Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory

and Applied Analysis 3 A3 lim inf|x|→ ∞ ∇F t, x , x − 2F t, x /|x|μ ≥ Q > 0 uniformly for some Q > 0 and a.e. t ∈ 0, T , where r > 2 and μ > r − 2. We state our first existence result as follows. Theorem 1.1. Assume that (A1)–(A3) hold and that F t, x satisfies the condition (A). Then BVP 1.1 has at least one solution on E. 1.2. The Asymptotically Quadratic Case For the asymptotically quadratic case, we assume the following. A2′ lim sup|x|→ ∞F t, x /|x| ≤ M < ∞ uniformly for some M > 0 and a.e. t ∈ 0, T . A4 There exists τ t ∈ L1 0, T ;R such that ∇F t, x , x −2F t, x ≥ τ t for all x ∈ R and a.e. t ∈ 0, T . A5 lim|x|→ ∞ ∇F t, x , x − 2F t, x ∞ for a.e. t ∈ 0, T . Our second and third main results read as follows. Theorem 1.2. Assume that F t, x satisfies (A), (A1), (A2’), (A4), and (A5). Then BVP 1.1 has at least one solution on E. Theorem 1.3. Assume that F t, x satisfies (A), (A1), (A2’), and the following conditions: A4′ there exists τ t ∈ L1 0, T ;R such that ∇F t, x , x − 2F t, x ≤ τ t for all x ∈ R and a.e. t ∈ 0, T ; A5′ lim|x|→ ∞ ∇F t, x , x − 2F t, x −∞ for a.e. t ∈ 0, T . Then BVP 1.1 has at least one solution on E. 1.3. The Subquadratic Case For the subquadratic case, we give the following multiplicity result. Theorem 1.4. Assume that F t, x satisfies the following assumption: A6 F t, x : a t |x|γ , where a t ∈ L∞ 0, T ;R and 1 < γ < 2 is a constant. Then BVP 1.1 has infinitely many solutions on E. 2. Preliminaries In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space E are also listed for the convenience of readers. 4 Abstract and Applied Analysis Definition 2.1 see 1 . Let f t be a function defined on a, b and q > 0. The left and right Riemann-Liouville fractional integrals of order q for function f t denoted by aD −q t f t and tD −q b f t , respectively, are defined by aD −q t f t 1 Γ ( q ) ∫ t


Introduction and Main Results
Consider the fractional boundary value problem BVP for short of the following form: T are the left and right Riemann-Liouville fractional integrals of order 0 ≤ β < 1, respectively, F : 0, T × R N → R satisfies the following assumptions.A F t, x is measurable in t for every x ∈ R N and continuously differentiable in x for a.e.t ∈ 0, T , and there exist a ∈ C R , R , b ∈ L 1 0, T; R , such that A3 lim inf |x| → ∞ ∇F t, x , x − 2F t, x /|x| μ ≥ Q > 0 uniformly for some Q > 0 and a.e.t ∈ 0, T , where r > 2 and μ > r − 2. We state our first existence result as follows.
Theorem 1.1.Assume that (A1)-(A3) hold and that F t, x satisfies the condition (A).Then BVP 1.1 has at least one solution on E α .

The Asymptotically Quadratic Case
For the asymptotically quadratic case, we assume the following.
A4 There exists τ t ∈ L 1 0, T; R such that ∇F t, x , x − 2F t, x ≥ τ t for all x ∈ R N and a.e.t ∈ 0, T .
Our second and third main results read as follows.
Then BVP 1.1 has at least one solution on E α .

The Subquadratic Case
For the subquadratic case, we give the following multiplicity result.
Theorem 1.4.Assume that F t, x satisfies the following assumption: A6 F t, x : a t |x| γ , where a t ∈ L ∞ 0, T; R and 1 < γ < 2 is a constant.
Then BVP 1.1 has infinitely many solutions on E α .

