Existence Results for a Class of Nonlinear Hadamard Fractional with p-Laplacian Operator Differential Equations

Recently, fractional differential equations have been acquired much attention due to its applications in a number of fields such as physics, mechanics, chemistry, biology, signal and image processing, see for example the books (Baleanu et al., 2012; Kilbas et al., 2006; Lakshmikantham et al., 2009; Yang et al., 2015). Some recent works on fractional differential equations involving Riemann Liouville and Caputo-type fractional derivatives are studied using nonlinear analysis methods such as Krasnoselskii fixed-point Theorems (Agarwal and O'Regan, 1998; Ghanmi and Horrigue, 2018; Guo et al., 2007; Guo et al., 2008), Leray-Schauder alternative (Ghanmi and Horrigue, 2019; Qi et al., 2017), sub-solution and super-solution methods (Wang et al., 2019; Mâagli et al., 2015) and iterative techniques (Liu et al., 2013). Hadamard (1892) introduced an important fractional derivative, which differs from the above-mentioned ones because its definition involves logarithmic function of arbitrary exponent and named as Hadamard derivative. In the last few decades many authors are paying more and more attention to fractional differential equation involving Hadamard derivative, the study of the topic is still in its primary stage. For details and recent developments on Hadamard fractional differential equations, see (Huang and Liu, 2018; Wang et al., 2018; Zhai et al., 2018) and references therein. Recently, some researches have extensively interested in the study of the fractional differential equations with p-Laplacian operators see for examples (Chamekh et al., 2018; Ding et al., 2015). From the above review of the literature concerning fractional differential equations, most of the authors investigated only the existence of solutions or positive solutions for Hadamard fractional differential equations without considering the pi-Laplacian operator. A very few authors established results along with p-Laplacian operator, us example in (Wang and Wang, 2016), the authors considered the following nonlinear Hadamard fractional differential problem:


Introduction
Recently, fractional differential equations have been acquired much attention due to its applications in a number of fields such as physics, mechanics, chemistry, biology, signal and image processing, see for example the books (Baleanu et al., 2012;Kilbas et al., 2006;Lakshmikantham et al., 2009;Yang et al., 2015).
Some recent works on fractional differential equations involving Riemann Liouville and Caputo-type fractional derivatives are studied using nonlinear analysis methods such as Krasnoselskii fixed-point Theorems (Agarwal and O'Regan, 1998;Ghanmi and Horrigue, 2018;Guo et al., 2007;Guo et al., 2008), Leray-Schauder alternative Qi et al., 2017), sub-solution and super-solution methods (Wang et al., 2019;Mâagli et al., 2015) and iterative techniques (Liu et al., 2013). Hadamard (1892) introduced an important fractional derivative, which differs from the above-mentioned ones because its definition involves logarithmic function of arbitrary exponent and named as Hadamard derivative. In the last few decades many authors are paying more and more attention to fractional differential equation involving Hadamard derivative, the study of the topic is still in its primary stage. For details and recent developments on Hadamard fractional differential equations, see (Huang and Liu, 2018;Wang et al., 2018;Zhai et al., 2018) and references therein. Recently, some researches have extensively interested in the study of the fractional differential equations with p-Laplacian operators see for examples (Chamekh et al., 2018;Ding et al., 2015).
From the above review of the literature concerning fractional differential equations, most of the authors investigated only the existence of solutions or positive solutions for Hadamard fractional differential equations without considering the pi-Laplacian operator. A very few authors established results along with p-Laplacian operator, us example in (Wang and Wang, 2016), the authors considered the following nonlinear Hadamard fractional differential problem: where fore an appropriate ξ, D ξ is the Hadamard fractional derivative of order ξ, 1 <  ≤ 2, 0 < σ ≤ 1, γ > 0, λ  ℝ and f ∈ C([1,T]× ℝ, ℝ). By using the Schauder fixed point Theorem, the existence of solutions is obtained. In Li and Lin (2013), the authors used the Guo-Krasnosel'skii fixed point Theorem to prove the existence and uniqueness of positive solutions of the following Hadamard fractional boundary value with p−Laplacian operator: where, 2 < α ≤ 3, 1 < σ ≤ 2, f ∈ C([1,e] × [0,∞),[0,∞)) and the function φp (p > 1), is called p-Laplacian and is defined in R by φp(s) = |s| p−2 s. The authors in Zhang et al. (2018) established some existence of positive solutions for the following nonlinear Hadamard fractional differential equations with p-Laplacian operator:
For the sake of computational convenience, we set: And we assume the following conditions. (H1) There exist nonnegative functions a(t), b(t)  C ([1,e], ℝ), such that: There exist a positive continuous nondecreasing function g on [0,∞) and a function p ∈ C([1,e],R + ) such that: where w1, w2 and w3 is given respectively by:  This study is organized as follows, in Section 2 we present some preliminaries and usefully results which will be used in the proofs of the main results. Section 3 is devoted to the proof of Theorem 1.1 and Theorem 1.2. In Section 4, we present some important examples in order to illustrate the main results of this article.

