Existence Results for a Class of p-Laplacian Fractional Differential Equations with Integral Boundary Conditions

In this paper, we investigate the existence and uniqueness of solutions for a class of integral boundary value problems of nonlinear fractional differential equations with p-Laplacian operator. We obtain some existence and uniqueness results concerned with our problem by using Schaefer’s fixed-point theorem and Banach contraction mapping principle. Finally, we present some examples to illustrate our main results.

In the past few decades, fractional differential equations have been widely applied to many fields in natural and social sciences, because they are important tools in mathematically describing many phenomena of science and engineering such as aerodynamics, control theory, signal and image processing, plasma dynamics, blood flow phenomena, and viscoelas-tic and non-Newtonian fluid mechanics (see [1][2][3][4][5] and their references). In 1983, Leibenson [2] proposed the following integer-order differential equation model with p-Laplacian operator to study the turbulent flow in a porous medium in his work: where φ p is a p-Laplacian operator. Based on Leibenson's results, many researchers have generalized his model into a variety of models with fractional order and obtained many valuable existence and uniqueness results for p-Laplacian fractional differential equations with two-point boundary conditions (see [6][7][8][9][10][11][12][13][14]), multipoint boundary conditions (see [15][16][17][18]), and nonlocal boundary conditions (see [19][20][21][22][23]). Since the 1 st derivatives of unknown functions existed in Leibenson's model, it became a special study to discuss the existence results for p-Laplacian fractional differential equations where the fractional orders were in the neighborhood of 1. So, Chai [7] used the fixed-point theorem on cones to investigate the existence and multiplicity of positive solutions for fractional differential equations with p-Laplacian operator: are the Riemann-Liouville fractional derivatives. Jong [15] established the existence and uniqueness of positive solutions for multipoint boundary value problems of nonlinear fractional differential equations with p-Laplacian operator by using the Banach contraction mapping principle where 1 < α, β ≤ 2, 0 < γ ≤ 1, and f ∈ Cð½0, 1 × ½0,+∞Þ, ½0,+∞ÞÞ.
Integral boundary value problems for differential equations have arisen in the study of various fields such as underground water flow, blood flow problems, and thermoelasticity. So, the study on the existence of solutions for the p-Laplacian integral boundary value problems has attracted the attention of many researchers recently (see [20][21][22][23]). Zhang et al. [23] considered the existence of symmetric positive solutions of the problem for the following nonlinear fourth-order p-Laplacian differential equations with integral boundary conditions: where ω, g, h ∈ L½0, 1 are the nonnegative, symmetric functions and f ∈ Cðð0, 1 × ½0,+∞Þ, ½0,+∞ÞÞ. Zhang and Cui [22] investigated the existence of positive solutions for nonlinear fourth-order singular p-Laplacian differential equations with the integral boundary conditions where f ∈ Cðð0, 1Þ × ð0,+∞Þ × ð0,+∞Þ, ½0,+∞ÞÞ, f may be singular at t = 0, 1, u = 0, and g, h ∈ L½0, 1 are nonnegative. The existence results on solutions of problem (2) are established by employing upper and lower solution methods together with maximal principle. Jiang [20] used the generalized continuous theorem to investigate the existence of solutions to the integral boundary value problem of p-Laplacian multiterm fractional differential equations at resonance where 0 < β ≤ 1, 1 < α ≤ 2, and Ð 1 0 hðtÞt α−1 dt = 1. Summarizing previous results, very few papers dealt with the existence of solutions for integral boundary value problems of p-Laplacian fractional differential equations, especially, Zhang and Cui [22] who established the existence of positive solutions for fourth-order singular p-Laplacian differential equations under p ≥ 2 and Jiang [20] who considered the existence of solutions for nonlinear multiterm fractional differential equations with 0 < β ≤ 1. Moreover, due to the nonlinearity of p-Laplacian operator φ p , it is more difficult to study for the case 1 < β ≤ 2 rather than for the case 0 < β ≤ 1. Motivated by the above facts, this paper deals with the existence and uniqueness of solutions of problem (1) in which the fractional derivatives are Caputo fractional derivatives with 1 < α, β ≤ 2.The structure of this paper is organized as follows.
In Section 2, we recall some definitions and lemmas. In Section 3, we prove the existence and uniqueness of solutions for nonlinear integral boundary value problem with p-Laplacian operator by using Schaefer's fixed-point theorem and Banach contraction mapping principle. Finally, we give two examples to illustrate our main results in Section 4.

Preliminaries
The Riemann-Liouville fractional integral and the Caputo fractional derivative of order α > 0 of a function f : ð0, ∞Þ ⟶ R is given by 2 Abstract and Applied Analysis where n = ½α + 1, provided that the right-hand side is pointwise defined on ð0, ∞Þ (see [4,5]).

