d\varpi\Bigg) \Bigg]\\& \quad \times(\log \eta_{1})^{\gamma_{2}-2} \log \Bigg(\frac{\eta_{1}}{\eta_{2}} \Bigg)\Biggr\} + \mathcal{K}_{1} \int_{1}^{\tau} \frac{\varrho({\varpi})}{\varpi} d\varpi \\& \quad + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\tau} \Bigg( \log \frac{\tau}{\varpi} \Bigg)^{\psi_{1}-1} \frac{[\kappa_{0} +\kappa_{1}|\varrho| + \kappa_{2}|\varphi|]}{\varpi}d\varpi\\ &\leq \mathfrak{W}_{1}{[\kappa_{0} +\kappa_{1}|\varrho| + \kappa_{2}|\varphi|]}+ \mathfrak{W}_{2}{[\hat{\kappa_{0}} +\hat{\kappa_{1}}|\varrho| +\hat{ \kappa_{2}}|\varphi|]}, \end{align} | (3.17) |
which implies that
\begin{align} ||\varrho|| = \sup\limits_{\tau \in [1, \mathfrak{T}]}|\varrho(\tau)| \leq \mathfrak{W}_{1} \kappa_{0} + \mathfrak{W}_{2} \hat{\kappa_{0}} + ( \mathfrak{W}_{1} \kappa_{1} + \mathfrak{W}_{2} \hat{\kappa_{1}})||\varrho|| + ( \mathfrak{W}_{1} \kappa_{2} + \mathfrak{W}_{2} \hat{\kappa_{2}})||\varphi||. \end{align} | (3.18) |
Similarly, one can find that
\begin{align} ||\varphi|| \leq \hat{\mathfrak{W}_{1}} \kappa_{0} + \hat{\mathfrak{W}_{2}} \hat{\kappa_{0}} + ( \hat{\mathfrak{W}_{1}} \kappa_{1} + \hat{\mathfrak{W}_{2}} \hat{\kappa_{1}})||\varrho|| + ( \hat{\mathfrak{W}_{1} }\kappa_{2} + \hat{\mathfrak{W}_{2}} \hat{\kappa_{2}})||\varphi||. \end{align} | (3.19) |
From (3.18) and (3.19), we obtain
\begin{align*} ||\varrho||+||\varphi|| \leq& (\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}})\kappa_{0} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}})\hat{\kappa_{0}} + (\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}})\kappa_{1} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}})\hat{\kappa_{1}} ||\varrho|| \\& + (\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}})\kappa_{2} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}})\hat{\kappa_{2}} ||\varphi||. \end{align*} |
Which, by ||(\varrho, \varphi)|| = ||\varrho|| + ||\varphi|| , yields
\begin{align*} ||(\varrho, \varphi)|| \leq \frac{1}{\Phi} [(\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}})\kappa_{0} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}})\hat{\kappa_{0}}]. \end{align*} |
As a result, \Theta(\Omega) is constrained within bounds. Consequently, the conclusion of Lemma 3.1 is applicable, implying that the operator \Omega 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 (\mathcal{H}_{2}) is met, and the inequality
\begin{align} (\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}}) \mathcal{L}_{1} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}}) \mathcal{L}_{2} < 1, \end{align} | (3.20) |
holds, where \mathfrak{W}{i} and \hat{\mathfrak{W}{i}} are defined in (3.3)–(3.6), then problems (1.1) and (1.2) possess unique solutions over the interval [1, \mathfrak{T}] .
