Journal of Function Spaces
Volume 2019 (2019), Article ID 2381530, 11 pages
https://doi.org/10.1155/2019/2381530

Research Article

Nontrivial Solutions for a Higher Order Nonlinear Fractional Boundary Value Problem Involving Riemann-Liouville Fractional Derivatives

Keyu Zhang,¹ Donal O’Regan,² Jiafa Xu,³ and Zhengqing Fu⁴

¹School of Mathematics, Qilu Normal University, Jinan 250013, China
²School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
³School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
⁴College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Keyu Zhang; keyu_292@163.com

Received 18 March 2019; Revised 11 April 2019; Accepted 14 April 2019; Published 19 June 2019

Academic Editor: Raúl E. Curto

Copyright © 2019 Keyu Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper using topological degree we study the existence of nontrivial solutions for a higher order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives. Here, the nonlinearity can be sign-changing and can also depend on the derivatives of unknown functions.

1. Introduction

In this paper, we investigate the existence of nontrivial solutions for the higher order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives:

where , are the Riemann-Liouville fractional derivatives, , and the function is continuous; here .

There are a large number of papers in the literature studying fractional differential equations because of applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, and control theory; we refer the reader to [1–35] and the references therein. In [1], using fixed point index theory in cones, the authors studied the existence and multiplicity of positive solutions for the nonlocal fractional differential equation boundary value problem

where is a nonnegative continuous function on and satisfies the conditions:

sublinear growth case: , uniformly on ,

superlinear growth case: , uniformly on ,

where is the first eigenvalue for the problem:

Conditions similar to and can be found in [2–9]. In [2], the authors studied the existence of positive solutions for the nonlinear fractional differential equation:

where is a linear functional on given by a Stieltjes integral. The nonlinearity satisfies the conditions:

and

In [10–14], the authors used Lipschitz conditions to study the existence and uniqueness of solutions for various fractional differential equations; for example, in [10], the author used the -positive operator method to obtain the unique solution for the fractional boundary value problem:

where is a Lipschitz continuous function, with the Lipschitz constant associated with the first eigenvalue for the relevant operator. In [15], the authors used the mixed monotone method to obtain a unique positive solution for the fractional boundary value problem:

where the nonlinearity and have different monotone properties. In [16, 17], the authors extended the methods in [15] to fractional -Laplacian equations and established some appropriate iterative sequences which converge to their solutions.

There are also some papers in the literature devoted to sign-changing nonlinearities; we refer the reader to [7–14, 18, 19, 22, 23, 26–33, 35]. For example, in [18] using coincidence degree theory, the author investigated the existence of nontrivial solutions for the coupled system of nonlinear fractional differential equations:

where satisfy Carathéodory conditions.

Motivated by the aforementioned papers and some integer order equations [36–39], in this paper we use topological degree to study the existence of nontrivial solutions for the higher order nonlinear fractional boundary value problem (1). Our nonlinearity can be sign-changing and can also depend on the derivatives of unknown functions. Moreover, our conditions improve the conditions in (5) and (6); see (H4)-(H5) in Section 3.

2. Preliminaries

We present some basic material involving Riemann-Liouville fractional derivatives; for details we refer the reader to the books [40–42].

Definition 1 (see [40–42]). The Riemann-Liouville fractional derivative of order of a continuous function is given by

where , denotes the integer part of number , provided that the right side is pointwise defined on .

Definition 2 (see [40–42]). The Riemann-Liouville fractional integral of order of a function is given by

provided that the right side is pointwise defined on .

Lemma 3 (see [3, Lemma  2.2]). Let , then for , we have

where is the smallest integer greater than or equal to .

In what follows, we present Green’s function associated with problem (1). The following lemma is from [3]; for completeness we provide the proof.

Lemma 4 (see [3, Lemma  2.3]). Let . Then (1) can be transformed into the following nonlinear integrodifferential equation:

Consequently, we obtain that (13) is equivalent to the following Hammerstein type integral equation:

where

Proof. Let . Then from [4, equations (2.11)], we have

Also we obtain , and . Hence, by , we have (13). Consequently, Lemma 3 implies that (13) can be reduced to an equivalent integral equation:

for some . From the condition , we have . Then we obtain

where we use to replace . Then solves the above equation, and we obtain

Therefore, we have

This completes the proof.

