# Existence and H-U Stability of Solution for Coupled System of Fractional-Order with Integral Conditions Involving Caputo-Hadamard Derivatives, Hadamard Integrals

Authors: Muath Awadalla; Muthaian Subramanian; Murugesan Manigandan; Kinda Abuasbeh
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, the primary focus of our study is to investigate the existence, uniqueness, and Ulam-Hyers stability results for coupled fractional differential equations of the Caputo-Hadamard type that are supplemented with Hadamard integral boundary conditions. We employ adequate conditions to achieve existence and uniqueness results for the presented problems by utilizing the Banach contraction principle and the Leray-Schauder fixed point theorem. We also show Ulam-Hyers stability using the standard functional analysis technique. Finally, examples are used to validate the results.


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## Body

## 1. Introduction

In the past two decades, scientists and researchers have published results on fractional calculus analysis and concluded that integer-order derivatives are not always reliable. The study of turbulent fluid flows, control theory, blood flow through biological tissues, porous media, and signal and image processing, among other fields, have all benefited greatly from the use of fractional calculus. The recent study on fractional calculus, including theory and applications, can be found in [1–13]. Their research is especially pertinent since coupled systems with fractional differential equations are used to address a wide range of real-world problems. Over the past few decades, FDEs have also been the topic of substantial research in the field of stability analysis. Different types of stability, such as Mittag-Leffler and Lyapunov, have been researched in the literature. Very few researches have investigated the Ulam-Hyers stability of a linked system of FDEs. Ulam and Hyers [14, 15] identified the novel type of stability known as Ulam-Hyers stability. Understanding biological processes, fluid motion, semiconductors, population dynamics, heat conduction, and elasticity can all be helped by this kind of research. Meanwhile, the researchers have focused on the differences and results of mathematical models created by these operators and have used a variety of fractional derivation operators in their studies as a result of the diversity of fractional operators described by mathematicians. Different forms of fractional mathematical models, in which the effects of the order of fractional derivatives on the dynamic behavior of the solutions of the assumed systems are rigorously simulated, are some of the well-known works on this topic. The following is just one illustration: Caputo derivatives are used in [16, 17], Caputo-conformable derivatives in [18, 19], generalized derivatives in [20, 21], quantum Caputo derivatives in [22], nonsingular Caputo-Fabrizio derivatives in [23], and nonsingular Mittag-Leffler kernel-type derivatives in [24, 25]. The features of the Caputo and Hadamard operations are combined to define the Caputo-Hadamard fractional derivative, one of the fractional derivatives. By using this operator, very few