Research Article | Open Access
Xinran Du, Yuan Meng, Huihui Pang, "Iterative Positive Solutions to a Coupled Hadamard-Type Fractional Differential System on Infinite Domain with the Multistrip and Multipoint Mixed Boundary Conditions", Journal of Function Spaces, vol. 2020, Article ID 6508075, 16 pages, 2020. https://doi.org/10.1155/2020/6508075
Iterative Positive Solutions to a Coupled Hadamard-Type Fractional Differential System on Infinite Domain with the Multistrip and Multipoint Mixed Boundary Conditions
This paper is devoted to the existence of positive solutions for a nonlinear coupled Hadamard fractional differential system, with multistrip and multipoint mixed boundary conditions on an infinite interval. Based on the Arzelá-Ascoli theorem, we establish an important lemma to prove the complete continuity of operators on the infinite interval. Using the monotone iterative technique, the existence criteria for positive extremal solutions can be acquired, and an example is given to illustrate the feasibility of the above study as well.
Fractional calculus, which is organically united with integral calculus, extends the concept of classical integral calculus to the whole real number line and even the complex plane. In latest researches, it has been fully shown that fractional calculus definitely has some excellent properties. This kind of calculus is nonlocal in nature, which can accurately describe several materials and processes with traits of heredity and memory [1, 2]. There are numerous applications in a variety of disciplinary fields such as characterization of anomalous diffusion , random processes , viscoelasticity , non-Newtonian fluid mechanics , and biomathematics . Moreover, differential equations can succinctly establish the relationship between variables and their derivatives. Therefore, the study of differential equations with fractional calculus is of significance theoretically and practically. In fact, fractional differential equations have been widely focused on and studied in depth. For the recent development of the topic, many researchers studied the existence [8–10], uniqueness [11, 12], and multiplicity of the solutions to fractional differential problems [13, 14].
Meanwhile, it is observed that there have been many studies based on Riemann-Liouville- and Caputo-type fractional differential equations. However, there is another kind of fractional differential equation, which is based on the Hadamard-type fractional calculus definition, found in the literature due to Hadamard (see ). It is worth mentioning that Hadamard’s construction of fractional integrodifferential, including a logarithmic function of arbitrary exponent, is a fractional power of the type . Therefore, it is well suited to the case of the half-axis and is invariant relative to dilation. As a result, the Hadamard fractional definition will be a strong tool to expand the interval further. From a finite interval to an infinite semi-infinite interval, this is theoretically a high generalization of many existing models, more general and innovative. This model has already been used in applied mathematics and physics. A detailed description about the boundary value problems of the Hadamard fractional derivative and integral on infinite intervals can be seen in [16–20].
In , Thiramanus et al. considered the positive solution for Hadamard fractional differential equations on infinite intervals by applying the idea of Leggett-Williams and Guo-Krasnoselskii fixed point theorems where denotes the Hadamard fractional derivative of order , and is the Hadamard fractional integral of order , and are the given constants.
In , Pei et al. not only established the existence of positive solutions but also sought the positive minimal and maximal solutions and got two explicit monotone iterative sequences which converge to the extremal solutions where denotes Hadamard fractional derivative, and is the Hadamard fractional integral. are the given constants and satisfy .
Furthermore, the system of fractional differential equations boundary value problems has also received much attention. More and more researchers invest effort in this direction; see [20, 21] and the references therein.
More recently in , Yang investigated the extremal iterative solutions for a coupled system of nonlinear Hadamard fractional differential equations with Cauchy initial value conditions by using the comparison principle and the monotone iterative technique combined with the method of upper and lower solutions where , and are the left-sided Hadamard fractional derivative and integral of order , respectively.
In , Ahmad and Ntouyas proved the existence and uniqueness of positive solutions of a class of boundary value problems of fractional differential equations by utilizing the Leray-Schauder alternative principle and the Banach contraction mapping principle where and , respectively, denote the Hadamard fractional derivative and Hadamard fractional integral, and are given continuous functions.
Inspired by the aforementioned work, to get more extensive results, we investigate the existence of iterative positive solutions to the following Hadamard fractional differential systems subject to the coupled fractional-order integral and discrete mixed boundary conditions where denotes the Hadamard fractional derivative of order and is the Hadamard fractional integral of order , for , and , for .
The main aim of this paper is to investigate the coupled nonlinear fractional differential system of the Hadamard type on infinite intervals subject to the coupled multistrip and multipoint mixed boundary conditions. This kind of condition is a linear combination of values at the multiple band integrals and the different discrete points, which highly summarizes the characteristics of boundary conditions in the existing study. There are two unknown functions which influence each other in this system. And the nonlinear part, and , contain lower order derivative operators as well. Based on the above model, the present research results are not abundant or even almost blank; therefore, we fill the gap in this paper.
The proofs of our main result are derived by using the monotone iterative method, which gets two significant benefits. Firstly, by selecting the appropriate initial function vector of concise form, the existence of solution can be guaranteed with the effective and succinct process. Moreover, we can seek the approximate positive solution under a different level of precision, which has more practical application value. Through comprehensive consideration, we construct the initial iterative function vector which satisfies the multiconstraints from the cone and the monotonicity of the complete continuous operator and its respective derivatives. More researches on the details of the method can be found in these references [22–27].
This article is structured as follows: Section 2 contains numerous relative definitions and lemmas. Also, there are some main results and their proofs. In Section 3, we present the existence results of monotone iterative positive solutions. Besides, the main results are illustrated by a practical example.
In this section, we will present here the definitions, some lemmas from the theory of fractional calculus, and some auxiliary results for the proof of our main results.
Definition 1 . If , then the Hadamard fractional integral of order for a function is defined as provided the integral exists. Such a space, which we denote by , consists of those complex-valued Lebesgue measurable functions on for which , with In particular, when the space coincides with the -space:
Definition 2 . The Hadamard derivative of fractional order for a function is defined as where denotes the integer part of the real number and .
Lemma 3 . If , then
Lemma 4 . Let and . Then, the Hadamard fractional differential equation has the solutions and the following formula holds: where , and where , and .
For convenience, we denote
In the forthcoming analysis, we always need the following assumptions: (F1)(F2)(F3), where are defined by (15)(F4) is continuous
Lemma 5. Assume that (F1)-(F3) hold. For with , the fractional differential system with boundary condition (6) has an integral representation where and for ,
Then, we can get
This completes the proof of the lemma.
Proof. (1)For , we haveFor , we have (2)According to (33), we can easily getThis completes the proof of the lemma.
For convenience, we denote
Proof. (1)According to (F3), Lemma 6 and the definition of , we obtainwhere is defined by (37). Similarly, we get
where is defined by (38).
(2)According to (F3), Lemma 6, and the definition of , we infer thatwhere is defined by (39). Analogously, we get
where is defined by (40).
This completes the proof of the lemma.
Let , and , and is endowed with the norm where .