Approximate iterative sequences for positive solutions of a Hadamard type fractional differential system involving Hadamard type fractional derivatives

Abstract: In this paper, we focus on a class of Hadamard type fractional differential system involving Hadamard type fractional derivatives on an infinite interval. By utilizing the monotone iterative technique and Banach’s contraction mapping principle, some explicit monotone iterative sequences for approximating the extreme positive solutions and the unique positive solution for the system are constructed.

Note that most of the results on the current works are based on Riemann-Liouville type and Caputo type fractional differential equations in the past ten years. Hadamard type fractional derivative is first introduced in 1892 [19], which contains logarithmic function of arbitrary exponent in the kernel of integral appearing in its definition. Hadamard type integrals arise in the formulation of many problems in mechanics such as in fracture analysis. For details and applications of Hadamard type fractional derivative and integral, see [3,[20][21][22]. Recently, more and more scholars pay special attention to Hadamard type fractional differential equations on the finite interval [23][24][25][26][27][28]. For example, by applying Leray-Schauder's alternative and Banach's contraction principle, Ahmad and Ntouyas [29] established the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with a fully Hadamard type integral boundary conditions: , v(t)), 1 < t < e, 1 < α ≤ 2, H D β v(t) = g(t, v(t), u(t)), 1 < t < e, 1 < β ≤ 2, u(1) = 0, u(e) = H I r u(σ 1 ) = 1 Γ(r) where γ > 0, 1 < σ 1 , σ 2 < e, H D (·) are the Hadamard type fractional derivative and H I r is the Hadamard type fractional integral of order r, f, g : [1, e] × R × R are given continuous functions. In [30] by means of comparison principle and the monotone iterative technique combined with the method of upper and lower solutions, Yang investigated the extremal iterative solutions for the following coupled system of nonlinear Hadamard type fractional differential equations: ( H D α a + y)(t) = g(t, x(t), y(t)), 0 < α ≤ 1, a < t ≤ b, ( H J 1−α a + x)(a + ) = x * , ( H J 1−α a + y)(a + ) = y * , where f, g ∈ C([a, b] × R × R, R), H D α a + and H J α a + are the left-sided Hadamard type fractional derivative and Hadamard type fractional integral of order α, respectively.
On the other hand, some authors have also focused on the existence of solutions for Hadamard type fractional differential equations on the infinite intervals, see [31][32][33][34][35][36] and the references quoted therein. In another study [37], by applying standard fixed point theorems, Tariboon et al. obtained the existence of positive solutions of the Hadamard type fractional differential system with coupled integral boundary conditions: In [38] Zhang and Liu focused on a class of Hadamard type fractional differential equation with nonlocal boundary conditions on an infinite interval: where H D α 1+ , H I β i 1+ are the Hadamard type fractional derivative of order α and the Hadamard type fractional integral of order β i > 0 (i = 1, 2, 3, · · · , m), 1 < η < ξ 1 < ξ 2 < · · · < ξ n . b, α i , σ j ≥ 0 (i = 1, 2, 3, · · · , m; j = 1, 2, 3, · · · , n) are given constants satisfy certain prior conditions. By using various fixed point methods, the authors not only obtained the existence and uniqueness of solutions, but also the iterative sequences of approximate solutions.
Motivated by the mentioned results above, a nature and meaningful question is if we know the existence of solution for the following Hadamard type fractional differential system (1.5), how can we seek it? This idea lead us to develop the research of approximate sequences of positive solutions for the following Hadamard type fractional differential system with Hadamard type fractional integral boundary conditions: where J = [1, +∞), H D α j , H I β ji are the common Hadamard type fractional derivative of order α j and the Hadamard type fractional integral of order β ji > 0, f j ∈ C(J × R × R × R × R, R + ), λ ji > 0 are given constants and satisfy Γ(α j +β ji ) (log η j ) α j +β ji −1 > 0, j = 1, 2; i = 1, 2, · · · , m j , m j ∈ N + . In this paper, we emphasize that the nonlinearity terms f j of the system (1.5) involve multiple unknown functions and the lower-order Hadamard type fractional derivative of multiple unknown functions. By utilizing the monotone iterative method, we establish some explicit monotone iterative sequences for approximating the extreme positive solutions and the unique positive solution, which are more valuable and interesting than just constructing the existence of solutions. Further we extend the iterative methods that are often used in a single equation to the system which is different from [31,34,[38][39][40][41][42]. Finally we give some examples to verify the application of main results.