Proof. Denoting \mathfrak{K}{1} = \{ \sup_{\tau \in [1, \mathfrak{T}]} |\rho_{1}(\tau, 0, 0)| < \infty\} and \mathfrak{K}{2} = \{ \sup_{\tau \in [1, \mathfrak{T}]} |\rho_{2}(\tau, 0, 0)| < \infty\} , it can be inferred from assumption ( \mathcal{H}_{1} ) that
\begin{align*} |\rho_{1}(\tau, \varrho, \varphi)|& \leq \mathcal{L}_{1}(||\varrho||+||\varphi||)+\mathfrak{K}_{1} \\&\leq \mathcal{L}_{1} ||(\varrho, \varphi)|| +\mathfrak{K}_{1}, \end{align*} |
and
\begin{align*} |\rho_{2}(\tau, \varrho, \varphi)| \leq \mathcal{L}_{2} ||(\varrho, \varphi)|| +\mathfrak{K}_{2}. \end{align*} |
First, we show that \Omega\mathcal{B}_{\mathfrak{r}} \subset \mathcal{B}_{\mathfrak{r}} , where \mathcal{B}_{\mathfrak{r}} = \{ (\varrho, \varphi) \in \mathcal{X} \times \mathcal{X} : ||(\varrho, \varphi)|| \leq \mathfrak{r}\} , with
\begin{align} \mathfrak{r} \geq \frac{(\mathfrak{W}_{1}+\hat{(\mathfrak{W}_{1})})\mathfrak{K}_{1}+(\mathfrak{W}_{2}+\hat{(\mathfrak{W}_{2})})\mathfrak{K}_{2}}{1-(\mathfrak{W}_{1}+\hat(\mathfrak{W}_{{1})})\mathcal{L}_{1}+(\mathfrak{W}_{2}+\hat{(\mathfrak{W}_{2})})\hat{\mathcal{L}}}. \end{align} | (3.21) |
For (\varrho, \varphi) \in \mathcal{B}_{\mathfrak{r}} , we have
\begin{align} ||\Omega_{1}(\varrho, \varphi)|| = & \sup\limits_{\tau \in [1, \mathfrak{T}]}|\Omega_{1}(\varrho, \varphi)(\tau)| \\ \leq & (\log \tau)^{\gamma_{1}-2} \times \frac{1}{\Delta} \Biggl\{ \Bigg[\lambda_{1} {}^{\mathcal{H}}\mathcal{I}_{1+}^{\delta_{1}}\Biggl\{ \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{\varphi({\varpi})}{\varpi}d\varpi - \mathcal{K}_{2} \int_{1}^{\eta_{1}} \frac{\varphi({\varpi})}{\varpi}d\varpi \\& - \Bigg(\frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \\& + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{1}} \Bigg( \log \frac{\eta_{1}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \Biggr\} \\& + \mathcal{K}_{1} \int_{1}^{\mathfrak{T}}\frac{\varrho({\varpi})}{\varpi}d\varpi - \frac{1}{\varGamma(\psi_{1})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \Bigg] \\& \times (\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\& - \Bigg[\Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{\varphi({\varpi})}{\varpi}d\varpi - \mathcal{K}_{2} \int_{1}^{\mathfrak{T}} \frac{\varphi({\varpi})}{\varpi}d\varpi \\& - \Bigg(\frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \\& - \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi + \lambda_{2} \mathcal{I}_{1+}^{\delta_{2}} \Bigg(\mathcal{K}_{1} \int_{1}^{\eta_{3}} \frac{\varrho({\varpi})}{\varpi}d\varpi \\& - \frac{1}{\varGamma(\psi_{2})}\int_{1}^{\eta_{3}} \Bigg( \log \frac{\eta_{3}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \Bigg) \Bigg](\log \eta_{1})^{\gamma_{2}-2} \log \Bigg(\frac{\eta_{1}}{\eta_{2}} \Bigg)\Biggr\} \\& - \mathcal{K}_{1} \int_{1}^{\tau} \frac{\varrho({\varpi})}{\varpi} d\varpi + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\tau} \Bigg( \log \frac{\tau}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \end{align} | (3.22) |
and
