The existence of solutions of Hadamard fractional di ff erential equations with integral and discrete boundary conditions on infinite interval

: In this article, the properties of solutions of Hadamard fractional di ff erential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids.


Introduction
The two most important aspects for studying differential equations are fitting and solving.First, for a practical problem, we want to determine whether we can fit a suitable equation that can realistically portray the practical problem.Second, for a practical problem, we want to determine whether we can find suitable solutions to verify the reasonableness of the equation, thus reflecting the practicability of the practical problem.
Fractional differential equations (FDEs) are popular in the fields of physics, engineering, and biology because they can well characterize complex processes such as heritability and memory properties; see the related literature [1][2][3][4][5].In order to better fit the constructed equations, we can add corresponding boundary-value problems (BVPs) to the equations, such as the integral BVP and the multipoint BVP; we can also regard the problem as a system of equations by means of coupling, or we can add the semilinear Laplace operators.In these ways, different practical problems can be better transformed into equations [6][7][8].After we obtain the equation, we need to verify the practical significance of the solutions.And, there are a number of ways to determine the properties of solutions, such as the monotonic iterative methods [9,10], and others.More detailed studies are as follows.
Hadamard [11] established the concept of fractional derivatives in 1892.Hadamard derivatives differ significantly from Riemann-Liouville and Caputo derivatives in terms of fractional powers; specifically, the kernel of the integral contains the logarithmic function of an arbitrary exponent.Hadamard derivatives have stable characterizations in terms of expansion and well matched the problems on the half-open interval; see [4].At the same time, the Hadamard FDE has an important role in the mechanical behavior of viscoelastic materials and turbulence phenomena in fluid dynamics; see [12,13].Since the problem we discuss in this paper is restricted to half-open intervals, we consider the use of Hadamard fractional derivatives.
Integral boundary conditions are instrumental in computational fluid dynamics studies related to blood flow problems.When dealing with these problems, the usual approach is to assume that the cross-section of the blood vessels is circular, which is not always reasonable.In order to optimize this detailed problem and make the results more detailed and convincing, the integral boundary conditions can be included to develop an efficient and applicable method.More details can be found in [14].In addition, integral boundary conditions have other uses in physics and biology; see [15].
The authors of [16] found that the study of fractional BVPs for m-points on infinite intervals is almost non-existent, so they studied the related problem by referencing [17] for the first time.The authors of [17] found that the intrinsic equations of viscoelastic fluids in the models of physics and biology are closely connected with the FDEs; see [18].Therefore, multipoint boundary problems are beginning to be studied.
In recent years, Hao et al. [19] considered a Hadamard FDE with integral boundary conditions.By applying Schauder's fixed-point theorem and Banach's contraction principle, they obtained the unique solution of the equation.Li et al. [20] considered the two integral boundary conditions of the Riemann-Liouville FDE.By employing Krasnoselskii's fixed-point theorem and Banach's contraction principle, they obtained the existence of the solution.
Zhang and Liu [21] applied Banach's contraction mapping principle, the monotone iterative method, and the Avery-Peterson fixed-point theorem to show the existence, uniqueness, and multiplicity results of solutions: where H D α 1+ is the Hadamard-type fractional derivative of order α; In [22], the authors investigated the fractional BVP with By employing the Bai-Ge fixed-point theorem, they obtained the solutions of the Hadamard-type FDE.
By considering the above two boundary conditions, research questions and the need for practical problem-solving, this article investigates the existence of positive solutions to the following Hadamardtype FDE with an integral and multipoint discrete BVP: where H D θ 1+ is the Hadamard-type fractional derivative of order θ, In previous works, the fixed-point theorem is usually used to determine the existence of the solutions of the equations, while the innovation of this paper is the use of the monotone iterative method to solve Hadamard FDEs containing multiple lower-order derivative terms, which results in not only obtaining the existence of the two positive solutions, but also deriving the error estimation formula for the unique positive solution.This paper enriches the use of monotone iterative methods and has potential application to the development of blood flow modeling and properties of viscoelastic fluids for practical applications.

Preliminaries and lemmas
In this section, it is essential to present some important lemmas.
and the following formula holds: where Then, the solution of the Hadamard-type FDE given by can be expressed as where ) ) Proof.Because of Lemma 2, (2.3) has a solution: (2.8) From x(1) = x ′ (1) = 0, we know that c 2 = c 3 = 0. From Lemma 1, we have .

Considering the boundary condition
.
Consequently, substituting c 1 , c 2 , and c 3 into (2.8),we have Next, after piecing together and organizing the first and third terms of the above equation, we get with the help of (2.2) and some arrangement, we obtain Thus, Lemma 4. The function K(t, s) defined in (2.5) satisfies the following conditions: We obviously get that conditions 1) and 2) above.Next, we show that condition 3) holds.For all (t, s) ∈ [1, +∞) × [1, +∞), we deduce the following: The proof is completed.

