Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions

In this paper we use the ﬁxed point index to study the existence of positive solutions for a system of nonlinear Hadamard fractional diﬀerential equations involving coupled integral boundary conditions. Here we use appropriate nonnegative matrices to depict the coupling behavior for our nonlinearities.


Introduction
In this paper we consider the system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions 1 (t, u(t), v(t)) = 0, 1 < t < e, D β v(t)+f 2 (t, u(t), v(t)) = 0, 1 < t < e, where β ∈ (2, 3], D β is the Hadamard fractional derivative of fractional order β,a n df i (i =1,2),h, g satisfy the following conditions: where D α t , D β t , D γ t are the Riemann-Liouville fractional derivatives, and they not only obtained existence and uniqueness of positive solutions for (1.2), but also constructed an iteration sequence for the unique positive solution. In , the authors used fixed point methods to study the existence of (positive) solutions fractional order equations. In [10] Mawhin's continuation theorem was used to study the following fractional order boundary value problem at resonance: x(T)=β ρ I p x(ξ ), 0 < ζ , ξ ≤ T, where c D q is the Caputo fractional derivative, I γ ,δ η is a Erdélyi-Kober type integral, and ρ I p denotes the generalized Riemann-Liouville type integral boundary conditions. For fractional differential systems, see [23][24][25][26][27][28][29][30][31][32]. In [23], using the Leray-Schauder alternative and the Banach contraction principle, the authors studied existence and uniqueness of solutions for the system of nonlinear Caputo type sequential fractional integro-differential equations (t, u(t), v(t), c D p 1 v(t), I q 1 v(t)), t ∈ (0, 1), ( c D β + µ c D β-1 )v(t)=g(t, u(t), c D p 2 u(t), I q 2 u(t), v(t)), t ∈ (0, 1), (1.4) Hadamard fractional differential equations are also popular in the literature; see [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] and the references therein. In [33], the authors used the Banach contraction principle, the Leray-Schauder's alternative, and Krasnoselskii's fixed-point theorem to study the existence and uniqueness of solutions for the coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions where C D p i , H D q i are respectively the Caputo and Hadamard fractional derivatives. In [34] the authors established positive solutions for the coupled Hadamard fractional integral where α, β ∈ (n -1,n]andn ≥ 3, D α , D β are the Hadamard fractional derivatives and their nonlinearities f , g satisfy the following conditions: =+∞. Motivated by the above, in this paper we study the existence of positive solutions for the system of nonlinear Hadamard fractional differential equations (1.1)i n v o l v i n gc o upled integral boundary conditions. We use appropriate nonnegative matrices to depict the coupling behavior for our nonlinearities, and note that they can grow both superlinearly and sublinearly. We remark here that our conditions for nonlinear terms are not as restrictive as those in (H) Yang1 and (H) Yang2 ; see (H3)-(H6) in Sect. 3.

2P r e l i m i n a r i e s
In this section, we first provide some material for Hadamard fractional calculus; for details, see the book [49].

Definition 2.1
The Hadamard derivative of fractional order q for a function g :[1,∞) → R is defined as where n =[q]+1;[q] denotes the integer part of the real number q and log(·)=log e (·).

Definition 2.2
The Hadamard fractional integral of order q for a function g :[1,∞) → R is defined as provided the integral exists.
In what follows, we calculate the Green's functions associated with (1.1)andstudysome properties of these Green's functions.
can be transformed into the following Hammerstein type integral equations: here, d g,h , d g , d h are three positive constants defined in the proof.
From Lemma 2.3,wenote(1.1) is equivalent to the Hammerstein type integral equations

Lemma 2.4
The function G 1 (t, s) satisfies the following inequalities: Proof We note a result from [14]. Let β ∈ (n -1,n]withn ∈ N, n ≥ 3. Then the function has the following properties: . Now, we turn our attention to G 1 .Iflog t, log s are regarded as z, l, then from (R2), (R3) we have Thus (I1), (I2) hold. This completes the proof.
Lemma 2.8 (see [50]) Let E be a real Banach space and P be a cone on E. Suppose that Ω ⊂ Eisaboundedopensetwith0 ∈ Ω and that A : Ω ∩ P → P is a continuous compact operator. If then i(A, Ω ∩ P, P)=1.

Main results
Let Now we list our assumptions for the nonlinearities f i (i =1,2).