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

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.


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][2][3][4] and the references therein. Some recent contributions to the theory of fractional differential equation can be seen in [5][6][7][8][9][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][14][15][16][17][18][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][25][26][27][28][29][30][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: Þ , a:e:t ∈ 0, T ½ , where 0 D α t and t D α T are the left and right fractional Riemann-Liouville derivatives of order 1/2 < α < 1, respectively, f : ½0, 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 w is an element of fractional Sobolev space E α . First, by using variational methods, we obtain the existence of solutions for (2). Then, under the assumption that f 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 a ∈ L 1 ð0, T ; ℝ + Þ, λ is a parameter, and g ∈ Cðℝ ; ℝÞ. 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 λ and a 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.

Preliminary
Let us briefly recall the property of a fractional derivative which will be used to construct the variational functional.
is absolutely continuous, and g : ð½a, b, ℝÞ is absolutely continuous with gðbÞ = 0, then Now, we recall some properties of the basic function space which have been studied in [32].
Throughout this paper, let 1/2 < α < 1. The fractional derivative space E α is defined by the completion of C ∞ 0 ð0, TÞ with respect to the norm kuk = ð Ð T 0 juðtÞj 2 dt + Ð T 0 j 0 D α t uðtÞj 2 dtÞ 1/2 , where 0 D α t is the α-order left Riemann-Liouville fractional derivative. Then, E α is a reflexive and separable Hilbert space. And the Riemann-Liouville fractional derivative exists for the elements in E α [22]. Lemma 2 [32]. For all u ∈ E α , we have According to (5), one can consider E α with respect to the equivalent norm Lemma 3 [32]. If the sequence fu k g converges weakly to u in By the proof of Proposition 4.1 in [8], we have the following property.

Journal of Function Spaces
In order to derive a weak solution of (1), we suppose that u is a solution of (1), and multiplying (1) by an arbitrary v ∈ C ∞ 0 ð0, TÞ and by Lemma 1, we have Since (11) is well defined for u, v ∈ E a , the weak solution of (1) may be defined as follows.
for every v ∈ E α .
For a given w ∈ E α , we consider the functional φ w : E α ⟶ ℝ, defined by In view of assumption ðH 0 Þ, we know that φ is continuously differentiable and for u, v ∈ E α . Hence, a critical point of φ w gives us a weak solution of (2).

Lemma 7.
If u is a weak solution of (1), then u is also a solution of (1).
Proof. Let u be a weak solution of (1), then u ∈ E α , so 0 D t α uðtÞ and t D T α uðtÞ exist and uð0Þ = uðTÞ = 0. Since Then, there exists a constant C such that thus, Lemma 8. Suppose ðH 0 Þ and ðH 1 Þ hold, then φ w satisfies the (PS) condition.
Proof. Let fu n g ⊂ E α , fφ w ðu n Þg is bounded, and φ w ′ ðu n Þ ⟶ 0; we first show that fu n g is bounded.
It follows from ðH 1 Þ that which implies that fu n g is bounded. 3

Journal of Function Spaces
From the reflexivity of E α , we may extract a weakly convergent subsequence that, for simplicity, we call fu n g, u n ⇀ u then ku n − uk ∞ ⟶ 0. Next, we will prove that fu n g strongly converges to u. By ðH 0 Þ, we know that Note that By φ ′ ðu n Þ ⟶ 0 and u n ⇀ u, we obtain that Thus, ku n − uk α ⟶ 0 as n⟶∞. Therefore, φ w satisfies the (PS) condition.
It follows from ðH 1 Þ that there exist c 1 , c 2 > 0 such that Choosingũ ∈ E α satisfies kũk α = 1, and we obtain which implies that φðrũÞ ⟶ −∞ as r ⟶ ∞. Hence, we obtain that there exists a β > 0 independent of u 1 and w such that φ w ðuÞ ≤ 0 for all kuk α > β.
The above discussions combined with the Mountain Pass theorem show that (2) has at least one nontrivial solution u w which can be characterized as where Γ = fg ∈ Cð½0, 1, E α Þjgð0Þ = 0, gð1Þ = βũg. In order to obtain the existence of solutions for (1), we need the following Lipschitz condition. ðH 3 Þ There exist L 1 , L 2 > 0 such that where r 1 = c 2 T α−ð1/2Þ /ΓðαÞð2α − 1Þ 1/2 ; C 2 will be determined later.
If φðu n Þ ⟶ φðuÞ, thenφðuÞ > σ, and since φð0Þ = 0, we obtain u≡0. In order to show φðu n Þ ⟶ φðuÞ, we only need to show Next, we show that for any v ∈ E α , 5 Journal of Function Spaces It remains only to show Note that where C is a constant. Thus, we obtain a nontrivial solution of problem (1).

Existence Result for (3)
In Section 3, condition ðH 3 Þ 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.
Proof. We first verify that for a given w ∈ E α with kwk α ≤ R 1 , (40) has at least a nontrivial weak solution.
In order to use the Mountain Pass theorem, we first show that ϕ satisfies the (PS) condition. Let fu n g ⊂ E α , fϕ w ðu n Þg is bounded, and ϕ w ′ ðu n Þ ⟶ 0, we show that fu n g is bounded.
In fact,