On a Hadamard-type fractional turbulent flow model with deviating arguments in a porous medium

Fractional differential equation models are important and useful in a number of fields such as blood flow phenomena, earthquake, rheology, electrodynamics of a complex medium, aerodynamics, viscoelasticity, and optics and signal processing, and so on [10, 21, 23, 28, 38, 47]. With the development of fractional differential equation models in theory and practice, the subject has acquired great achievements. For some recent works on the subject, readers can see [4–9,12,13,15,19,25,27,29,31,33,36,37,40–42,44–46] and the references therein.

Differential equations with deviating arguments are often applied to investigations connected with economic, mechanics, mathematical physics, etc. [1,16,32]. In recent years, many researchers are taking notice of Hadamard-type fractional differential equation, not only Riemann-Liouville or Caputo-type fractional differential equation. It is another kind of fractional derivatives and be introduced by Hadamard in 1892 [18]. Hadamard derivative has predominant properties, and its definition includes logarithmic function of arbitrary. For the development of Hamamard fractional calculus, we refer to [2, 3, 11, 17, 20-22, 26, 34, 35].
As we all know, turbulent flow in a porous medium is a basic mechanics problem. In [24], in order to study this type of problem, Leibenson put forward the p-Laplacian equation as follows: where φ p (s) = |s| p−2 s, which is defined as p-Laplacian operator, and φ −1 p (s) = φ q (s), 1/p + 1/q = 1.
In [14], Chen, Liu and Hu studied the existence of solutions of the following fractional differential equation with p-Laplacian operator at resonance: where 0 < α, β 1, 1 < α + β 2, D α 0 + is a Caputo fractional derivative. By using the coincidence degree theory, a new existence result for the above problem is given.
In [39], Zhang studied the existence of solutions of the following two point boundary value problems with the fractional p-Laplacian operator: where 0 < α, β 1, D α 0 + is a Caputo fractional derivative, and f : [0, 1] × R 2 → R is continuous. With the help of Schauder fixed point theorem, author establishes two theorems on the existence of solutions for two point boundary value problems with the fractional p-Laplacian operator.
A is a function of bounded variation, and dA can be a signed measure. By employing the fixed point theorem of the mixed monotone operator, authors obtain the uniqueness of positive solution of the above fractional-order turbulent flow model. Inspired by the above mentioned work, in this paper, we study the uniqueness, the existence and nonexistence of solutions for a fractional-order turbulent flow model with deviating arguments Hadamard fractional derivative of order α and β, respectively. Since the fractional-order turbulent flow model involves Hadamard fractional derivative and deviating arguments has been seldom studied, in this paper, we study the uniqueness, the existence and nonexistence of solutions of the Hadamard-type fractional-order turbulent flow model with deviating arguments, i.e. problem (1). By using the Schauder's fixed point theorem and the contraction mapping principle, three new results on the Hadamard-type fractional-order turbulent flow model (1) are obtained.
We also use the following properties of the p-Laplacian operator: (L2) If q 2, |u|, |v| M , then

Preliminaries
For the convenience of the reader, we recall some basic definitions about Hadamard fractional calculus and important lemmas.
Definition 1. (See [21].) The Hadamard fractional integral of order q for a function g is defined as provided the integral exists.
Definition 2. (See [21].) The Hadamard derivation of fractional order q for a function g : denotes the integer part of the real number q, and log(·) = log e (·).
is given by Proof. By Lemma 1, we get where constants c 0 , c 1 ∈ R. Thus, where constants c 0 , c 1 ∈ R.
Employing the boundary value condition H D α u(1) = H D α u(e) = 0, we have

The boundary value condition
Then, substituting c 2 and c 3 into (4), we get equation (3). The proof is complete.
Proof. For any u, v ∈ B C , in view of (H1) and (H2) combined with Lemma 3, we have In view of (5), T is a contraction mapping. By the Banach contraction mapping principle, the operator T has a fixed point, which implies that Hadamard-type fractional turbulent flow model (1) has a unique solution.

Existence results
Now, we are in the position to present a existence result, which is based on the following Schaefer fixed point theorem.
Theorem 2. (See [30].) Let X be a Banach space. Assume that Ω is an open bounded subset of X with θ ∈ Ω, and let T : Ω → X be a completely continuous operator such that T u u for all u ∈ ∂Ω. Then T has a fixed point in Ω. (H4) There exists a constant M > 0 such that Then problem (1) has at least one solution.
Proof. Firstly, we prove that the operator T is completely continuous. Clearly, continuity of the operator T follows from the continuity of f . Let Ω ⊂ C[1, e] be bounded. For any u, v ∈ Ω, there exist positive constant L such that |f (t, u, v)| L. We can get

Therefore, T u
N . For all t 1 , t 2 ∈ [1, e] and 1 t 1 t 2 e, we have Thus, by the Arzela-Ascoli theorem, the operator T is completely continuous. Next, we consider the set V = {u ∈ C[1, e]: u = µT u, µ ∈ (0, 1)} and show that the set V is bounded. Thus, This, together with condition (H4), gives u < M. So, the set V is bounded. Thus, by the Schaefer's fixed point theorem, the operator T has at least one fixed point, which implies that Hadamard-type fractional turbulent flow model has at least one solution.

Nonexistence results
In this section, we introduce a parameter in problem (1), then we give some sufficient conditions of nonexistence of solution to a class of Hadamard-type fractional turbulent flow model with a parameter. Precisely, we study the following problem: where λ is a positive parameter.
Thus, all the assumptions of Theorem 4 are satisfied. Consequently, the conclusion of Theorem 4 implies that problem (9) has no solution for 0 < λ < 1.6280.