Compactness of the operator
Next, let .
Lemma 6.Let U ⊂ F be a bounded set.Then, U is relatively compact in F if the following conditions hold: 1) For all x(t) ∈ U, x(t) 1+log t , and H D θ−1 1+ x(t) are equicontinuous on any compact interval of [1, +∞).
2) For all ε > 0, there is a constant T = T (ε) > 1, which satifies Proof.This proof has been proved by Lemma 4 in [19], so it is omitted here.Define a cone P : Next, we give two circumstances: . ., n.According to Lemma 3, let T : P → F be an operator with Thus, we can get the solution of (1.1) to be the fixed point of the operator T .
Lemma 7. Assuming that (H 1 ) and (H 2 ) are valid, then T : P → F is a continuous operator.
Proof.T x(t) ≥ 0 is obvious for all t ∈ [1, +∞) and x ∈ P.Then, we have the evidence that T is continuous: Thus, we know that T : P → F. Next, we give the proof that T : P → F is a continuous operator.Let Next, we aim to prove that ∥T x n − T x 0 ∥ → 0 as n → ∞.By (H 1 ), (H 2 ), the Lebesgue dominated convergence theorem, and continuity of f , we can get and sup Hence, T is continuous.
Proof.We will show this lemma in the following three procedures.First, let Ω be a bounded subset of cone P to prove the truth that T Ω is a bounded set of cone P.There is a constant k 1 > 0, which guarantees that ∥x∥ ≤ k 1 for all x ∈ Ω.
Hence, T Ω is bounded in P.
Second, we prove that condition 1) in Lemma 6 is established.For all t 1 , t 2 ∈ [L 1 , L 2 ] with t 1 < t 2 and x ∈ Ω mentioned above, we get Because K(t,s) 1+(log t) θ−1 , (log t) θ−1 1+(log t) θ−1 , and log t 1+log t are uniformly continuous on any compact set , and [L 1 , L 2 ], respectively, we have Furthermore, in view of the function does not depend on t.Hence, we know that condition 1) in Lemma 6 is established.Finally, we prove that condition 2) in Lemma 6 is established.For any ε > 0, since 0 < +∞ 1 r(s) ds s < +∞, we can get that there exists

Since lim
for all ε > 0, there exist constants M 4 > 0 and M 5 > M 1 such that, for all t 1 , t 2 > M 4 , Let M > max{M 2 , M 3 , M 4 , M 5 }; for all t 1 , t 2 > M, we deduce the following result by substituting (2.5)-(2.7)and applying Lemma 4: Next, we substitute the specific form of K ⋆ (t, s) and apply the properties of K ⋆ (t, s).We have In the same way, considering the the properties of K ⋆ (t, s), we obtain Hence, we know that condition 2) in Lemma 6 is established.By Lemma 6, T is a compact operator.
The proof is completed.
In consequence, according to Lemmas 7 and 8, the operator T is completely continuous.

Existence of solutions
where Theorem 1. ( [24]) Assume that (H 1 )-(H 4 ) hold; there are two positive solutions x * , y * of (1.1), where ∥x * ∥, ∥y * ∥ ∈ (0, ζ].Actually, the solutions can be established by applying the sequences {x n } ∞ n=1 and {y n } ∞ n=1 , which satisfy where with the initial values x 0 (t) = ζ(log t) Hence, Suppose that the following holds: Then, we show that By (H 4 ), we can get In the same way, according to (H 4 ), (2.9) and (2.10), we get ), we derive the following: Due to the complete continuity of T and the existence of x * in P ζ , we obtain that x n k → x * as n → ∞.So, we can get that there is a convergent subsequence {x n k } ∞ k=1 of {x n } ∞ n=1 .This demonstrates that lim n→∞ x n = x * .Given that x n+1 = T x n and T is continuous, we prove that T x * = x * , which demonstrates that x * is a fixed point of T .
Since y 0 (t) = 0 and ∥y 0 ∥ ≤ Because of the properties of T , there exists a y * ∈ P ζ , which satifies that y n k → y * as n → ∞.So, we can get that there is a convergent subsequence {y n k } ∞ k=1 of {y n } ∞ n=1 .This demonstrates that lim n→∞ y n = y * .Given that y n+1 = T y n and T is continuous, we have that T y * = y * , which implies that y * is a fixed point of T .By induction, we obtain According to (3.7), (3.10), and (3.13), we have This, together with x * = lim n→∞ x n and y * = lim n→∞ y n , yields that Because f (t, 0, 0, 0) 0, it implies that x = 0 is not a solution of (1.1).Therefore, x * and y * are the solutions of (1.1).From the above steps, the conditions (H 1 )-(H 5 ) hold; according to Theorem 1, the BVP (1.1) has twin positive solutions x * and y * such that ∥x * ∥, ∥y * ∥ ∈ (0, 8].

Conclusions
In this article, we first transformed the solutions of the equation into fixed points of the operator by means of a nonlinear alternative, while proving that the operator is completely continuous.Then, we applied two iterative sequences to find two positive solutions of the equation via the monotone iterative method.Finally, we derived the unique solution of the Hadamard fractional BVP.The significance of this article lies in the fact that we can use the monotone iterative method to discuss the existence and uniqueness of the positive solution to the equation containing multiple fractional-order derivative terms.

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