\begin{align*} ||\Omega(\varrho, \varphi)|| \leq &(\mathcal{L}_{1}\mathfrak{r}+\mathfrak{K}_{1}) \Biggl\{ \frac{\log \mathfrak{T}^{\gamma_{1}-2}}{\Delta} \Bigg[\mathcal{K}_{1}(\log \mathfrak{T}) + \frac{(\log \mathfrak{T})^{\psi_{1}}}{\varGamma(\psi_{1}+1)}\Bigg](\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\& + \Bigg[\lambda_{2} \mathcal{K}_{1} \frac{(\log \eta_{1})^{\delta_{2}}}{\varGamma(\delta_{2}+2)} + \lambda_{2} \frac{(\log \eta_{3})^{\delta_{2}+\psi_{1}}}{\varGamma(\delta_{1}+\psi_{1}+1)} \Bigg] (\log \eta_{1})^{\gamma_{1}-2} \log \Bigg( \frac{\eta_{1}}{\eta_{2}}\Bigg) + \mathcal{K}_{1} (\log \mathfrak{T}) \\& +\frac{(\log \mathfrak{T})^{\psi_{1}}}{\varGamma(\psi_{1}+1)} + \frac{\log \mathfrak{T}^{\gamma_{1}-2}}{\Delta}\Bigg[\lambda_{1}\mathcal{K}_{2} \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}}\Bigg)^{\gamma_{2}-2} \frac{(\log \eta_{2})^{\delta_{1}}}{\varGamma(\delta_{1}+2)} + \lambda_{1}\mathcal{K}_{2}\frac{(\log \eta_{2})^{\delta_{1}}}{\varGamma(\delta_{1}+2)} \\& + \lambda_{1} \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}}\Bigg)^{\gamma_{2}-2} \frac{(\log \eta_{2})^{\psi_{2}}}{\varGamma(\delta_{1}+ \psi_{2}+1)} + \frac{(\log \eta_{2})^{\psi_{2}}}{\varGamma(\delta_{1}+ \psi_{2}+1)} \Bigg](\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\& +\Bigg[\Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} (\log \eta_{2}) + \mathcal{K}_{2} (\log \mathfrak{T}) + \Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{(\log \eta_{2})^{\psi_{2}}}{\varGamma(\psi_{2}+1)} + \frac{(\log \mathfrak{T})^{\psi_{2}}}{\varGamma(\psi_{2}+1)} \Bigg] \\& \times (\log \eta_{1})^{\gamma_{1}-2} \log \Bigg( \frac{\eta_{1}}{\eta_{2}}\Bigg)\Biggr\}. \end{align*} |
Making use of the notation of (3.3)–(3.6), we get
\begin{align} ||\Omega_{1}(\varrho, \varphi)|| \leq (\mathcal{L}_{1}\mathfrak{W}_{1}+\mathcal{L}_{2}\mathfrak{W}_{2})\mathfrak{r} + \mathfrak{W}_{1}\mathfrak{K}_{1} + \mathfrak{W}_{2}\mathfrak{K}_{2}. \end{align} | (3.23) |
Likewise, we can find that
\begin{align} ||\Omega_{2}(\varrho, \varphi)|| \leq (\mathcal{L}_{1}\hat{\mathfrak{W}_{1}}+\mathcal{L}_{2}\hat{\mathfrak{W}_{2}})\mathfrak{r} + \mathfrak{W}_{1}\hat{\mathfrak{K}_{1}} + \hat{\mathfrak{W}_{2}}\mathfrak{K}_{2}. \end{align} | (3.24) |
Then, it follows from (3.23)–(3.24) that
\begin{align*} ||\Omega(\varrho, \varphi)||\leq ||\Omega_{1}(\varrho, \varphi)|| + ||\Omega_{2}(\varrho, \varphi)||\leq \mathfrak{r}. \end{align*} |
Therefore, \Omega\mathcal{B}_{\mathfrak{r}} \subset \mathcal{B}_{r} as (\varrho, \varphi) \in \mathcal{B}_{\mathfrak{r}} is an arbitrary element.
To confirm the contraction property of the operator \Omega , consider (\varrho_{i}, \varphi_{j}) \in \mathcal{B}_{\mathfrak{r}} for i = 1, 2 . Subsequently, we obtain
\left\| \Omega_{1}(\varrho_{1}, \varphi_{1}) - \Omega_{1}(\varrho, \varphi)\right\| \\ \leq \left| (\log \tau)^{\gamma_{1}-2} \right| \times \frac{1}{\Delta} \Biggl\{ \Bigg[\lambda_{1} {}^{\mathcal{H}}\mathcal{I}_{1+}^{\delta_{1}}\Biggl\{ \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \left| \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{\varphi_{1}({\varpi})-\varphi({\varpi})}{\varpi}d\varpi\right| |
+ \left| \mathcal{K}_{2} \int_{1}^{\eta_{1}} \frac{\varphi_{1}({\varpi})-\varphi({\varpi})}{\varpi}d\varpi\right| \\ + \left| \Bigg(\frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2}\right| \frac{1}{\varGamma(\psi_{2})} \left| \int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right| |
+ \left| \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{1}} \Bigg( \log \frac{\eta_{1}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi \right| \Biggr\} \\ + \mathcal{K}_{1} \left| \int_{1}^{\mathfrak{T}}\frac{\varrho_{1}({\varpi})-\varrho({\varpi})}{\varpi}d\varpi\right| \\ + \frac{1}{\varGamma(\psi_{1})}\left| \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right| \Bigg] |
