Results of Positive Solutions for the Fractional Differential System on an Infinite Interval

*e chief topic of this paper is to investigate the fractional differential system on an infinite interval. By introducing an appropriate compactness criterion in a special function space and applying the Schauder fixed-point theorem and the Banach contraction mapping principle, we established the results for the existence and uniqueness of positive solutions. An example is then given to show the utilization of the main results.


Introduction
In this paper, we investigate the following fractional differential system on an infinite interval: where 1 ≤ n − 1 < α 1 ≤ n, 1 ≤ m − 1 < α 2 ≤ m, and n, m ≥ 2, D α i 0 + is the Riemann-Liouville derivative operator. μ i > 0 is a constant,    Numerous models in physics, chemistry, biology, medicine, and other fields have promoted the research of differential equations, for instance, evaluation of water quality on receiving water [1], and the advection-dispersion equation can be formulated as shown for the case of one-dimensional flow: where C is the concentration of a generic pollutant, t is the time, x is the longitudinal displacement, u is the velocity, D L is the diffusion coefficient, and f(C) is a generic term for reactions involving the pollutant C. Westerlund [2] established a one-dimensional model to describe the transmission of the electromagnetic wave: where μ, ε, and ζ are constants and D ] t E(x, t) � (z ] E(x, t))/zt ] is a fractional derivative. In the process of establishing the model, k-Hessian equations [3], Sobolev equations [4], and Schrödinger elliptic equations [5,6], there are also huge applications.
Under the proper initial or boundary conditions, to study the positive solution of the above models is very necessary; especially, for the boundary value problems on the infinite interval, many authors put their interest in it [7][8][9][10][11][12][13][14][15][16]. Liang and Zhang [17] applied the fixed-point theorem to obtain the existence of positive solutions for the following fractional differential equation: For all we know, there are few studies on fractional differential systems of infinite intervals, although it is necessary to do so. In this paper, we aimed at getting the existence and uniqueness of positive solutions for system (1) on infinite interval. Compared with the existing literature, the innovations of this paper are as follows. Firstly, the paper which we discuss is the system rather than a single equation. Secondly, we study the system with integral boundary value conditions on infinite intervals, which are more general than those of two-point, three-point, and multipoint boundary value condition. At last, we use two different techniques: the Schauder fixed-point theorem and the Banach contraction mapping principle, for system (1), not only to obtain the existence of positive solutions but also the uniqueness of positive solutions.
has a unique integral representation 2

Journal of Function Spaces
Proof. By Lemma 1, the equations in system (8) can be transformed into the equivalent integral equations that is, Since we have So, We also have we obtain Combining (15) and (18), we have ( 19) and (20) are multiplied by a 1 (t) and a 2 (t), respectively, and then solved the integral from 0 to +∞ with respect to A 1 (t) and A 2 (t); then, we have Journal of Function Spaces erefore, So, (9) holds. e proof is completed.

Lemma 3.
e Green function in Lemma 2 has the following properties: e space X � E 1 × E 2 will be used in the study of system (1), where

(25)
Clearly, (X, ‖·‖) is a Banach space with the norm ‖u, v‖ � ‖u‖ + ‖v‖. Define nonlinear integral operators T i : X ⟶ E i and T : X ⟶ X by

us, the existence of solutions to system (1) is equivalent to the existence of solutions in X for operator equation T(u, v) � (u, v) defined by (27).
Lemma 4 (see [20,21]

). Let E be defined as (24) and M be any bounded subset of E. en, M is relatively compact in E if
x ∈ M is equicontinuous on any finite subinterval of J, and for any given ε > 0, there exists N > 0 such that | (x(t 1 )/1 + t α− 1 1 ) − (x(t 2 )/1 + t α− 1 2 )| < ε uniformly with respect to all x ∈ M, and t 1 , t 2 > N.

Main Results
We list the conditions to be used later: (H 1 ) there exist nonnegative functions p i (t), g i (t), h i (t) ∈ L 1 [0, +∞) and where

Journal of Function Spaces
In fact, by (H 1 ), for any (u, v) ∈ X, we have (32)

Theorem 1. Assume that (H 1 ) holds; then, T : X ⟶ X is a completely continuous operator.
Proof. First, we show that T : , and there exists a constant r > 0 such that ‖(u n , v n )‖ ≤ r and ‖(u, v)‖ ≤ r. By (H 1 ) and (30), we have From (H 1 ) and (33), for any ε > 0, there exists sufficiently large M 0 such that On the contrary, by the continuity of )r], there exists N > 0 such that when n > N and t ∈ [0, M 0 ], we have Hence, for any t ∈ [0, +∞) and n > N, we obtain