fractional models and problems were produced. Examples can be seen in [26–29]. However, the Hadamard fractional derivative (HFD) is the most frequently used [30]. Butzer et al. [31] investigated a variety of properties of HFD, which are more general than HFDs. In [32], the authors investigated a hybrid fractional Caputo-Hadamard boundary value problem with hybrid Hadamard integral boundary value conditions. The authors in [33] studied topological degree theory and Caputo-Hadamard fractional boundary value problems. In [34], Etemad et al. investigated a fractional Caputo-Hadamard inclusion problem with sum boundary value conditions by using approximate endpoint property. In 2020, Etemad et al. [35] discussed a fractional Caputo-Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators:
(1)κCHD1+ϱut+1−κCHD1+ϖwt=αψt,wt+βHI1+μφt,wtw1=0,CHD1+δwe=0,CHD1+δw1=0,HI1+ϑwe=0,where t∈1,e, ρ,ϖ∈3,4, δ∈1,2, κ∈0,1, and μ,ϑ>0 with δ+ϑ≠0 and also α,β∈ℝ. In 2021, Rezapour et al. [36] investigated Caputo-Hadamard fractional boundary value problem via mixed multi-order integro-derivative conditions:
(2)λCHD1+ςut+CHD1+θ∗ut=A^t,ut,u1=0,μ1∗CHD1+γ1∗uM+∗CHD1+γ2∗uη=δ1,μ2∗HI1+q1∗uM+∗HI1+q2∗uη=δ2,so that λ,μ1∗,μ2∗∈0,1, γ1∗,γ2∗∈0,ς−θ∗ with 2<θ∗<ς<3, q1∗,q2∗∈ℝ+, δ1,δ2∈ℝ, and t∈1,M. Recently, in [37], the authors derived existence and uniqueness results for a nonlinear coupled system of Caputo-type FDEs equipped with new coupled boundary conditions given by
(3)CDαut=ft,ut,vt,t∈J≔0,T,T>0CDβvt=gt,ut,vt,t∈J≔0,T,u+v0=−u+vT,∫ηξu+vsds=A,0<η<ξ<T,where CD0+· denote the CFDs of order ·, α,β∈0,1, f,g:0,T×Re2⟶Re are continuous functions and A is real constant. In 2022, Belbali et.al [38] existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard fractional initial value problem using the Lyapunov method. By using main ideas of the aforementioned articles, we investigate the Caputo-Hadamard coupled system of FDEs with the Hadamard fractional integral conditions and present its existence, uniqueness, and Ulam-Hyers stability results. We study the following system:
(4)CDςPϑ=Y1ϑ,Pϑ,Uϑ,ϑϵS1,T,CDϱUϑ=Y2ϑ,Pϑ,Uϑ,ϑϵS1,T,P+U1=−P+UT,1Γρ∫1φlogφνρ−1P+Uνdνν=ϕ,1<φ<T,where Y1,Y2:S×R2⟶R are continuous functions; HIρ is the HFI of order ρ defined by
(5)HIρPϑ=1Γρ∫1ϑlogϑνρ−1Pνdνν,ρ>0,and HDς denotes HFD of order ς and is defined by
(6)HDςPϑ=1Γn−ςϑddϑn∫1ϑlogϑνn−ς−1Pνdνν,n−1<ς<n,n=ς+1,see [39, 40], where CD· denote the Caputo-Hadamard fractional derivatives (CHFDs) of order ·, 0<ς,ρ≤1, and ϕ is real constant. In this article, authors have extend the aforementioned articles [35–37] to nonlinear coupled system of Caputo-Hadamard fractional differential equations having the value of the sum of unknown functions P and U at the interval endpoints 1,T being zero, whereas the value of the sum of the unknown functions on an arbitrary domain 1,φ of the given interval 1,T remains constant. The remainder of the paper is as follows: Section 2 introduces some fundamental definitions, lemmas, and theorems that support our results. In Section 3, we prove the existence and uniqueness of solutions to the given system (4) using various conditions and some regular fixed-point theorems. Finally, examples are given to explain the main results.