\times (\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\ + \Bigg[\Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \left| \int_{1}^{\eta_{2}} \frac{\varphi_{1}({\varpi})-\varphi({\varpi})}{\varpi}d\varpi \right| + \mathcal{K}_{2} \left| \int_{1}^{\mathfrak{T}} \frac{\varphi_{1}({\varpi})-\varphi({\varpi})}{\varpi}d\varpi\right| \\ + \Bigg(\frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})} \left| \int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right| |
+ \frac{1}{\varGamma(\psi_{2})} \left| \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right| \\ + \lambda_{2} \mathcal{I}_{1+}^{\delta_{2}} \Bigg(\mathcal{K}_{1} \left| \int_{1}^{\eta_{3}} \frac{\varrho_{1}({\varpi})-\varrho({\varpi})}{\varpi}d\varpi\right| \\ + \frac{1}{\varGamma(\psi_{2})} \left| \int_{1}^{\eta_{3}} \Bigg( \log \frac{\eta_{3}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right| \Bigg) \Bigg] |
\times (\log \eta_{1})^{\gamma_{2}-2} \log \Bigg(\frac{\eta_{1}}{\eta_{2}} \Bigg)\Biggr\} + \mathcal{K}_{1}\left| \int_{1}^{\tau} \frac{\varrho_{1}({\varpi})-\varrho({\varpi})}{\varpi} d\varpi\right| \\ + \frac{1}{\varGamma(\psi_{2})}\left| \int_{1}^{\tau} \Bigg( \log \frac{\tau}{\varpi} \Bigg)^{\psi_{1}-1} \frac{\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho({\varpi}), \varphi({\varpi}))({\varpi})}{\varpi}d\varpi\right|, |
\leq (\log \tau)^{\gamma_{1}-2} \times \frac{1}{\Delta} \Biggl\{ \Bigg[\lambda_{1} {}^{\mathcal{H}}\mathcal{I}_{1+}^{\delta_{1}}\Biggl\{ \Bigg( \frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{|\varphi_{1}({\varpi})-\varphi_{2}({\varpi})|}{\varpi}d\varpi \\ + \mathcal{K}_{2} \int_{1}^{\eta_{1}} \frac{|\varphi_{1}({\varpi})-\varphi_{2}({\varpi})|}{\varpi}d\varpi |
+ \Bigg(\frac{\log \eta_{1}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{|\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \\ + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{1}} \Bigg( \log \frac{\eta_{1}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{|\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \Biggr\} \\ + \mathcal{K}_{1} \int_{1}^{\mathfrak{T}}\frac{|\varrho_{1}({\varpi})-\varrho_{2}({\varpi})|}{\varpi}d\varpi |
+ \frac{1}{\varGamma(\psi_{1})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{|\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \Bigg] \\ \times (\log \mathfrak{T})^{\gamma_{2}-2} \log \Bigg(\frac{\mathfrak{T}}{\eta_{2}} \Bigg) \\ + \Bigg[\Bigg( \frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \mathcal{K}_{2} \int_{1}^{\eta_{2}} \frac{|\varphi_{1}({\varpi})-\varphi_{2}({\varpi})|}{\varpi}d\varpi + \mathcal{K}_{2} \int_{1}^{\mathfrak{T}} \frac{|\varphi_{1}({\varpi})-\varphi_{2}({\varpi})|}{\varpi}d\varpi |
+ \Bigg(\frac{\log \mathfrak{T}}{\log \eta_{2}} \Bigg)^{\gamma_{2}-2} \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{2}} \Bigg( \log \frac{\eta_{2}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{|\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \\ + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\mathfrak{T}} \Bigg( \log \frac{\mathfrak{T}}{\varpi} \Bigg)^{\psi_{2}-1} \frac{|\rho_{2}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{2}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \\ + \lambda_{2} \mathcal{I}_{1+}^{\delta_{2}} \Bigg(\mathcal{K}_{1} \int_{1}^{\eta_{3}} \frac{|\varrho_{1}({\varpi})-\varrho_{2}({\varpi})|}{\varpi}d\varpi |
