Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative

Li, Yaning; Zhang, Quanguo; Sun, Baoyan

doi:https://doi.org/10.1155/2020/7139795

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Research Article | Open Access

Volume 2020 | Article ID 7139795 | https://doi.org/10.1155/2020/7139795

Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative

Yaning Li,¹Quanguo Zhang,²and Baoyan Sun³

Academic Editor: Liguang Wang

Received25 Mar 2020

Accepted24 Apr 2020

Published13 May 2020

Abstract

In this paper, we deal with two fractional boundary value problems which have linear growth and quadratic growth about the fractional derivative in the nonlinearity term. By using variational methods coupled with the iterative methods, we obtain the existence results of solutions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term.

1. Introduction

It has been seen that fractional differential equations have better effects in many realistic applications than the classical ones. Qualitative theory and its applications in physics, engineering, economics, biology, and ecology are extensively discussed and demonstrated in [1–4] and the references therein. Some recent contributions to the theory of fractional differential equation can be seen in [5–10].

Some classical tools such as fixed point theorems [5], the method of upper and lower solutions, and monotone iterative technique [11, 12] have been widely used to study the fractional differential equation. Recently, the study of fractional differential equations has attracted much attention by using variational methods, for example, [7, 8, 13–19]. We also mention that in the recent works [20, 21], the authors have developed a general approach concerting the existence of solutions.

Fractional differential equations containing left and right fractional differential operators have received attention from scientists due to their applications in physical phenomena exhibiting anomalous diffusion. In [7], appropriate fractional derivative spaces were defined and existence and uniqueness results for a fractional boundary value problem were proven using the Lax-Milgram theorem.

Jiao and Zhou [8] showed the variational structure of a fractional boundary value problem under an appropriate functional space; they used the least action principle and the Mountain Pass theorem to obtain the existence of at least one solution. Sun and Zhang [22] obtained the existence result for a fractional boundary value problem by using the Mountain Pass method and an iterative technique. In [23], the authors discussed the existence of a fractional boundary value problem with linear growth about the fractional derivative in a nonlinearity term. Compared with some integral-order partial differential equations such as [6, 24–31], the fractional derivatives have hereditary and nonlocal properties so that they are much more suitable for describing long-memory processes than the classical integer-order derivatives.

Motivated by the above papers, in this paper, we first investigated the existence of solutions for the following fractional boundary value problems: where and are the left and right fractional Riemann-Liouville derivatives of order , respectively, .

Note that problem (1) is not variational due to the fractional derivative contained in nonlinearity, so we cannot find a functional such that its critical point is the weak solution corresponding to (1). In order to overcome this difficulty, we consider the following fractional boundary value problem which is independent on the fractional derivative of the solution where is an element of fractional Sobolev space . First, by using variational methods, we obtain the existence of solutions for (2). Then, under the assumption that is linear growth about the fractional derivative and based on iterative methods, we show there exists a solution for (1). Our conditions are weaker than that in [23].

We also discuss the following fractional boundary value problem: where , is a parameter, and . Compared with (1), the nonlinearity of (3) is quadratic growth about a fractional derivative. By using variational methods and an iterative technique, we obtain that there exists solutions for (3) when and satisfy suitable conditions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term .

The paper is organized as follows. In Section 2, we will list some important properties of the basic functional space. We show the existence results for (1) and (3) in Section 3 and Section 4, respectively.

2. Preliminary

Let us briefly recall the property of a fractional derivative which will be used to construct the variational functional.

Lemma 1 [1]. For , if with , is absolutely continuous, and is absolutely continuous with , then

Now, we recall some properties of the basic function space which have been studied in [32].

Throughout this paper, let .

The fractional derivative space is defined by the completion of with respect to the norm , where is the α-order left Riemann-Liouville fractional derivative. Then, is a reflexive and separable Hilbert space. And the Riemann-Liouville fractional derivative exists for the elements in [22].

Lemma 2 [32]. For all , we have

According to (5), one can consider with respect to the equivalent norm

Lemma 3 [32]. If the sequence converges weakly to in , i.e., , then in , i.e., as .
By the proof of Proposition 4.1 in [8], we have the following property.

Lemma 4. For any ,

3. Existence Result for (1)

We assume that satisfies the following conditions:

is measurable in for every and continuous in for , and there exist , such that for all .

There are constants and such that, for , where .

uniformly for and .

In order to derive a weak solution of (1), we suppose that is a solution of (1), and multiplying (1) by an arbitrary and by Lemma 1, we have

Since (11) is well defined for , the weak solution of (1) may be defined as follows.

Definition 5. A weak solution of (1) is a function such that for every .