## 2. Preliminaries

In this section, we discuss some relevant definitions and lemmas that will be needed later in our proof [11, 39, 40].Lemma 1.
Ifb,ς,ρ>0 then
(7)HIbςlogϑbρ−1P=ΓρΓρ+ςlogpbρ+ς−1.Definition 2.
Let0<b<c<∞, Reς≥0, n=Reς+1. The left and right CHFDs of order ς are, respectively, defined by
(8)CDb+ςPϑ=Db+ςPν−∑k=0n−1δkPbk!logνbkϑ,CDc−ςPϑ=Dc−ςhν−∑k=0n−1−1kδkPck!logcνkϑ.Definition 3.
LetReς>0, n=Reς+1 and p∈Cb,c. If Reς≠0 or ς∈ℕ, then
(9)CDb+ςIb+ςPϑ=Pϑ,CDc−ςIc−ςPϑ=Pϑ.Definition 4.
Letp∈ACδnb,c or Cδnb,c and ς∈ℂ, then
(10)Ib+ςCDb+ςPϑ=Pϑ−∑k=0n−1δkPbk!logϑbk,Ic−ςCDc−ςPϑ=Pϑ−∑k=0n−1δkPck!logcϑk.Definition 5.
LetK,L∈C1,e and P,U∈ACS. Then, the solution of the following linear coupled system:
(11)CDςPϑ=Kϑ,0<ς<1,ϑϵS,CDϱUϑ=Lϑ,0<ϱ<1,ϑϵS,P+U1=−P+Ue,1Γρ∫1φlogφνρ−1P+Uνdνν=ϕ,1<φ<T,(12)Pϑ=1Γς∫1ϑlogϑνς−1Kνdνν+12ϕη−121Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν−1η1Γς+ρ∫1φlogφνς+ρ−1Kνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Lνdνν,(13)Uϑ=1Γρ∫1ϑlogϑνρ−1Lνdνν+121η1Γς+ρ∫1φlogφνς+ρ−1Kνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Lνdνν−ϕη−121Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν,where
(14)η=logφΓρ+1≠0.Proof.
Using Lemma 2.3 and the operatorsHIς and HIρ on both sides of FDEs in (11), we obtain
(15)Pϑ=HIςKϑ+a0,(16)Uϑ=HIρLϑ+b0,where a0,b0∈Re, are arbitrary constants. Using the boundary Condition (11) in (15) and (16), we obtain
(17)a0+b0=−121Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν,(18)a0−b0=1ηφ−1Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν.Solving the system (17)-(18) for a0,b0, we get
(19)a0=12ϕη−121Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν−1η1Γς+ρ∫1φlogφνς+ρ−1Kνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Lνdνν,b0=121η1Γς+ρ∫1φlogφνς+ρ−1Kνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Lνdνν−ϕη−121Γς∫1elogeνς−1Kνdνν+1Γρ∫1elogeνρ−1Lνdνν,where η is given by (14). Substituting the values of a0 and b0 yields the result (12) and (13).


