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Research Article | Open Access

Volume 2019 |Article ID 2381530 | https://doi.org/10.1155/2019/2381530

Keyu Zhang, Donal O’Regan, Jiafa Xu, Zhengqing Fu, "Nontrivial Solutions for a Higher Order Nonlinear Fractional Boundary Value Problem Involving Riemann-Liouville Fractional Derivatives", Journal of Function Spaces, vol. 2019, Article ID 2381530, 11 pages, 2019. https://doi.org/10.1155/2019/2381530

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

Academic Editor: Raúl E. Curto
Revised11 Apr 2019
Accepted14 Apr 2019
Published19 Jun 2019

#### 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 [135] 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 problemwhere 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 [29]. 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:andIn [1014], 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 [714, 18, 19, 22, 23, 2633, 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 [3639], 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 [4042].

Definition 1 (see [4042]). 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 [4042]). 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 haveThis 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 thatHence, we obtainUsing 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 thatfor all .
For any given with , and using (H3) there exists such thatHence, (H2), (35), and (36) enable us to obtain for all , .
Let Then we obtainfor all . Note that can be chosen arbitrarily small, and let where .
Now we claimwhere is the positive eigenfunction of corresponding to the eigenvalue , i.e., , and from (30), .
Suppose (41) is false. Then there exist and such thatLetThen we haveConsequently, we have Note from (43) and (25), , and then implies that Then we haveTherefore, (26), (39), and (43) enable us to obtain