Existence of Solutions for Fractional Differential Equations with p-Laplacian Operator and Integral Boundary Conditions

where 1 < α ≤ 2, 0 < β ≤ 1, 0 < ξ, b, λ < 1, D0+ and cD β 0+ are the Caputo fractional derivative. φpðsÞ = jsjp−2s is the p-Laplacian operator such that ð1/pÞ + ð1/qÞ = 1, p > 1, and φ−1 p ðsÞ = φqðsÞ, and f ðt, uÞ: 1⁄20, 1 × 1⁄20,∞Þ⟶ 1⁄20,∞Þ is a given continuous function. In recent years, boundary value problems of fractional differential equations have significantly been discussed by some researchers because fractional calculus theory and methods have been widely used in various fields of natural sciences and social sciences. In the field of physical mechanics, fractional calculus not only provides suitable mathematical tools for the study of soft matter but also provides new research ideas and plays an irreplaceable role in the modeling of soft matter [1–3]. Some nonlinear analysis tools such as coincidence degree theory [4, 5], upper and lower solution method [6–8], fixed point theorems [9–11], and variational methods [12–14] have been widely used to discuss existence of solutions for boundary value problems of fractional differential equations. On the other hand, it is well known that differential equation models with p-Laplacian operators are often used to simulate practical problems such as tides caused by celestial gravity and elastic deformation of beams and rich results of fractional differential equations with a p-Laplacian operator have been obtained [15–18]. In particular, in [15], by using the fixed point theorem, Yan et al. studied the existence of solutions for boundary value problems of fractional differential equations with a p-Laplacian operator:

In recent years, boundary value problems of fractional differential equations have significantly been discussed by some researchers because fractional calculus theory and methods have been widely used in various fields of natural sciences and social sciences. In the field of physical mechanics, fractional calculus not only provides suitable mathematical tools for the study of soft matter but also provides new research ideas and plays an irreplaceable role in the modeling of soft matter [1][2][3]. Some nonlinear analysis tools such as coincidence degree theory [4,5], upper and lower solution method [6][7][8], fixed point theorems [9][10][11], and variational methods [12][13][14] have been widely used to discuss existence of solutions for boundary value problems of fractional differential equations.
On the other hand, it is well known that differential equation models with p-Laplacian operators are often used to simulate practical problems such as tides caused by celestial gravity and elastic deformation of beams and rich results of fractional differential equations with a p-Laplacian operator have been obtained [15][16][17][18]. In particular, in [15], by using the fixed point theorem, Yan et al. studied the existence of solutions for boundary value problems of fractional differential equations with a p-Laplacian operator: where 1 < α ≤ 2, 3 < β ≤ 4, 0 < η < 1, 0 < b < η ð1−αÞ/ðp−1Þ , ð1/ pÞ + ð1/qÞ = 1, 1 < p, ϕ −1 p ðsÞ = ϕ q ðsÞ,D α 0+ , D hot issue for scholars and some good results have been achieved [19][20][21][22][23]. In [24], by using the method of the upper and lower solutions and Schauder's and Banach's fixed points theorem, Abdo et al. obtained the existence and uniqueness of a positive solution of the fractional differential equations with integral boundary equations: where is the standard Caputo derivative, and f : ½0, 1 × ½0,∞Þ ⟶ ½0,∞Þ is a given countinuous function.
Motivated by the works mentioned above, we concentrate on the solutions for the nonlinear fractional differential equation (1). We obtain the existence result of the fractional differential equations with integral boundary equations by using the Schauder fixed point theorem and other mathematical analysis techniques.
The rest of this paper is organized as follows. In Section 2, we give some notations and lemmas. Section 3 is devoted to study existence of solutions for boundary value problems of fractional differential equations. Finally, we provide an example to illustrate our results.

Preliminaries
In the section, we present some definitions and lemmas, which are required for building our theorems.
Definition 1 (see [1]). The fractional integral of order αðα > 0Þ of function f : ½0,∞Þ ⟶ R is given by where ΓðαÞ is the Gamma function, provided the right side is pointwise defined on ð0, +∞Þ.
Lemma 5 (see [1]). Let X be a Banach space and Ω ⊂ X a convex, closed, and bounded set. If T : Ω ⟶ Ω is a continuous operator such that TΩ ⊂ X, TΩ is relatively compact, then T has at least one fixed point in Ω. Let ϕ p ð c D α 0+ uðtÞÞ = vðtÞ, then vð1Þ = b p−1 vðξÞ. We now consider the following equations: Lemma 6. Lety ∈ C½0, 1, then (7) has a unique solution v t ð Þ = where

Journal of Function Spaces
Proof. Suppose v satisfies boundary value problem (7), by (i) of Lemma 4, we can obtain Using the boundary condition vð1Þ = b p−1 vðξÞ, we can obtain From the above analysis, the equation is equivalent to Lemma 7. Lety ∈ C½0, 1. Then (14) has a unique solution: where Proof. By (i) of Lemma (4), we can obtain uðtÞ = ð1/ΓðαÞÞ Hðs, τÞyðτÞdτÞds − c 2 , using the boundary condition u ′ ð1Þ = 0, we can obtain Another, because we have Now, we express Ð 1 0 uðsÞds , let

Journal of Function Spaces
We obtain Therefore, Reverse, if by (ii) of Lemma (4), we can obtain that uðtÞ is a solution of (14). The proof is completed.

Main Results
In this section, we will show the existence results for boundary value problem (1) by the Schauder fixed point theorem. Let I = ½0, 1, U = fuðtÞ | uðtÞ ∈ CðIÞg, and definite the normkuk = max t∈½0,1 j uðtÞj, ðU, k•kÞ is a Banach space.
then the problem (1) has at least one solution.