## 3. Existence Results for the Problem (4)

Defined byΩ=CS,Re×CS,Re the Banach space endowed with the norm P,U=supϑ∈SPϑ+supϑ∈SUϑ, for P,U∈Ω. In spite of Lemma 2.4, the following operator Ξ:Ω⟶Ω is associated with the problem (4):
(20)ΞP,Uϑ=Ξ1P,Uϑ,Ξ2P,Uϑ,(21)Ξ1P,Uϑ=1Γς∫1ϑlogϑνς−1Y1ν,Pν,Uνdνν+12ϕη−121Γς∫1elogeνς−1Y1ν,Pν,Uνdνν+1Γρ∫1elogeνρ−1Y2ν,Pν,Uνdνν−1η1Γς+ρ∫1φlogφνς+ρ−1Y1ν,Pν,Uνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Y2ν,Pν,Uνdνν(22)Ξ2P,Uϑ=1Γρ∫1ϑlogϑνρ−1Y2ν,Pν,Uνdνν+121η1Γς+ρ∫1φlogφνς+ρ−1Y1ν,Pν,Uνdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Y2ν,Pν,Uνdνν−ϕη−121Γς∫1elogeνς−1Y1ν,Pν,Uνdνν+1Γρ∫1elogeνρ−1Y2ν,Pν,Uνdνν.Following that, we introduce the assumptions necessary to construct the paper’s primary results.LetY1,Y2:S×Re2⟶Re functions be continuous.F1∃ continuous positive functions ωi,ω^i∈CS,Re+,i=1,2,3, such that
(23)Y1ϑ,P,U≤ω1ϑ+ω2ϑP+ω3ϑU,∀ϑ,P,U∈S×Re2,Y2ϑ,P,U≤ω^1ϑ+ω^2ϑP+ω^3ϑU,∀ϑ,P,U∈S×Re2.F2∃ positive constants ϖi,ϖ^ii=1,2 such that
(24)Y1ϑ,P1,U1−Y1ϑ,P2,U2≤ϖ1P1−P2+ϖ2U1−U2,Y2ϑ,P1,U1−Y2ϑ,P2,U2≤ϖ^1P1−P2+ϖ^2U1−U2,∀ϑ∈S,Pi,Ui∈Re,i=1,2.We use the notation: For computational ease.(25)Y1=14Γς+1+12ηlogφς+ρΓς+ρ+1,(26)Y2=14Γρ+1+12ηlogφρ+ρΓρ+ρ+1,(27)Δ=min1−2Y1+logTςΓς+1ω2+2Y2+logTρΓρ+1ω^2,(28)1−2Y1+logTςΓς+1ω3+2Y2+logTρΓρ+1ω^3.Our first existence result for the problem (4) is based on the following fixed point theorem ([41, 42]).Lemma 6.
LetG:H⟶H be a completely continuous operator in the Banach space H, and the set Π=P∈HP=λGP,0<λ<1 is bounded. Then, G has a fixed point in H.Theorem 7.
Suppose that (F1) hold. Then, the problem (4) has at least one solution on S provided that
(29)2Y1+logTςΓς+1ω2+2Y2+logTρΓρ+1ω^2<1,(30)2Y1+logTςΓς+1ω3+2Y2+logTρΓρ+1ω^3<1.where Yjj=1,2 are defined by (25)-(26).Proof.
We begin by demonstrating that the operatorΞ:Ω⟶Ω defined by (20) is completely continuous, i.e., that Ξ is continuous and maps any bounded subset of Ω to a relatively compact subset of Ω. Since the functions Y1 and Y2 are continuous, the operator Ξ:Ω⟶Ω is also continuous. Now, let Ψr^⊂Ω be bounded. Then, ∃ positive constants TY1 and TY2 such that
