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

1. Introduction

In this paper, we consider the nonlinear fractional differential equations with a p-Laplacian operator and integral boundary conditions

[mathematical expression not reproducible], (1)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] are the Caputo fractional derivative.

[mathematical expression not reproducible] is the p-Laplacian operator such that (1/p) + (1/q) = 1, p > 1, and [[phi].sup.-1.sub.p](s) = [[phi].sub.q](s), and f(t, u): [0, 1] x [0, [infinity]) [right arrow] [0, [infinity]) 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 , 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:

[mathematical expression not reproducible], (2)

where 1 < [alpha] < 2, 3 < [beta] [less than or equal to] 4, 0 < [eta] < 1, 0 < b < [[eta].sup.(1-a)/(p-1)], (1/p) + (1/q) = 1, 1 < p, [[phi].sup.-1.sub.p](s) = [[phi].sub.q](s), [D.sup.[alpha].sub.0+], [D.sup.[beta].sub.p+] is the standard Riemann-Liouville derivative, and f (t, u): (0,1) * (0, [infinity]) [right arrow] [0, [infinity]) is a given countinuous function.

Moreover, during the last decade, the integral boundary value problem of fractional differential equations is also a hot issue for scholars and some good results have been achieved [19-23]. In , 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:

[mathematical expression not reproducible], (3)

where [mathematical expression not reproducible] is the standard Caputo derivative, and f : [0,1] * [0, [infinity]) [right arrow] [0, [infinity]) is a given countinuous function.

In , Bai and Qiu discuss the existence of positive solutions for boundary value problems of fractional differential equations:

[mathematical expression not reproducible], (4)

where 2 < [alpha] [less than or equal to] 3, f [member of] C([0,1] * [0, +[infinity]), [0, +[infinity])), and [mathematical expression not reproducible] is the standard Caputo derivative.

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.

2. Preliminaries

In the section, we present some definitions and lemmas, which are required for building our theorems.

Definition 1 (see ). The fractional integral of order [alpha]([alpha] > 0) of function f : [0, [infinity]) [right arrow] R is given by

[mathematical expression not reproducible], (5)

where [GAMMA]([alpha]) is the Gamma function, provided the right side is pointwise defined on (0, +[infinity]).

Definition 2 (see ). The Caputo fractional derivative of order [alpha]([alpha] > 0) of function f : [0, [infinity]) [right arrow] R is given by

[mathematical expression not reproducible], (6)

where t > 0, n = [[alpha]] + 1, [GAMMA]([alpha]) is the Gamma function.

Lemma 3 (see ). For [alpha] > 0, the solution of fractional differential equation [mathematical expression not reproducible] is given by u(t) = [c.sub.0] + [c.sub.1]t + ... + [c.sub.n-1][t.sup.n-1], [c.sub.i] [member of] R, i = 0, 1, 2, ..., n - 1, n = [[alpha]] + 1, and [[alpha]] denotes the integer part of the real number [alpha].

Lemma 4 (see ). For [alpha] > 0, then [mathematical expression not reproducible] and (ii) [mathematical expression not reproducible].

Lemma 5 (see ). Let * be a Banach space and [OMEGA] [subset] * a convex, closed, and bounded set. If T : [OMEGA] [right arrow] [OMEGA] is a continuous operator such that T[OMEGA] [subset] X, T[OMEGA] is relatively compact, then T has at least one fixed point in [OMEGA].

Let [mathematical expression not reproducible], then v(1) = [b.sup.p-1]v([xi]). We now consider the following equations:

[mathematical expression not reproducible]. (7)

Lemma 6. Lety [member of] C[0,1], then (7) has a unique solution

[mathematical expression not reproducible], (8)

where

[mathematical expression not reproducible]. (9)

Proof. Suppose v satisfies boundary value problem (7), by (i) of Lemma 4, we can obtain

[mathematical expression not reproducible]. (10)

Using the boundary condition v(1) = [b.sup.p-1]v([xi]), we can obtain

[mathematical expression not reproducible]. (11)

Thus,

[mathematical expression not reproducible]. (12)

From the above analysis, the equation

[mathematical expression not reproducible] (13)

is equivalent to

[mathematical expression not reproducible]. (14)

Lemma 7. Lety [member of] C[0,1]. Then (14) has a unique solution:

[mathematical expression not reproducible], (15)

where

[mathematical expression not reproducible]. (16)

Proof. By (i) of Lemma (4), we can obtain [mathematical expression not reproducible]. [mathematical expression not reproducible], using the boundary condition u1 (1) =0, we can obtain

[mathematical expression not reproducible]. (17)

Another, because

[mathematical expression not reproducible], (18)

[mathematical expression not reproducible], (19)

we have

[mathematical expression not reproducible]. (20)

Now, we express [mathematical expression not reproducible] let

[mathematical expression not reproducible]. (21)

We obtain

[mathematical expression not reproducible]. (22)

Therefore,

[mathematical expression not reproducible]. (23)

Reverse, if

[mathematical expression not reproducible], (24)

by (ii) of Lemma (4), we can obtain that u(t) is a solution of (14).