+ \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\eta_{3}} \Bigg( \log \frac{\eta_{3}}{\varpi} \Bigg)^{\psi_{1}-1} \frac{|\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi \Bigg) \Bigg] \\ \times (\log \eta_{1})^{\gamma_{2}-2} \log \Bigg(\frac{\eta_{1}}{\eta_{2}} \Bigg)\Biggr\} + \mathcal{K}_{1} \int_{1}^{\tau} \frac{|\varrho_{1}({\varpi})-\varrho_{2}({\varpi})|}{\varpi} d\varpi \\ + \frac{1}{\varGamma(\psi_{2})} \int_{1}^{\tau} \Bigg( \log \frac{\tau}{\varpi} \Bigg)^{\psi_{1}-1} \frac{|\rho_{1}(s, \varrho_{1}({\varpi}), \varphi_{1}({\varpi}))({\varpi})-\rho_{1}(s, \varrho_{2}({\varpi}), \varphi_{2}({\varpi}))({\varpi})|}{\varpi}d\varpi. |
Which, by ( \mathcal{H}_{2} ), yields
\begin{align} ||\Omega_{1}(\varrho_{1}, \varphi_{1})-\Omega_{1}(\varrho_{2}, \varphi_{2})|| \leq (\mathfrak{W}_{1}\mathcal{L}_{1}+\mathfrak{W}_{2}\mathcal{L}_{2})[||\varrho_{1}-\varrho_{2}+||\varphi_{1}-\varphi_{2}||]. \end{align} | (3.25) |
Similarly, we can discover that
\begin{align} ||\Omega_{2}(\varrho_{1}, \varphi_{1})-\Omega_{1}(\varrho_{2}, \varphi_{2})|| \leq (\hat{\mathfrak{W}_{1}}\mathcal{L}_{1}+\hat{\mathfrak{W}_{2}}\mathcal{L}_{2})[||\varrho_{1}-\varrho_{2}+||\varphi_{1}-\varphi_{2}||]. \end{align} | (3.26) |
Consequently, it follows from (3.25) and (3.26) that
\begin{align} ||\Omega(\varrho_{1}, \varphi_{1})-\Omega(\varrho_{1}, \varphi_{1})|| & = ||\Omega_{1}(\varrho_{1}, \varphi_{1})-\Omega_{1}(\varrho_{1}, \varphi_{1})||+||\Omega_{2}(\varrho_{1}, \varphi_{1})-\Omega_{1}(\varrho_{1}, \varphi_{1})||\\& \leq ([\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}}]\mathcal{L}_{1}+[{\mathfrak{W}_{2}}+\hat{\mathfrak{W}_{2}}]\mathcal{L}_{2})[||\varrho_{1}-\varrho_{2}+||\varphi_{1}-\varphi_{2}||]. \end{align} | (3.27) |
This, in line with condition (3.20), implies that \Omega acts as a contraction. As a result, the operator \Omega 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, \mathfrak{T}] .
The sequential fractional differential system under consideration, involving the coupled Hilfer-Hadamard operators, is expressed as:
\begin{align} \begin{cases} ( {}^{\mathcal{HH}} \mathcal{D}^{\psi_{1}, \beta_{1}}_{1^{+}} + \mathcal{K}_{1} {}^{\mathcal{HH}} \mathcal{D}^{\psi_{1}-1, \beta_{1}}_{1^{+}} ) \varrho(\tau) = \rho_{1} ( \tau, \varrho(\tau), \varphi(\tau)), \ \ 1 < \psi_{1} \leq 2, \ \ \tau \in \mathcal{E} : = [1, \mathfrak{T}], \\ ( {}^{\mathcal{HH}} \mathcal{D}^{\psi_{2}, \beta_{2}}_{1^{+}} + \mathcal{K}_{2} {}^{\mathcal{HH}} \mathcal{D}^{\psi_{2}-1, \beta_{2}}_{1^{+}} ) \varphi(\tau) = \rho_{2} ( \tau, \varrho(\tau), \varphi(\tau)), \ \ 2 < \psi_{2} \leq 3, \ \ \tau \in \mathcal{E} : = [1, \mathfrak{T}], \end{cases} \end{align} | (4.1) |
supplemented with nonlocal coupled Hadamard integral boundary conditions:
\begin{align} \begin{cases} \varrho(1) = 0, \quad \varrho(\mathfrak{T}) = \lambda_{1} {}^{\mathcal{H}} \mathcal{I}_{1^{+}}^{\delta_{1}} \varphi(\eta_{1}), \\ \varphi(1) = 0, \quad \varphi(\eta_{2}) = 0, \quad \varphi(\mathfrak{T}) = \lambda_{2} {}^{\mathcal{H}} \mathcal{I}_{1^{+}}^{\delta_{2}} \varrho(\eta_{3}), \ \ \ \ 1 < \eta_{1}, \eta_{2}, \eta_{3} < \mathfrak{T}. \end{cases} \end{align} | (4.2) |
Here, \psi_{1} = \frac{5}{4}, \psi_{1} = \frac{3}{2}, \beta_{1} = \frac{1}{2}, \beta_{2} = \frac{1}{2}, \mathfrak{T} = 10, \delta_{1} = \frac{1}{3}, \delta = \frac{3}{4}, \eta_{1} = 6, \eta_{2} = \frac{4}{3}, \eta_{3} = 5, \lambda_{1} = 3, \lambda_{2} = 2, \gamma_{1} = \frac{11}{16}, \gamma_{2} = \frac{11}{16}, \mathcal{K}_{1} = \frac{1}{7}, \mathcal{K}_{2} = \frac{1}{9}, \Delta = 0.114465 with the given data, and it is found that \mathfrak{W}_{1} = 2.79137199, \mathfrak{W}_{2} = 1.574688, \hat{\mathfrak{W}}_{1} = 6.799260, \hat{\mathfrak{W}}_{2} = 0.91745564.
