Existence of Positive Solutions of Nonlocal p ( x ) -Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

: Motivated by the idea which has been introduced by Boulaaras and Guefaiﬁa (Math. Meth. Appl. Sci. 41 (2018), 5203–5210 and, by Afrouzi and Shakeri (Afr. Mat. (2015) 26:159–168) combined with some properties of Kirchhoff type operators, we prove the existence of positive solutions for a class of nonlocal p ( x ) -Kirchhoff evolutionary systems by using the sub and super solutions concept.

In this article, we are interested in the p(x)-Kirchhoff parabolic systems of the form where Ω ⊂ R N is a bounded smooth domain with C 2 boundary ∂Ω, 1 < p (x) ∈ C 1 Ω is a functions with 1 < p − := inf Ω p (x) ≤ p + := sup Ω p (x) < ∞, p(x) u = ÷ |∇u| p(x)−2 ∇u is called p(x)-Laplacian, λ, λ 1 , λ 2 , µ 1 , and µ 2 are positive parameters, I 0 (u) = Problem (1) is a generalization of a model introduced by Kirchhoff [13]. More precisely, Kirchhoff proposed a model given by the equation where ρ, P 0 , h, E, L are constants, which extends the classical D'Alembert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [14][15][16][17][18][19][20][21], in which the authors have used a variational method and topological method to get the existence of solutions.
In this paper, motivated by the ideas introduced in [22] and the properties of Kirchhoff type operators in [22], we study the existence of positive solutions for system (2) by using the sub-and super solutions techniques. To our best knowledge, this is a new research topic for nonlocal problems. The remainder of this paper is organized as follows. In Section 2, we present some preleminary results on the variable exponent Sobolev space W 1,p(x) 0 (Ω) and the method of sub-and super solutions. Section 3 is devoted to stating and proving the main result.

Preliminary Results
In order to discuss problem (1), we need some theories on W 1,p(x) 0 (Ω) which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces W 1,p(x) 0 (Ω) which will be used later (for details, see [3]).
(2) We say that (u, v) is called a sub solution (respectively a super solution) of the problem defined in (3) if Lemma 2 (see [22]). Let (H1) hold. η > 0 and let u be the unique solution of the problem and when where C * and C * are positive constants depending p + , p − , N, |Ω| , C 0 and m 0 .
Here and hereafter, we will use the notation d (x, ∂Ω) to denote the distance of x ∈ Ω to denote the distance of Ω. Denote d (x) = d (x, ∂Ω) and we have the following Lemma: Lemma 3 (see [23]). If positive parameter η is large enough and ω is the unique solution of (5), then we have (i) For any θ ∈ (0, 1) , there exists a positive constant C 1 such that (ii) There exists a positive constant C 2 such that

Main Result
In the following, when there is no misunderstanding, we always use C i to denote positive constants. Proof. We shall establish Theorem 1 by constructing a positive subsolution (φ k , φ 1 ) and supersolution (z k , z 1 ) of the problem defined in (1) such that φ k ≤ z k and φ 1 ≤ z 1 . That is, (φ k , φ 1 ) and (z k , z 1 ) satisfies (Ω) with q ≥ 0. According to the sub-super solution method for p(x)-Kirchhoff type equations (see [22]), then the problem defined in (1) has a positive solution.
Step 1. We will construct a subsolution of (1). Let σ ∈ (0, δ) be small enough. Denote It is easy to see that φ k , φ 1 ∈ C 1 Ω . Denote By some simple computations, we can obtain From (H4), there exists a positive constant L > 1 such that Let σ = 1 k ln L. Then, σk = ln L.
If k is sufficiently large, from the problem defined in (6), we have Let λζ From the problem defined in (7), we have Since d(x) ∈ C 2 ∂Ω 3δ , there exists a positive constant C 3 such that If k is sufficiently large, let λζ m ∞ = k α. Then, we have Then, Since Combining two problems which defined in (8) and (9), we can conclude that From the problems defined in (10) and (11), we can see that (φ k , φ 1 ) is a subsolution of problem (3).
Step 2. We will construct a supersolution of problem (3). We consider in Ω, . We shall prove that (z k , z 1 ) is a supersolution of problem (3).
For q ∈ W 1,p(x) 0 (Ω) with q ≥ 0, it is easy to see that By (H6), for large enough µ, using Lemma 2, we have Hence, In addition, By (H4), (H5) and Lemma 2, when µ is sufficiently large, we have Then, According to the problems (14) and (15), we can conclude that (z k , z 1 ) is a supersolution of problem (3). It only remains to prove that φ k ≤ z k and φ 1 ≤ z 1 .
In the definition of v 1 (x), let We claim that From the definition of v 1 , it is easy to see that If v 1 (x 0 ) − φ k (x 0 ) < 0, it is easy to see that 0 < d (x) < δ and then From the definition of v 1 , we have It is a contradiction to ∇v 1 (x 0 ) − ∇φ k (x 0 ) = 0.
Obviously, there exists a positive constant C 3 such that γ ≤ C 3 λ.
When η ≥ λ p + is large enough, we have According to the comparison principle, we have From problems (16) and (17), when η ≥ λ p + and λ ≥ 1 is sufficiently large, we have According to the comparison principle, when µ is large enough, we have v 1 (x) ≤ ω (x) ≤ z k (x) , ∀x ∈ Ω.
Combining the definition of v 1 (x) and the problem defined in (18), it is easy to see that When µ ≥ 1 and λ is large enough, from Lemma 2.6 (see [22]), we can see that β (λ p + (λ 1 a 2 + µ 1 c 2 ) µ) is large enough, and then is large enough. Similarly, we have φ 1 ≤ z 1 . This completes the proof.
Author Contributions: All authors contributed equally.