Preliminaries
In this section, we recall some background materials in fractional differential equation and critical point theory.The properties of space E α are also listed for the convenience of readers.
Definition 2.1 see 1 .Let f t be a function defined on a, b and q > 0. The left and right Riemann-Liouville fractional integrals of order q for function f t denoted by a D −q t f t and t D −q b f t , respectively, are defined by provided the right-hand sides are pointwise defined on a, b , where Γ is the gamma function.
Definition 2.2 see 1 .Let f t be a function defined on a, b and q > 0. The left and right Riemann-Liouville fractional derivatives of order q for function f t denoted by a D q t f t and t D q b f t , respectively, are defined by where t ∈ a, b , n − 1 ≤ q < n and n ∈ N.
The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives.In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by AC a, b , then the left and right Caputo fractional derivative of order q for function f t denoted by Moreover, if α > 1/p and 1/p 1/q 1, then Making use of Property 2.4 and Definition 2.3, for any u ∈ AC 0, T , R N , BVP 1.1 is equivalent to the following problem: where α 1 − β/2 ∈ 1/2, 1 .
In the following, we will treat BVP 2.10 in the Hilbert space E α E α,2 0 with the corresponding norm u α u α,2 .The variational structure of BVP 2.10 on the space E α has been established.Lemma 2.9 see 32 . where If 1/2 < α ≤ 1, then the functional defined by 12 is continuously differentiable on E α , and ∀u, v ∈ E α , we have

2.13
Definition 2.10 see 32 .A function u ∈ AC 0, T , R N is called a solution of BVP 2.10 if i D α u t is derivative for almost every t ∈ 0, T , ii u satisfies 2.10 , where Lemma 2.11 see 32 .Let 1/2 < α ≤ 1 and ϕ be defined by 2.12 .If assumption (A) is satisfied and u ∈ E α is a solution of corresponding Euler equation ϕ u 0, then u is a solution of BVP 2.10 which corresponding to the solution of BVP 1.1 .
By Lemma 2.11, it means that the solutions for BVP 1.1 correspond to the critical points of the functional ϕ.We need the following estimate and known results for the sequel.Proposition 2.12 see 32 .If 1/2 < α ≤ 1, then for any u ∈ E α , one has

2.14
Lemma 2.13 see 23 .Let X be a real Banach space, Φ : Lemma 2.14 Mountain Pass theorem 24 .Let X be a real Banach space and Φ : X → R is differentiable and satisfies the (PS) condition.Suppose that i Φ 0 0, ii there exist ρ > 0 and σ > 0 such that Φ z ≥ σ for all z ∈ X with z ρ, iii there exists z 1 in X with z 1 ≥ ρ such that Φ z 1 < σ.
Then Φ possesses a critical value c ≥ σ.Moreover, c can be characterized as where Lemma 2.15 Clark theorem 24 .Let X be a real Banach space, Φ ∈ C 1 X, R with Φ even, bounded below, and satisfying the (PS) condition.Suppose Φ 0 0, there is a set K ⊂ X such that K is homeomorphic to S m−1 , m ∈ N, by an odd map, and sup K Φ < 0. Then Φ possesses at least m distinct pairs of critical points.

Proof of the Theorems
For u ∈ E α , where is a reflexive Banach space with the norm defined by It follows from Lemma 2.9 that the functional ϕ on E α given by

3.4
Recall that a sequence {u n } ∈ E α is said to be a C sequence of ϕ if ϕ u n is bounded and 1 u n α ϕ u n α → 0 as n → ∞.The functional ϕ satisfies condition C if every C sequence of ϕ has a convergent subsequence.This condition is due to Cerami 21 .

Proof of Theorem 1.1
We will first establish the following lemma and then give the proof of Theorem 1.1.Lemma 3.1.Assume (A), (A2), and (A3) hold, then the functional ϕ satisfies condition (C).
Proof of Lemma 3.1.Let {u n } ⊂ E α be a C sequence of ϕ, that is, ϕ u n is bounded and 1 u n α ϕ u n α → 0 as n → ∞.Then there exists M 0 such that

3.9
On the other hand, by A3 , there exist η > 0 and M 2 > 0 such that which, combining 3.9 , implies that u n α is bounded. where 3.17 Then we obtain u n → u in E α by use of the same argument of Theorem 5.2 in 32 .The proof of Lemma 3.1 is completed.
Proof of Theorem 1.1.By A1 , there exist 1 ∈ 0, | cos πα | and δ > 0 such that Then it follows from 2.8 that for all u ∈ E α with u α ρ.Therefore, we have

3.21
for all u ∈ E α with u α ρ.This implies that ii in Lemma 2.14 is satisfied.
It is obvious from the definition of ϕ and A1 that ϕ 0 0, and therefore, it suffices to show that ϕ satisfies iii in Lemma 2.14.