Lemma 5 (see [3, Lemma  2.4]). The function has the following properties:
(i) is continuous,
(ii) for all , , where .
Let Then is a real Banach space, and are cones on . From Lemma 4, we can define an operator as follows:

where is denoted by (15). Note that our functions are continuous, so the operator is a completely continuous operator. Moreover if there is a as a fixed point of , i.e., (13) has s solution , then from , we have that is a solution for (1). Therefore we study the existence of fixed points of the operator . For nonnegative real numbers which are not all zero, let

where for , and are

Let denote the spectral radius of , and from Gelfand’s theorem we can prove (the proof is standard; see [6, Lemma 5]).
Define an operator as follows:

Then from Lemma 5(ii), we have

Lemma 6 (see [43, Theorem  19.3]). Let be a reproducing cone in a real Banach space and let be a compact linear operator with . Let be the spectral radius of . If , then there exists such that .
Therefore, from Lemma 6 there exists such that

Hence, we obtain

Using Lemma 5(ii) and the definitions of , we have

and

This implies

Lemma 7 (see [44]). Let be a Banach space and a bounded open set in . Suppose that is a continuous compact operator. If there exists such that

then the topological degree .

Lemma 8 (see [44]). Let be a Banach space and a bounded open set in with . Suppose that is a continuous compact operator. If

then the topological degree .

3. Main Results

Let be two families of nonnegative real numbers which are not all zero. Now we list our assumptions for as follows:

(H1) is continuous,

(H2) there exist with and such that

(H3) /

,

(H4) , , … , /, uniformly for ,

(H5) , , … , /, uniformly for , where

Theorem 9. Assume that (H1)-(H5) are satisfied. Then (1) has at least one nontrivial solution.

Proof. From (H4), there exist and such that

for all .
For any given with , and using (H3) there exists such that

Hence, (H2), (35), and (36) enable us to obtain

for all , .
Let

Then we obtain

for all . Note that can be chosen arbitrarily small, and let

where .
Now we claim

where is the positive eigenfunction of corresponding to the eigenvalue , i.e., , and from (30), .
Suppose (41) is false. Then there exist and such that

Let

Then we have

Consequently, we have

Note from (43) and (25), , and then implies that

Then we have

Therefore, (26), (39), and (43) enable us to obtain

Consequently, we obtain

From (47), , , and from (44), we have

Now from (48), (49), and (50), we find

Therefore, using (42) and (51), we have

Define . Then we have and . From , we obtain

Hence

which contradicts the definition of . Therefore, (41) holds, and from Lemma 7, we obtain

From (H5), there exist and such that

for all with Consequently, we obtain

Now for this , we claim

Suppose the contrary. Then there exist and such that Let Then and

By induction, we have , for As a result, we have

Therefore, by Gelfand’s theorem, we have

This contradicts

Hence (58) holds, and from Lemma 8, we have

Now (55) and (63) together imply that

Therefore the operator has at least one fixed point in . Equivalently, (1) has at least one nontrivial solution. This completes the proof.

Example 10. For convenience in our example, we let . Let

where , , and , with . Let , , and . We now show that conditions (H1)-(H5) are satisfied. Indeed, for the function , we obtain

and hence, (H3) holds. For the function , it is clear that (H1)-(H2) hold. Also note we have

and

Then (H4)-(H5) hold. Therefore, all the conditions in Theorem 9 are satisfied, and thus (1) has at least one nontrivial solution.

Now, we consider a special case of this example. Let , , and consider the operators

Then for all , from Lemma 5(ii), we have

and so

For all , from Lemma 5(ii), we have

and so

From Gelfand’s theorem, we obtain that the spectral radius of , denoted by , satisfies

Note the definition of , and from Lemma 5(ii), we have

Thus , for . Using a similar method, we have that the spectral radius of , denoted by , satisfies

Consequently, the spectral radius of the operator satisfies

Thus

Now take .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by Talent Project of Chongqing Normal University (Grant No. 02030307-0040), the National Natural Science Foundation of China (Grant No. 11601048), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0181), Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24), and the Natural Science Foundation of Shandong Province (ZR2018MA009, ZR2015AM014).

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