(31)Y1ϑ,Pϑ,Uϑ≤TY1,Y1ϑ,Pϑ,Uϑ≤TY2,∀P,U∈Ψr^.
So, for anyP,U∈Ψr^, ϑ∈S, we get
(32)Ξ1P,Uϑ≤TY1logTςΓς+1+Y1+Y2TY2+ϕη,Ξ2P,Uϑ≤Y1TY1+TY2logTρΓρ+1+Y2+ϕη.
Thus,(33)ΞP,U=Ξ1P,U+Ξ2P,U≤logTςΓς+1+2Y1TY1+logTρΓρ+1+2Y2TY2+2ϕη.
Thus, the operatorΞ is uniformly bounded as a result of the preceding inequality. Let Ξ prove that it determines bounded sets into equicontinuous sets of Ω, let ϑ1,ϑ2∈S, ϑ1<ϑ2, and P,U∈Ψr^. Then,
(34)Ξ1P,Uϑ2−Ξ2P,Uϑ2≤1Γς∫1ϑ1logϑ2νς−1−logϑ1θς−1Y1ν,Pν,Uνdνν+∫ϑ1ϑ2logϑ2νς−1Y1ν,Pν,Uνdνν≤TY12logϑ2−logϑ1ς+logϑ2ς−logϑ1ςΓς+1.
Take note that in the limitϑ1⟶ϑ2, the RHS of the preceding inequalities tends to zero independently of P,U∈Ψr^. Then, The Arzela-Ascoli theorem implies that the operator Ξ:Ω⟶Ω is completely continuous. Following that, we consider the set Λ=P,U∈ΩP,U=κΞP,U,0<κ<1 and demonstrate that it is bounded. Let P,U∈Λ, then P,U=κΞP,U,0<κ<1. For any ϑ∈S, we have
(35)Pϑ=κΞ1P,Uϑ,Uϑ=κΞ2P,Uϑ.
UsingYii=1,2 given by (25)-(26), we find that
(36)Pϑ=κΞ1P,Uϑ≤ω1+ω2P+ω3UlogTςΓς+1+Y1+ω^1+ω^2P+ω^3UY2+ϕη,Uϑ=κΞ2P,Uϑ≤ω1+ω2P+ω3UY1+ω^1+ω^2P+ω^3UlogTρΓρ+1+Y2+ϕη.
In consequence, we get(37)P+U≤ω12Y1+logTςΓς+1+ω^12Y2+logTρΓρ+1+2ϕη+ω22Y1+logTςΓς+1+ω^22Y2+logTρΓρ+1P+ω32Y1+logTςΓς+1+ω^32Y2+logTρΓρ+1U,
Thus, in Conditions (29)-(30), we obtain
(38)P,U≤ω12Y1+logTς/Γς+1+ω^12Y2+logTρ/Γρ+1+2ϕ/ηΔ
T demonstrates that P,U is constrained for ϑ∈S. As a result, the set Λ is bounded. As a result, the inference of Lemma 6 applies, and the operator Ξ has at least one fixed point, corresponding to a solution of the problem (4).The existence of a unique solution to the problem (4) is demonstrated using Banach’s contraction mapping theorem in the following result.Theorem 8.
Suppose thatF2 hold. Then, the problem (4) has a unique solution on S if
(39)ϖlogTςΓς+1+2Y1+ϖ^logTρΓρ+1+2Y2<1,where ϖ=maxϖ1,ϖ2, ϖ^=maxϖ^1,ϖ^2, and Yi, i=1,2 are defined by (25)-(26).Proof.