In order to demonstrate Theorem 3.2, we use
\begin{align} &\rho_{1} ( \tau, \varrho(\tau), \varphi(\tau)) = \sqrt{2\tau+1}+ \frac{|\mathfrak{u}(\tau)|}{25(1+|\varrho(\tau)|)}+ \frac{\cos \varphi(\tau)}{5\tau+10}, \\& \rho_{2} ( \tau, \varrho(\tau), \varphi(\tau)) = e^{-2\tau}+ \frac{\tan^{-1}\varrho(\tau)}{30\tau)}+ \frac{1}{45}\sin \varphi(\tau). \end{align} | (4.3) |
It is evident that condition (\mathcal{H}{1}) is fulfilled with parameter values: \kappa_{0} = \sqrt{3} , \kappa_{1} = \frac{1}{25} , \kappa_{2} = \frac{1}{15} , \hat{\kappa_{0}} = \frac{1}{e^{2}} , \hat{\kappa_{1}} = \frac{1}{30} , and \hat{\kappa_{2}} = \frac{1}{45} . Moreover, we have
\begin{align} (\mathfrak{W}_{1} +\mathfrak{W}_{2})\frac{1}{25} + (\hat{\mathfrak{W}_{1}} +\hat{\mathfrak{W}_{2}})\frac{1}{30} \approx 0.4318658333 < 1, \end{align} | (4.4) |
and
\begin{align} (\mathfrak{W}_{1} +\mathfrak{W}_{2})\frac{1}{15} + (\hat{\mathfrak{W}_{1}} +\hat{\mathfrak{W}_{2}})\frac{1}{45} \approx 0.462553055 < 1. \end{align} | (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 \rho_{1} and \rho_{2} specified in (4.3), possess at least one solution over the interval [1,10].
To demonstrate Theorem 3.3, we take into account
\begin{align} &\rho_{1} ( \tau, \varrho(\tau), \varphi(\tau)) = \frac{1}{\tau^{2}+4}+\frac{1}{10\sqrt{2\tau+7}}(\sin \varrho(\tau)+|\varphi(\tau)|), \\& \rho_{2} ( \tau, \varrho(\tau), \varphi(\tau)) = e^{-2\tau}+\frac{1}{5(\tau+4)}(\tan^{-1} \varrho(\tau)+\cos\varphi(\tau)). \end{align} | (4.6) |
Put simply, we discover that \mathcal{L} = \frac{1}{30} and \hat{\mathcal{L}} = \frac{1}{25} , and
\begin{align*} (\mathfrak{W}_{1}+\hat{\mathfrak{W}_{1}}) \frac{1}{30} + (\mathfrak{W}_{2}+\hat{\mathfrak{W}_{2}})\frac{1}{25}\approx0.41937332 < 1. \end{align*} |
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 \rho_{1} and \rho_{2} 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 \lambda_{1} = \lambda_{2} = 0 , we get
\begin{align*} \begin{cases} \varrho(1) = 0, \quad \varrho(\mathfrak{T}) = 0, \\ \varphi(1) = 0, \quad \varphi(\eta_{2}) = 0, \quad \varphi(\mathfrak{T}) = 0, \ \ \ \ 1 < \eta_{1}, \eta_{2} < \mathfrak{T}. \end{cases} \end{align*} |
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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