The proof is completed.

Lemma 8. The functions H is continuous on [0,1] * [0,1] and has the following properties:

(1) H(t, [tau]) [less than or equal to] H([tau], [tau]), t, [tau] [member of] [0,1]

(2) [mathematical expression not reproducible]

Proof. (1) For any t, [tau] [member of] [0,1], by (9), it is obvious that H (t, [tau]) [less than or equal to] H([tau], [tau]). (2) For any t, [tau] [member of] [0,1], by (9), we conclude that

[mathematical expression not reproducible]. (25)

Therefore,

[mathematical expression not reproducible]. (26)

This completes the proof.

3. Main Results

In this section, we will show the existence results for boundary value problem (1) by the Schauder fixed point theorem.

Let I = [0,1], U = {u(t) | u(t) [member of] C(I)}, and definite the norm[parallel]u[parallel] = [max.sub.t[member of][0,1]][absolute value of (u(t))], (U, [parallel]*[parallel]) is a Banach space.

Theorem 9. Assume that the following conditions (H1) and ([H.sub.2]) are satisfied:

([H.sup.1]) f(t, u): [0,1] * [0, [infinity]) [right arrow] [0, [infinity]) is continuous

([H.sub.2]) There exists a constant k > 0, satisfying f (t, u) [less than or equal to] L[[phi].sub.p] (u), t [member of] [0,1], [parallel]u[parallel] [less than or equal to] k, where

[mathematical expression not reproducible], (27)

then the problem (1) has at least one solution.

Proof. Let P = {u(t) | [parallel]u(t)[parallel] < k, t [member of] [0,1]}; thus, P [subset] U is convex, bounded, and closed.

Define an operator T : P [right arrow] U by

[mathematical expression not reproducible]. (28)

For any u [member of] P, then by ([H.sub.2]), we have

f(t, u) [less than or equal to] L[[phi].sub.p](u) [less than or equal to] L[[phi].sub.p](k). (29)

By Lemma (8), we conclude that

[mathematical expression not reproducible]. (30)

Thus, T(P) [??] P. By ([H.sub.2]), we have

[mathematical expression not reproducible]. (31)

We express

[mathematical expression not reproducible]. (32)

For each [t.sub.1], [t.sub.2] [member of] [0, 1], [t.sub.1] < [t.sub.2], we get

[mathematical expression not reproducible]. (33)

As [t.sub.2] [right arrow] [t.sub.1], the right-hand side of the previous inequality is independent of u and tends to zero; thus, T(P) is equi-continuous. From the Arzela-Ascoli Theorem, T is compact. Applying Schauder's fixed point theorem, T has at least one fixed point u [member of] P. Therefore, the problem (1) has at least one positive solution u in P.

4. Applications

In this section, we will give an example to illustrate our main results.

Example 1. Consider the following equation:

[mathematical expression not reproducible], (34)

where [beta] = 1/2, [alpha] = 3/2, p = 4/3, q = 4, [lambda] = [xi] = b = 1/2, and f (t, u) = [[phi].sub.p][absolute value of (u(t))]/[(t + 5).sup.2]. Since f is continuous and

[absolute value of (f (t, u))] [less than or equal to] 1/25 [[phi].sub.p] [absolute value of (u(t))], (35)

for (t, u) [member of] [0,1] * [0, [infinity]), we obtain

L = 1/25. (36)

Then,

[mathematical expression not reproducible]. (37)

It is obvious that

[mathematical expression not reproducible]. (38)

By Theorem (9), we conclude that the problem (34) has at least one solution.

https://doi.org/10.1155/2020/4739175

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors' Contributions

Both authors made an equal contribution.

Acknowledgments

This research is funded by the National Natural Science Foundation of China (No:11661037) and Scientific Research Fund of Jishou University (No:Jdy19004).

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Jingli Xie [ID] and Lijing Duan [ID]

College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China

Correspondence should be addressed to Lijing Duan; duanlijing1995@163.com

Received 10 February 2020; Revised 17 April 2020; Accepted 1 May 2020; Published 10 June 2020

Academic Editor: Liguang Wang
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