Consider the operatorΞ:Ω⟶Ω denoted by (20) and fix
(40)π>W1logTς/Γς+1+2Y1+W2logTρ/Γρ+1+2Y21−ϖlogTς/Γς+1+2Y1+ϖ^logTρ/Γρ+1+2Y2,where W1=supϑ∈SY1ϑ,0,0 and W2=supϑ∈SY2ϑ,0,0. Then, we show that ΞBπ⊂Bπ, where Bπ=P,U∈Ω:P,U≤π. For P,U∈Bπ, we have
(41)Ξ1P,Uϑ≤1Γς∫1ϑlogϑνς−1Y1ν,Pν,Uν−Y1ν,0,0+Y1ν,0,0dνν+12ϕη−121Γς∫1elogeνς−1Y1ν,Pν,Uν−Y1ν,0,0+Y1ν,0,0dνν+1Γρ∫1elogeνρ−1Y2ν,Pν,Uν−Y2ν,0,0+Y2ν,0,0dνν−1η1Γς+ρ∫1φlogφνς+ρ−1Y1ν,Pν,Uν−Y1ν,0,0+Y1ν,0,0dνν−1Γρ+ρ∫1φlogφνρ+ρ−1Y2ν,Pν,Uν−Y2ν,0,0+Y2ν,0,0dνν≤ϖlogTςΓς+1+Y1+ϖ^Y2P+U+W1logTςΓς+1+Y1+W2Y2,which leads to
(42)Ξ1P,U≤ϖlogTςΓς+1+Y1+ϖ^Y2P+U+W1logTςΓς+1+Y1+W2Y2,when the norm for ϑ∈S. Equivalently, for P,U∈Bπ, one can obtain
(43)Ξ2P,U≤ϖY1+ϖ^logTρΓρ+1+Y2P+U+W1Y1+W2logTρΓρ+1+Y2.Therefore, for anyP,U∈Bπ, we have
(44)ΞP,U=Ξ1P,U+Ξ2P,U≤ϖlogTςΓς+1+2Y1+ϖ^logTρΓρ+1+2Y2P+U,W1logTςΓς+1+2Y1+W2logTρΓρ+1+2Y2.which demonstrates thatΞ maps Bπ into itself. To demonstrate that the operator Ξ is a contraction, let P1,U1,P2,U2∈Ω,ϑ∈S. Then, in view of F2, we obtain
(45)Ξ1P1,U1ϑ−Ξ1P1,U1ϑ≤1Γς∫1ϑlogϑνς−1Y1ν,P1ν,U1ν−Y1ν,P2ν,U2νdνν+12ϕη−121Γς∫1elogeνς−1Y1ν,P1ν,U1ν−Y1ν,P2ν,U2νdνν+1Γρ∫1elogeνρ−1Y2ν,P1ν,U1ν−Y2ν,P2ν,U2νdνν+1η1Γς+ρ∫1φlogφνς+ρ−1Y1ν,P1ν,U1ν−Y1ν,P2ν,U2νdνν+1Γρ+ρ∫1φlogφνρ+ρ−1Y2ν,P1ν,U1ν−Y2ν,P2ν,U2νdνν≤ϖlogTςΓς+1+Y1+ϖ^Y2P+U,Ξ2P1,U1ϑ−Ξ2P1,U1ϑ≤1Γρ∫1ϑlogϑνρ−1Y2ν,P1ν,U1ν−Y2ν,P2ν,U2νdνν+121η1Γς+ρ∫1φlogφνς+ρ−1Y1ν,P1ν,U1ν−Y1ν,P2ν,U2νdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Y2ν,P1ν,U1ν−Y2ν,P2ν,U2νdνν−ϕη−121Γς∫1elogeνς−1Y1ν,P1ν,U1ν−Y1ν,P2ν,U2νdνν+1Γρ∫1elogeνρ−1Y2ν,P1ν,U1ν−Y2ν,P2ν,U2νdνν≤ϖY1+ϖ^logTρΓρ+1+Y2P+U.Clearly, the preceding inequalities imply that(46)ΞP1,U1−ΞP2,U2=Ξ1P1,U1−Ξ1P2,U2+Ξ2P1,U1−Ξ2P2,U2≤ϖlogTςΓς+1+2Y1+ϖ^logTρΓρ+1+2Y2P1−P2,U1−U2,which, in view of (40) means that Ξ is a contraction mapping. As a result of Banach’s contraction mapping theorem, Π has a unique fixed point. This demonstrates that the problem (4) has a unique solution on S.