Preliminaries
First we recall some Hadamard type fractional calculus definitions and lemmas that are helpful to the proof of main results.
Definition 2.1 (see [1]). The Hadamard type fractional derivative of order q for a integrable function g : [1, ∞) → R is given by where n = [q] + 1, [q] denotes the integer part of the real number q and log(·) = log e (·). Definition 2.2 (see [1]). The Hadamard type fractional integral of order q for a integrable function g is given by provided the integral exists. Lemma 2.1 (see [1,32]). If a, α, β > 0, then and Ω j > 0, j = 1, 2, then the following Hadamard type fractional differential system with Hadamard type fractional integral boundary conditions has a unique solution: and and (2.7) For convenience, we introduce the following notations: Lemma 2.3 (see [32]). The Green's function G j (t, s) defined by (2.3) has the following properties: (A1): G j (t, s) ≥ 0 and G j (t, s) are continuous for all (t, s) ∈ J × J, j = 1, 2; (A2): The Green's function G j (t, s) and G * j (t, s) defined by (2.3) and (2.6) still have the following properties: so (B1) holds. And from (2.6) and (2.7), it is easy to that (B2) holds. Lemma 2.4 (see [32,33]). Let U ⊂ X be a bounded set. Then U is a relatively compact in X if the following conditions hold: (i) For any u ∈ U, u(t) 1 + (log t) α−1 and H D α−1 u(t) are equicontinuous on any compact interval of J; Next we present some assumptions that will play an important role in subsequent discussion.