Lemma 3.
Let σ ∈ C½0, 1. Then, the fractional boundary value problem has a unique solution which is given by where Proof. In view of Lemma 2, we have that By means of the property of the fractional integral of a continuous function, we obtain that zð0Þ = c 0 .
Since zð0Þ = zð1Þ = Ð 1 0 hðsÞzðsÞds, from (12), we obtain Hence, (12) can be written as Thus, we can easily get In the right side of (15), the term Ð 1 0 hðsÞzðsÞds can be rewritten as so we get Therefore, the unique solution of (9) is given by Conversely, let z ∈ C½0, 1 be the function which is expressed by (10). Putting we get 3 Abstract and Applied Analysis Then, we have that Since σ ∈ C½0, 1, applying c D β 0+ to both sides of (21) and using Lemma 1, we can obtain On the other hand, multiplying (10) by hðtÞ and integrating on ½0, 1, we have Hence, (10) can be written as Since G β ð0, sÞ = G β ð1, sÞ = 0, we can also have that Therefore, we can know that zðtÞ is a solution of problem (9) and (9) has a unique solution which is given by (10). The proof is completed.
Lemma 4. Let f ∈ Cð½0, 1 × R, RÞ, then p-Laplacian integral boundary value problem (1) has a unique solution which is given by where q is the number that satisfies 1/p + 1/q = 1.
Proof. The proof of this lemma is divided into two steps.
(Step 1) Let r ∈ C½0, 1 and consider the boundary value problem By using Lemma 2, we have that From (28) and boundary condition of (27) we can obtain Then, (28) can be written as In a similar way to the proof of Lemma 3, it can be easily seen that Hence, we have that the unique solution of (27) is given by Conversely, let xðtÞ be the function which is expressed by (32). Then, from the definition of G α ðt, sÞ, (32) can be written as 4 Abstract and Applied Analysis Since r ∈ C½0, 1 and 1 < α ≤ 2, applying c D α 0+ to both sides of (33), we can obtain On the other hand, multiplying (32) by gðtÞ and integrating on ½0, 1, we have that So, we can rewrite (32) as Since G α ð0, sÞ = G α ð1, sÞ = 0, we can get Therefore, we can know that xðtÞ is the solution of (27). ( Step 2) Now let xðtÞ be the solution of (1). Putting yðtÞ ≔ c D α 0+ xðtÞ, then we have that Also denoting φ p ðyðtÞÞ by zðtÞ, then by Lemma 3, we can see that that Since it is well-known that φ −1 p = φ q , combining (38) and (39) yields The proof is completed.
Remark 5. From the definition of G α ðt, sÞ and G β ðt, sÞ, it is easy to know that those functions are continuous in ½0, 1 × ½0, 1.
Lemma 6 (see [24]). Schaefer's fixed-point theorem. Let X be the Banach space and T : X ⟶ X be completely continuous operator. If the set E ≔ fu ∈ X | u = ρTu, 0 < ρ < 1g is bounded, then T has at least one fixed point in X.
The basic properties of the p-Laplacian operator which will be used in the following studies are listed below (see [10]).

Main Results
In this section, we establish the existence and uniqueness of solutions of problem (1) by using Schaefer's fixed-point theorem and the Banach contraction mapping principle. Let us consider the Banach space X = C½0, 1 endowed with the norm kuk ≔ max 0≤t≤1 juðtÞj.

Abstract and Applied Analysis
Define an operator T : X ⟶ X by Then, Equation (26) is equivalent to the operator equation From Lemma 4, the existence of solutions for the problem (1) refers to the existence of fixed points of Equation (44). Therefore, it is sufficient to prove the existence of fixed points of (44).

Lemma 7. The operator T is completely continuous.
Proof. Since f ∈ Cð½0, 1 × R, RÞ and φ q is continuous, we can know that T : X ⟶ X is continuous.
Let Ω ⊂ X be a bounded subset, then for any u ∈ Ω, there exists M 0 > 0 such that kuk ≤ M 0 .
We will show that TðΩÞ is relatively compact in X. Since f is a continuous function, there exists M f > 0 such that j f ðt, uðtÞÞj ≤ M f , t ∈ ½0, 1, u ∈ Ω. Then, we have Tx t ð Þ j j≤ And since ðt, sÞ ∈ ½0, 1 × ½0, 1, evaluating the upper bound of jG β ðt, sÞj gives So, we obtain Then, we can get easily that In view of the definition of the p-Laplacian operator, we have that This shows that TðΩÞ is uniformly bounded in X. For any x ∈ Ω, 0 ≤ t 1 < t 2 ≤ 1, we have that Abstract and Applied Analysis And since the integral term Ð 1 0 jG α ðt 2 , sÞ − G α ðt 1 , sÞjds can be divided into the following three parts: In a similar way to this, we can obtain These inequalities yield Therefore, we get This shows that TðΩÞ is equicontinuous in X. By using the Arzela-Ascoli theorem, we can see that TðΩÞ is relatively compact in X. As a consequence of the above discussion, the operator T : X ⟶ X is completely continuous. The proof is completed.
Denote as follows: In this article, the following hypotheses will be used.

Abstract and Applied Analysis
Proof. Consider the following set: For any x ∈ E, it can be easily seen that On the other hand, from the condition (i) of the hypothesis (H1) and (49), we have that From (58), we get Since 1/p + 1/q = 1 implies ðp − 1Þðq − 1Þ = 1, we obtain Therefore, we have that and by using the condition (ii) of the hypothesis (H1), we can see that So, we can know that E is bounded. In view of Schaefer's fixed-point theorem (Lemma 6), the operator T : X ⟶ X has at least one fixed point which is the solution of the problem (1). The proof is completed.
Here, put r ≔ ðAkak/1 − AkbkÞ q−1 and list more hypotheses to obtain the uniqueness results for our problem.
Firstly, we will show that TðBÞ ⊂ B. In fact, from the hypothesis (H1) and (49), for any x ∈ B, we have that Next, we will prove the uniqueness of solutions for problem (1).
For any x ∈ B and any t ∈ ½0, 1, by the hypothesis (H1), we get Since 1 < p < 2, we can see q > 2. Hence, from (H2) and one basic property of p-Laplacian operator (42), for any x, y ∈ B and any t ∈ ½0, 1, we have that 8 Abstract and Applied Analysis