Preliminaries
In this section, we recall some results and we prove key lemmas which we will use later in section 3. Also, we give some definitions and properties related on Hadamard fractional calculus, we refer the reader to Kilbas et al. (2006) for more details.

Proof
As argued in Kilbas et al. (2006), the Hadamard differential Equation in (2.8) can be written as: x(1)) = 0, then b = 0. So, we obtain: By applying 2 I  on both sides of (2.12) for t = η2 and using the property (2.5), we obtain:

64
On the other hand, put t = e in Equation (2.12), we obtain: (2.14) By combining Equations (2.13), (2.14) and the second boundary condition in (2.8), we obtain: . : Then, the solution can be written us follows: Since x(1) = 0, then b' = 0 and we get: Now, if we apply 1 I  to (2.16) and we replace t by η1, then, using the property (2.5), we obtain: On the other hand, Equation ( Substituting the values of c3 and c4 in (2.16), we obtain (2.9). This completes the proof.
To prove the main results of this study, we recall the following theorems.
Theorem 2.4 Smart (1974) Let X be a Banach space. If the operator T: X → X is completely continuous and if the set: is bounded. Then T has a fixed point in X.
Theorem 2.5 Granas and Dugundji (2003) Let X be a Banach space, C be a closed, convex subset of X, U be an open subset of C and 0  C. Suppose that the operator F: U → C is continuous and compact. Then either (i). F has a fixed point in U, or (ii). There is u ∈ ∂U and λ ∈ (0,1), with u = λF(u).

Proof of the Main Results
This section is devoted to prove existence results for the nonlinear boundary value problem (1.1). Also, we shall prove existence and uniqueness results by using different methods. Let us de ne the operator Q: where c3 and c4 are given respectively by Equation (2.10) and (2.11). Notice that the existence of fixed points of the operator Q is equivalent to the existence of solutions for problem (1.1).

Proof of Theorem 1.1
In this subsection, by using the well known Banach's fixed point Theorem, we present the existence and uniqueness result for Problem (1.1). Let The proof is divided into tow steps.
Step 1: In this step, we will prove that the operator Q is completely continuous. Let O be an open bounded subset of C([1,e], ℝ). Since f is continuous, then, the operator Q is contunuous and so, Q(O) is bounded. Next, we will show that Q is equicontinuous.
On the other hand, we have:   67 Therefore, it follows from (2.7), (3.8), (3.9) and (3.10), that: Using the inequality (2.7), we have: This together with condition (H2), gives ||u||∞ < M1. That is D is bounded, so the operator Q has at lest one fixed point. Which implies that the problem (1.1) has at least one solution.

Proof of Theorem 1.2
In this subsection, by using Leray-Schauder's nonlinear alternative Theorem, we give the proof of Theorem 1.2. The proof is divided into several steps.
Step 1: In this step, We prove that Q maps bounded sets into equicontinuous sets of C ([1,e] Obviously the right-hand side of the above inequality tends to zero independently of x ∈ Br as t1 − t2 → 0. Step 2: In this step, we will prove that Q maps bounded sets (balls) into bounded sets in C([1, e], ℝ).