## 4. Ulam-Hyers Stability Results (4)

The U-H stability of the solutions to the BVPs (4) will be discussed in this section using the integral representation of their solutions defined by
(47)Pϑ=Ξ1P,Uϑ,Uϑ=Ξ2P,Uϑ,where ϑ1 and ϑ2 are given by (21) and (22). Consider the following definitions of nonlinear operators
(48)Q1,Q2∈CS,Re×CS,Re⟶CS,Re,CDςPϑ−Y1ϑ,Pϑ,Uϑ=Q2P,Uϑ,ϑϵS,CDϱPϑ−Y2ϑ,Pϑ,Uϑ=Q2P,Uϑ,ϑϵS.It considered the following inequalities for someλ^1,λ^2:(49)Q1P,U≤λ^1,Q2P,U≤λ^2.Definition 9.
The coupled system (4) is said to be U-H stable if V1,V2>0, and there exists a unique solution P,U∈CE,Re of a problem (4) with
(50)P,U−P∗,U∗≤V1λ^1+V2λ^2,
∀P,U∈CS,Re of inequality (49).Theorem 10.
Assume thatF2 holds. Then, the problem (4) is U-H stable.Proof.
LetCS,Re×CS,Re be the solution to (4) that satisfies (21) and (22). Let P,U be any solution that meets Condition (49):
(51)CDςPϑ−Y1ϑ,Pϑ,Uϑ+Q1P,Uϑ,ϑϵS,CDϱPϑ−Y2ϑ,Pϑ,Uϑ+Q2P,Uϑ,ϑϵS,so,
(52)P∗ϑ=Ξ1P∗,U∗ϑ+1Γς∫1ϑlogϑνς−1Q1P∗,U∗νdνν+12ϕη−121Γς∫1elogeνς−1Q1P∗,U∗νdνν+1Γρ∫1elogeνρ−1Q2P∗,U∗νdνν−1η1Γς+ρ∫1φlogφνς+ρ−1Q1P∗,U∗νdνν−1Γρ+ρ∫1φlogφνρ+ρ−1Q2P∗,U∗νdνν.
It follows that(53)Ξ1P∗,U∗ϑ−P∗ϑ≤1Γς∫1ϑlogϑνς−1λ^1dνν+12ϕη−121Γς∫1elogeνς−1λ^1dνν+1Γρ∫1elogeνρ−1λ^2dνν+1η1Γς+ρ∫1φlogφνς+ρ−1λ^1dνν+1Γρ+ρ∫1φlogφνρ+ρ−1λ^2dνν≤14Γς+1+12ηlogφς+ρΓς+ρ+1λ^1+14Γρ+1+12ηlogφρ+ρΓρ+ρ+1λ^2≤Y1λ^1+Y2λ^2.Ξ2P∗,U∗ϑ−U∗ϑ≤1Γρ∫1ϑlogϑνρ−1λ^2dνν+121η1Γς+ρ∫1φlogφνς+ρ−1λ^1dνν+1Γρ+ρ∫1φlogφνρ+ρ−1λ^2dνν+ϕη−121Γς∫1elogeνς−1λ^1dνν+1Γρ∫1elogeνρ−1λ^2dνν,≤14Γς+1+12ηlogφς+ρΓς+ρ+1λ^1+14Γρ+1+12ηlogφρ+ρΓρ+ρ+1λ^2≤Y1λ^1+Y2λ^2.where Y1 and Y2 are defined in (25)-(26), respectively. As an outcome, we deduce from operator Ξ’s fixed-point property, which is defined by (21) and (22)
(54)Pϑ−P∗ϑ=Pϑ−Ξ1P∗,U∗ϑ+Ξ1P∗,U∗ϑ−P∗ϑ≤Ξ1P,Uϑ−Ξ1P∗,U∗ϑ+Ξ1P∗,U∗ϑ−P∗ϑ≤Y1ϕ1+Y1ϕ^1+Y1ϕ2+Y1ϕ^2P,U−P∗,U∗+Y1λ^1+Y1λ^2,(55)Uϑ−U∗ϑ=Uϑ−Ξ2P∗,U∗ϑ+Ξ2P∗,U∗ϑ−U∗ϑ≤Ξ2P,Uϑ−Ξ2P∗,U∗ϑ+Ξ2P∗,U∗ϑ−U∗ϑ≤Y2ϕ1+Y2ϕ^1+Y2ϕ2+Y2ϕ^2P,U−P∗,U∗+Y2λ^1+Y2λ^2.
From the above Equations (54) and (55), it follows that
(56)P,U−P∗,U∗≤Y1+Y2λ^1+Y1+Y2λ^21−Y1+Y2ϕ1+ϕ2+Y1+Y2ϕ^1+ϕ^2≤V1λ^1+V2λ^2,with
(57)V1=Y1+Y21−Y1+Y2ϕ1+ϕ2+Y1+Y2ϕ^1+ϕ^2V2=Y1+Y21−Y1+Y2ϕ1+ϕ2+Y1+Y2ϕ^1+ϕ^2.
Hence, the problem (4) is U-H stable.