Definition 6. A function is called a solution of (1), if and exist, is derivable for almost every , and satisfies (1).

For a given , we consider the functional , defined by

In view of assumption , we know that is continuously differentiable and for , . Hence, a critical point of gives us a weak solution of (2).

Lemma 7. If is a weak solution of (1), then is also a solution of (1).

Proof. Let be a weak solution of (1), then , so and exist and . For every , we have Since , we have and , so Then, there exists a constant such that thus,

Lemma 8. Suppose and hold, then satisfies the (PS) condition.

Proof. Let , is bounded, and ; we first show that is bounded.

It follows from that which implies that is bounded.

From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call , then . Next, we will prove that strongly converges to . By , we know that

Note that

By and , we obtain that

Thus, . Therefore, satisfies the (PS) condition.

Lemma 9. Let and suppose , , and hold, then (2) has at least one nontrivial solution.

Proof. The proof relies on the Mountain Pass theorem [33, 34]. It is clear that , , and satisfies the (PS) condition from Lemma 8. By , for all , there is a such that

Let and choose ; by the above inequality and (6), we have

It suffices to choose and to get . Thus, there exist positive numbers and which are independent of such that for satisfies .

It follows from that there exist such that

Choosing satisfies , and we obtain which implies that as . Hence, we obtain that there exists a independent of and such that for all .

The above discussions combined with the Mountain Pass theorem show that (2) has at least one nontrivial solution which can be characterized as where .

In order to obtain the existence of solutions for (1), we need the following Lipschitz condition.

There exist such that where ; will be determined later.

Theorem 10. Let , suppose (H₀)–(H₃) hold, , and , then problem (1) has a nontrivial solution.

Proof. For and , we construct a sequence , where is the solution of the following problem:

Now, we assume ; by the mathematical induction, we will prove that . It follows from satisfying (29) that

Let then can achieve its maximum at , so . By and , we have

Then,

Thus, we can choose .

Since and , then , where is given by . From , we have

Combining the above estimates with (8) and (5), we obtain that is . Since , the above inequality implies that is a Cauchy sequence in . Thus, there is a such that converges strongly to in .

Now, we show that . By the proof of Lemma 9, we know .

If , then, and since , we obtain . In order to show , we only need to show . In fact,

Next, we show that for any ,

It remains only to show

Note that where is a constant. Thus, we obtain a nontrivial solution of problem (1).

4. Existence Result for (3)

In Section 3, condition implies that the nonlinearity is linear growth about a fractional derivative of solutions; this section will consider the fractional boundary value problem (3) in which the nonlinearity is quadratic growth about the fractional derivative of solutions.

Assume that satisfies the following conditions:

There are constants and such that, for, where .

Similar to Section 3, since (3) is not variational, given , we consider the following problem which is independent on the fractional derivative of the solution:

Then, the corresponding functional is given by where .

implies that there are constants such that

By , there is a such that implies

For convenience of our statement, let us give some notations and denote where is given in and is given in (43). Assuming , let

For a fixed function with , we denote

For , denote

We also need the following Lipschitz condition:

There exists such that for . Then we have the following result.

Theorem 11. Suppose – hold, ; if there exists, such that Then, (3) has a nontrivial weak solution.

Proof. We first verify that for a given with , (40) has at least a nontrivial weak solution.
In order to use the Mountain Pass theorem, we first show that satisfies the (PS) condition. Let , is bounded, and , we show that is bounded.
In fact, which implies that is bounded, and similar to the last part of Lemma 8, we get that satisfies the (PS) condition.
Let and choose with , then . From (43), (6), and (49), we have Then, we obtain that there exists such that for , uniformly for with .
Let with , then which implies there exists such that Then, from the Mountain Pass theorem, we get that has a nontrivial weak solution which can be characterized as where .
Let , we can obtain that has a nontrivial critical point . For , we construct a sequence , where is the critical point of . Now, we assume that ; by the mathematical induction, we will prove that . In fact, by (54), we have where is given in (47).
Hence, So, By (49), we obtain That is When , where and are given in (45), we have Then, Therefore, the above argument implies that .
Finally, we show that is convergent to a nontrivial solution of (3).
Since , we have, then by , we obtain So From (50), we know that is a Cauchy sequence in and is a weak solution of (3). Since for and β does not depend on , we obtain that is a nontrivial weak solution of (3).

Corollary 12. Suppose - hold; if the right-hand side of (50) is greater than 0, then there exists a constant , such that (1) has a solution when .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by NSF of China (11626132, 11801276) and the Scientific Research Foundation of Yantai University (2219008).

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Copyright © 2020 Yaning Li 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.

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