Main results
In this paper, we will use two Banach spaces which are define by Then the space (X, · X ) and (Y, · Y ) are two Banach spaces which can be shown similarly to Lemma 2.7 of the literature [32]. Moreover, the product space (X × Y, · X×Y ) is also a Banach space with the norm Proof. For any (u, v) ∈ X × Y, by assumption (C2), one can obtain Lemma 3.2 If assumption (C4) holds, then for any (u, v) ∈ X × Y, Proof. For any (u, v) ∈ X × Y, by assumption (C4), one can obtain Thus we have From Lemma 2.2, we can know that the system (2.2) is equivalent to the following system of Hammerstein-type integral equations: and for convenience, we set Therefore one can define an operator T : P 1 × P 2 → P 1 × P 2 as follows: Therefore, if (u, v) ∈ P 1 × P 2 /(0, 0) is a fixed point of the operator T , then (u, v) is a positive solution for the Hadamard type fractional differential system (1.5). It is obvious that the system (1.5) has a positive solution if and only if the operator equation (u, v) = T (u, v) has a positive fixed point in P 1 × P 2 , where T is given as (3.2). Next we will directly consider the existence of fixed points of the operator T . Lemma 3.3 If assumption (C1), (C2) and (C3) hold, then the operator T : Next we show in four steps that the operator T : P 1 × P 2 → P 1 × P 2 is completely continuous. Step For any (u, v) ∈ U, by Lemma 2.3, Lemma 3.1 and Remark 2.2, one can obtain and (3.5) Thus Similarly Then which implies that T U is uniformly bounded for any (u, v) ∈ U.
Step 2 Let I ⊂ J be any compact interval. Then, for all t 1 , t 2 ∈ I, t 2 > t 1 and (u, v) ∈ U, we have Noticing that G 1 (t, s)/1 + (log t) α 1 −1 is uniformly continuous for any (t, s) ∈ I × I. Moreover the function G 1 (t, s)/1 + (log t) α 1 −1 is only associated with t for s ≥ t, which implies that G 1 (t, s)/1 + (log t) α 1 −1 is uniformly continuous on I × (J \ I). That is, for all s ∈ J and t 1 , (3.7) By Lemma 3.1, for all (u, v) ∈ U, we have For all t 1 , t 2 ∈ I, t 2 > t 1 and (u, v) ∈ U, together (3.6), (3.7) and (3.8) mean that Note that In the same way, one can easily show that T 2 (u, v)(t)/1 + (log t) α 2 −1 and D α 2 −1 T 2 (u, v)(t) are equicontinuous. Hence T 1 and T 2 are equicontinuous on I. Then the operator T is equicontinuous for all (u, v) ∈ U on any compact interval I of J.
Step 3 Now we prove the operator T is equiconvergent at +∞. Due to one can infer that for any > 0, there exists a constant C = C( ) > 0, for any t 1 , t 2 ≥ C and s ∈ J, such that with the help of Lemma 3.1 and (3.6), which mean that T j (u, v)(t)/1 + (log t) α j −1 ( j = 1, 2) are equiconvergent at +∞. Meanwhile function G * j (t, s)( j = 1, 2) are independent of t, one can easily show that H D α j −1 T j (u, v)(t)( j = 1, 2) are equiconvergent at +∞. From Step 1, Step 2 and Step 3, Lemma 2.4 holds. So the operator T is relatively compact in P 1 × P 2 .
Step 4 Finally we prove that the operator T : P 1 × P 2 → P 1 × P 2 is continuous. Set (u n , v n ), (u, v) ∈ P 1 × P 2 and (u n , v n ) → (u, v)(n → ∞). So ||(u n , v n )|| X×Y < +∞, ||(u, v)|| X×Y < +∞. Similar to (3.4) and (3.5), one has and Via the Lebesgue dominated convergence theorem and continuity of function f 1 , we know Therefore, as n → ∞, This implies that the operator T 1 is continuous. At the same way, one can obtain than the operator T 2 is continuous. That is, the operator T is continuous. Summarize all of the above discussions, one can infer that the operator T : P 1 × P 2 → P 1 × P 2 is completely continuous. So the proof of Lemma 3.3 is completed.
Due to T (U R ) ⊂ U R , it is easy to see that (u n , v n ), (w n , z n ) ∈ T (U R ) for n = 1, 2, · · · . Thus we just need to show that there exist (u * , v * ) and (w * , z * ) satisfying lim n→∞ (u n , v n ) = (u * , v * ) and lim n→∞ (w n , z n ) = (w * , z * ), which are two monotone sequences for approximating positive solutions of the system (1.5).
For t ∈ J, (u n , v n ) ∈ U R , from Lemma 2.2 and (3.9), one has Next we consider the monotonicity of the Hadamard type fractional derivative of (u, v). By (3.13) we have Then, by (3.13) and (3.14), for any t ∈ J, via the monotonicity conditions (C3) of functions f j ( j = 1, 2), we do the second iteration For t ∈ J, by method of induction, the sequences With the help of iterative sequences (u n+1 , v n+1 ) = T (u n , v n ) and the complete continuity of the operator T , one can easily infer that (u n , v n ) → (u * , v * ) and T (u * , v * ) = (u * , v * ) . For the sequences {(w n , z n )} ∞ n=0 , we employ a similar discussion. For t ∈ J, we have Using the the monotonicity condition (C3) of functions f j , one has Moreover there exists an error estimate for the approximation sequence where m is defined by (3.15) and = max{ 1 , 2 }, j ( j = 1, 2) are defined by the aussumption (C4).
For any (u, v) ∈ U r , by Lemma 3.2 and Remark 2.2, we have So one has ||T (u, v)|| X×Y ≤ mr + Υ ≤ r, ∀(u, v) ∈ U r . Now we demonstrate that operator T is a contraction. For any (u 1 , v 1 ), (u 2 , v 2 ) ∈ U r , by assumption (C4), we have In the same way, one can obtain By (3.18) and (3.19), we gain Due to m < 1, then operator T is a contraction. With the help of the Banach fixed-point theorem, T has a unique fixed point (x, y) in U r . That is, the system (1.5) has a unique positive solution (x, y). Further, for any (u 0 , v 0 ) ∈ U r , (u n , v n ) − (x, y) X×Y → 0 as n → ∞, where u n = T 1 (u n−1 , v n−1 ), v n = T 2 (u n−1 , v n−1 ), n = 1, 2, · · · . From (3.20), we have (3.21) Taking j → +∞ on both sides of (3.21), one can obtain So the proof of Theorem 3.2 is completed.
Noting that which imply that assumption (C2) holds.
From the expression of function f j , we can infer that f j is increasing respect to the variables u 1 , u 2 , u 3 , u 4 , ∀t ∈ J, j = 1, 2. Hence assumption (C3) is also satisfied . By Theorem 3.1, it follows that the system (4.1) have two pairs of positive solutions (u * , v * ) and (w * , z * ), which can be constructed via the limit of two explicit monotone iterative sequences in (3.11) and (3.12).
Same to example (4.1), it is easy to verify that assumption (C1) hods.

Conclusions
In this paper, we consider a class of Hadamard type fractional differential system. By the aid of monotone iterative technique and Banach's contraction mapping principle, under certain nonlinear and linear increasing conditions, we construct some explicit monotone iterative sequences for approximating the extreme positive solutions and the unique positive solutions. Our results generalize iterative solution of a single equation to the case of a system, and the nonlinear term contains Hadamard type fractional derivative which can be used more widely. Further work is still needed including discussions on iterative solution for Hadamard type fractional differential system with coupling integral condition and additional studies on iterative solution for impulsive Hadamard type fractional differential system.