## 5. Examples

Example 11.
Consider the following problem:(58)CD2150Pϑ=Y1ϑ,Pϑ,Uϑ,ϑϵ1,2,CD1625Uϑ=Y2ϑ,Pϑ,Uϑ,ϑϵ1,2,P+U1=−P+U2,HI925P+U7950=4,where ς=21/50, ς=16/25, ρ=9/25, φ=79/50, ϕ=4, T=2, Y1ϑ,P,U, and Y2ϑ,P,U are set later. Using the provided data, we determine that Y1=0.852825 and Y2=0.72329, where Y1 and Y2 are denoted by (25) and (26), respectively. We will use
(59)Y1ϑ,P,U=1logϑ+52ϑ+2cosp+q4andY2ϑ,P,U=14logϑ2+400p4+2sinq+ϑ,to illustrate Theorem 7. Y1 and Y2 are continuous and fulfill the condition F1 with ω1ϑ=ϑ/logϑ+52, ω2ϑ=2/logϑ+52, ω3ϑ=1/4logϑ+52, ω^1ϑ=ϑ/4logϑ2+400, ω^2ϑ=1/16logϑ2+400, and ω^3ϑ=1/2logϑ2+400. Also 2Y1+logTς/Γς+1ω2+2Y2+logTρ/Γρ+1ω^2≈0.9527249493 and 2Y1+logTς/Γς+1ω3+2Y2+logTρ/Γρ+1ω^3≈0.9042700804. As a result, all of Theorem 10 conditions hold, and there exists at least one solution to the problem (60) involving the equations Y1ϑ,P,U and Y2ϑ,P,U specified in (59).Example 12.
Consider the following problem:(60)CD2150Pϑ=Y1ϑ,Pϑ,Uϑ,ϑϵ1,2,CD1625Uϑ=Y2ϑ,Pϑ,Uϑ,ϑϵ1,2,P+U1=−P+U2,HI925P+U7950=4,where ς=21/50, ρ=16/25, ρ=9/25, φ=79/50, ϕ=4, T=2, Y1ϑ,P,U, and Y2ϑ,P,U are set later. Using the provided data, we determine that Y1=0.852825 and Y2=0.72329, where Y1 and Y2 are denoted by (25) and (26), respectively. We chose
(61)Y1ϑ,P,U=130logϑ+22tan−1u+U1+U,(62)Y2ϑ,P,U=1logϑ2+4002cosP+tan−1U,to illustrate the implementation of Theorem 8. Remember the fact that Y1 and Y2 are continuous and satisfy the condition F2 with ϖ1=ϖ2=1/120=ϖ and ϖ^1=ϖ^2=1/20=ϖ^. Also ϖlogTς/Γς+1+2Y1+ϖ^logTρ/Γρ+1+2Y2≈0.9246609812. As a result, all of Theorem 3.2 conditions are met, and its inference applies to the problem (60) with Y1ϑ,P,U and Y2ϑ,P,U given by (61).Example 13.
Consider the following problem:(63)CD2150Pϑ=ϑ2+15ϑ+25Pϑ1+Pϑ+380cosUϑ,ϑϵ1,2,CD1625Uϑ=ϑ5+17300cosPϑ+170Uϑ1+Uϑ,ϑϵ1,2,P+U1=−P+U2,HI925P+U7950=4,We choose(64)fϑ,P1ϑ,U1ϑ−fϑ,P2ϑ,U2ϑ=1125P1ϑ−P2ϑ+380U1ϑ−U2ϑ,(65)gϑ,P1ϑ,U1ϑ−gϑ,P2ϑ,U2ϑ=17300P1ϑ−P2ϑ+170U1ϑ−U2ϑ.Withϕ1=1/125,ϕ2=3/80,ϕ^1=17/300, and ϕ^2=1/70, the functions P and U clearly satisfy the F2 condition. Next, we find that Y1=0.852825 and Y2=0.72329, where Y1 and Y2 are, respectively, given by (25) and (26), based on the data available. Thus, Y1+Y2ϕ1+ϕ2+Y1+Y2ϕ^1+ϕ^2≊0.183542<1, all the conditions of Theorem 10 are satisfied, and there is a unique solution for problem (3) on 0,1, which is stable for Ulam-Hyers, with P and U given by (4) and (5), respectively.Example 14.
Consider the following problem:(66)CD1945Pϑ=Y1ϑ,Pϑ,Uϑ,ϑϵ1,2,CD1320Uϑ=Y2ϑ,Pϑ,Uϑ,ϑϵ1,2,P+U1=−P+U2,HI2350P+U85=5,where ς=19/45, ρ=13/20, ρ=23/50, φ=8/5, ϕ=5, T=2, Y1ϑ,P,U, and Y2ϑ,P,U are set later. Using the provided data, we determine that Y1=0.5870324536104914 and Y2=0.492344074388438, where Y1 and Y2 are denoted by (25) and (26), respectively. We will use
(67)Y1ϑ,P,U=ϑ+sinp+q/34+logϑ2,Y2ϑ,P,U=cosq+p/2+ϑ5625+logϑ2,to illustrate Theorem 7. Y1 and Y2 are continuous and fulfill the condition F1 with ω1ϑ=ϑ/4+logϑ2, ω2ϑ=1/4+logϑ2, ω3ϑ=1/34+logϑ2, ω^1ϑ=ϑ/5625+logϑ2, ω^2ϑ=1/25625+logϑ2, and ω^3ϑ=1/5625+logϑ2. Also 2Y1+logTς/Γς+1ω3+2Y2+logTρ/Γρ+1ω^2≈0.13130829302659 and 2Y1+logTς/Γς+1ω3+2Y2+logTρ/Γρ+1ω^3≈0.05486795746034246. As a result, all of Theorem 3.1 conditions hold, and there exists at least one solution to the problem (66) involving the equations Y1ϑ,P,U and Y2ϑ,P,U specified in (67).


## 6. Conclusion

This article established and discussed the existence, uniqueness, and Ulam-Hyers stability of solutions for a coupled system of fractional-order nonlinear Caputo-Hadamard fractional differential equations with boundary conditions involving Hadamard fractional integrals. In addition, three examples are provided to demonstrate the applicability of the acquired results. This paper’s approach is novel and adds to the field of theory on boundary value problems for nonlinear fractional differential equations. Future research can expand the given fractional boundary value problem to include more complex structures, such as the boundary value requirements for finitely point multi-strip integrals provided by recently developed generalized fractional operators with non-singular kernels.



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