Proof of Theorem 1.2
The following lemmata are needed in the proof of Theorem 1.2.Lemma 3.2.Assume (A5), then for any ε > 0, there exists a subset E ε ⊂ 0, T with meas 0, T \ E ε < ε such that 27 Proof of Lemma 3.2.The proof is similar to that of Lemma 2 in 29 and is omitted.

3.29
We only need to show that {u n } is bounded in E α .If {u n } is unbounded, we may assume, without loss of generality, that u n α → ∞ as n → ∞.Put z n u n / u n α , we then have z n α 1. Going to a sequence if necessary, we assume that z n z weakly in E α , z n → z strongly in C 0, T , R N and L 2 0, T; R N .By A2 , it follows that there exist constants B 2 > 0 and M 4 > 0 such that

3.33
Abstract and Applied Analysis 13 from which, it follows that

3.34
Passing to the limit in the last inequality, we get which yields z / 0. Therefore, there exists a subset E ⊂ 0, T with meas E > 0 such that z t / 0 on E. By virtue of Lemma 3.2, for ε 1/2 meas E > 0, we can choose a subset

3.37
which leads to a contradiction and establishes the assertion.By A4 , we obtain thye following:

3.38
By 3.36 , 3.38 , and Fatou's lemma, it follows that 3.39 which contradicts 3.29 .This contradiction shows that u n α is bounded in E α , and this completes the proof.By virtue of Lemmas 3.2 and 3.3, the rest of the proof is similar to Theorem 1.1.Theorem 1.3 can be proved similarly.

Proof of Theorem 1.4
The proof of Theorem 1.4 is divided into a sequence of lemma.
Lemma 3.4.The functional ϕ is bounded below on E α .Proof of Lemma 3.4.By 2.8 and 2.14 , for every u ∈ E α , we have

3.40
where a 0 ess sup{a t : t ∈ 0, T }.The proof of Lemma 3.4 is complete.
Proof of Lemma 3.5.Let {u n } be a Palais-Smale sequence in E α , that is,

3.42
However, from 3.42 , we have thus u n α is a bounded sequence in E α .Since E α is a reflexive space, going, if necessary, to a subsequence, we can assume that u n u in E α , thus we have

3.49
It is easy to prove that the odd mapping Ψ : K m,β → S m−1 defined by is a homeomorphism between K m,β and S m−1 .Since E m ⊂ E α is a finite dimensional space, there exists ε m > 0 such that meas t ∈ 0, T :

3.51
Otherwise, for any positive integer n, there exists

3.53
Since dim E m < ∞, it follows from the compactness of the unit sphere of E m that there exists a subsequence, denoted also by {v n }, such that {v n } converges to some v 0 in E m .It is obvious that v 0 α 1.
By the equivalence of the norms on the finite dimensional space, we have 3.54 By 3.54 and H ölder inequality, we have −→ 0, as n −→ ∞.

3.60
for all positive integer n.Let n be large enough such that

Examples
In this section, we give some examples to illustrate our results.

Example
− 2, it follows from 3.9 that u n α is bounded too.Thus u n α is bounded in E α .By Proposition 2.8, the sequence {u n } has a subsequence, also denoted by {u n }, such that Moreover, according to 2.8 and Proposition 2.8, we have that{u n } is bounded in C 0, T , R N and u n − u ∞ → 0 as n → ∞.Combining 3.44 and 3.45 , it is easy to verify that u n − u α → 0 as n → ∞, and hence that u n → u in E α .Thus, {u n } admits a convergent subsequence.The proof of Lemma 3.5 is complete.For any m ∈ N, there exists a set K ⊂ E α which is homeomorphic to S m−1 by an odd map, and sup k ϕ < 0.
By Theorem 1.1, BVP 1.1 has at least one solution u ∈ E α .