Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

In this paper we consider a fractional differential system with coupled integral boundary value problems on a half-line, where the nonlinearity terms depend on unknown functions and the lower-order fractional derivative of unknown functions, and the fractional infinite boundary value conditions depend on the coupled infinite integral of unknown functions. By virtue of the monotone iterative technique, we find two explicit monotone iterative sequences which converge to the positive minimal and maximal solutions when the nonlinearities can satisfy certain nonlinear growth conditions.


Introduction
Fractional-order differential equations is a natural generalization of the case of integer order, which has become the focus of attention involving various kinds of boundary conditions because of the wide application in mathematical models and applied sciences. Some latest results on the topic can be found in a series of papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. In particular, a monotone iterative technique is believed to be an efficient and important method to deal with sequences of monotone solutions for initial and boundary value problems. For some applications of this method to nonlinear fractional differential equations, see [16][17][18][19][20][21][22][23][24]. We also note that there are some results about monotone iterative solution of a single fractional order equation on a half-line, see [25][26][27][28][29].
In [9] Jiang et al. utilized the fixed point index to construct the existence of positive solutions for the following system on a finite interval: where the nonlinearities f i (i = 1, 2) can grow superlinearly and sublinearly, and boundary value conditions depend on the coupled integral of unknown functions.
In [32] Aljoudi et al. studied the sequential fractional differential equations on a finite interval where D (·) and I (·) denote the Hadamard fractional derivative and Hadamard fractional integral, f , g : [1, e] × R 3 → R are given continuous functions, and boundary value conditions depend on the coupled fractional integral of unknown functions. Inspired by the works above, in this paper we utilize the monotone iterative technique to study the existence of positive extremal solutions of a fractional differential system on a half-line where D α , D β are the Riemann-Liouville fractional derivatives. Here we emphasize that the nonlinearity terms ϕ, ψ include not only unknown functions, but also the lower-order fractional derivative of unknown functions. By the way, the fractional infinite boundary value conditions depend on the coupled infinite integral of unknown functions. To the best of the authors' knowledge, the system with coupled infinite integral boundary value conditions is yet to be investigated. ϕ, ψ satisfy the following assumptions:  and (C3) ϕ(t, u, v, w) and ψ(t, u, v, z) are increasing with respect to the variables u, v, w and u, v, z, and ϕ(t, 0, 0, 0) ≡ 0, ψ(t, 0, 0, 0) ≡ 0, ∀t ∈ J. For convenience, we set

Preliminaries
In this section we only list some definitions and lemmas of the Riemann-Liouville fractional integral and derivative; for more details, we refer the readers to [1]. Definition 2.1 (see [1]) The Riemann-Liouville fractional integral of order q > 0 for an integrable function g is defined as provided that the integral exists. Definition 2.2 (see [1]) The Riemann-Liouville fractional derivative of order q > 0 for an integrable function g is defined as where n = [q] + 1, [q] is the smallest integer greater than or equal to q, provided that the right-hand side is pointwise defined on (0, +∞).
Lemma 2.4 Let x, y ∈ C(0, +∞) ∩ L(0, +∞) and assumption (C1) be satisfied. Then the fractional differential system with coupled integral boundary conditions has a solution which can take the integral representation Proof By Lemma 2.3, we can turn system (2.1) into an equivalent integral system That is, +∞ 0 x(s) ds, (2.8) Multiplying both sides of equality (2.9) by g(t) and h(t) and integrating from 0 to +∞, we have ⎧ ⎨ (2.10) Submitting (2.10) to (2.9), we have The proof is completed.
Proof From (2.3), it is obvious that By a similar calculation, we can prove other inequality results about K 2 (t, s), K 3 (t, s), and K 4 (t, s). So the proof is completed.
Remark 2.7 From Lemma 2.5, by a direct calculation, we can easily obtain that Define two spaces where 2 < α, β ≤ 3. C(J) denotes the space of all continuous functions defined on [0, +∞).
Proof The proof is similar to that of Lemma 2.4 in [29], so we omit it. Lemma 2.9 (see [44]) Let U ⊂ X be a bounded set. Then U is relatively compact in X if the following conditions hold: (i) For any u ∈ U, u(t) 1+t α-1 and D α-1 u(t) are equicontinuous on any compact interval of J; (ii) For any ε > 0, there is a constant C = C(ε) > 0 such that | u(t 1 ) 1+t α-1 2 | < ε and |D α-1 u(t 1 ) -D α-1 u(t 2 )| < ε for any t 1 , t 2 ≥ C and u ∈ U. Remark 2.10 Let U ⊂ X be a bounded set. According to Lemmas 2.8 and 2.9, it is clear that U is relatively compact in X if the following conditions hold: (i) For any u ∈ U, u(t) 1+t α+β-1 and D α-1 u(t) are equicontinuous on any compact interval of J;

Main results
We define the cone P ⊂ X × Y as P From Lemma 2.4 it is easy to know that the fractional differential system (1.6) is equivalent to the following system of Hammerstein-type integral equations: and for convenience, we set Then we can define an operator : P × P → P × P as follows: Therefore, if (u, v) ∈ (P × P)\{0} is a fixed point of , then (u, v) is a positive solution for the fractional differential system (1.6). Next, we will directly study the existence of fixed points of the operator . By Remark 2.6 and (3.1), we have
So it is obvious that : P → P.
Next we show that the operator : P → P is relatively compact.
Note that and H 1 (t, s), H 3 (t, s) ∈ C(J × J) do not depend on t, which infers that D α-1 1 (u, v)(t) is equicontinuous on I. In the same way, we can show that 2 (u, v)(t)/(1 + t α+β-1 ) and D β-1 2 (u, v)(t) are equicontinuous. Thus condition (i) of Remark 2.10 is satisfied. Then we show that operators 1 , 2 are equiconvergent at +∞. Since we can infer that, for any > 0, there exists a sufficiently large constant C = C( ) > 0, for any t 1 , t 2 ≥ C and s ∈ J, such that Therefore, by Lemma 3.1 and (3.6), we conclude that 1 (u, v)(t)/1 + t α+β-1 is equiconvergent at +∞. On the other hand, the functions H 1 (t, s), are equiconvergent at +∞. Thus condition (ii) of Remark 2.10 is satisfied. As can be seen from the above discussion, all the conditions of Remark 2.10 are satisfied. Thus the operator : P → P is relatively compact.
Finally, we prove that the operator : P → P is continuous. Let (u n , v n ), (u, v) ∈ P such that (u n , v n ) → (u, v)(n → ∞). Then (u n , v n ) X×Y < +∞, (u, v) X×Y < +∞. Similar to (3.3) and (3.4), we can obtain By the continuity of function ϕ, ψ and the Lebesgue dominated convergence theorem, we have Then, as n → ∞, and as n → ∞, So, as n → ∞, which implies that the operator 1 is continuous. By the same way, we can obtain that the operator 2 is continuous. That is, the operator is continuous.
In view of all above arguments, the operator : P → P is completely continuous. So the proof is completed.

Conclusions
In this paper, we apply the monotone iterative technique to study a fractional differential system with coupled integral boundary conditions in a half-line. We first transform system (1.6) into an equivalent operator equation (3.1), and then we construct some norm inequalities related to nonlinear terms ϕ, ψ and a new Banach space. Finally, some explicit monotone iterative sequences for approximating the extreme positive solutions are obtained.