Wang et al. Boundary Value Problems (2018) 2018:94 https://doi.org/10.1186/s13661-018-1012-0

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Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian Yang Wang1 , Yansheng Liu1* and Yujun Cui2 *

Correspondence: [email protected] 1 School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China Full list of author information is available at the end of the article

Abstract This paper deals with the existence of infinitely many solutions for a class of impulsive fractional boundary value problems with p-Laplacian. Based on a variant fountain theorem, the existence of infinitely many nontrivial high or small energy solutions is obtained. In addition, two examples are worked out to illustrate the effectiveness of the main results. MSC: 26A33; 34B15 Keywords: Fractional p-Laplacian; Variant fountain theorems; Impulsive effects; Infinitely many solutions

1 Introduction Consider the following nonlinear impulsive fractional boundary value problem (BVP, for short): ⎧ α c α p–2 ⎪ ⎨ DT – p ( D0+ u(t)) + |u(t)| u(t) = f (t, u(t)), t ∈ [0, T], t = tj , α–1 c α (DT – p ( D0+ u))(tj ) = Ij (u(tj )), j = 1, 2, . . . , m, ⎪ ⎩ u(0) = u(T) = 0,

(1.1)

where α ∈ ( p1 , 1], p > 1, p (s) = |s|p–2 s, DαT – represents the right Riemann–Liouville fractional derivative of order α and c Dα0+ represents the left Caputo fractional derivative of order α, 0 = t0 < t1 < · · · < tm+1 = T and c α  c α  +  c α  –   α–1 α–1  Dα–1 T – p D0+ u (tj ) = DT – p D0+ u tj – DT – p D0+ u tj , c α  +  c α  α–1 Dα–1 T – p D0+ u tj = lim+ DT – p D0+ u (t), t→tj

Dα–1 T – p

c

c α    Dα0+ u tj– = lim– Dα–1 T – p D0+ u (t). t→tj

f : [0, T] × R → R and Ij : R → R are continuous. Fractional differential equations have gained importance because of their numerous applications in various fields such as chemical physics, neural network model, signal processing and control, mechanics and engineering, fractal theory, and so on. For details, © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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see [1–5] and the references therein. Recently, the existence and multiplicity of solutions for nonlinear fractional differential equations have been studied extensively by using the theory of coincidence degree, some fixed point theorems, upper-lower solution method, monotone iterative method, etc. [6–9]. It should be noted that the critical point theory and variational methods have proved to be a very effective approach in dealing with the existence and multiple solutions for fractional boundary value problems, see [10–17]. On the other hand, the impulsive differential equation is used to describe the dynamics of processes in which sudden, discontinuous jumps occur. It has numerous applications in many fields such as population dynamics, ecology, optimal control, economics, and so on. For details, see [18–21] and the references therein. Recently, many authors have studied the existence of solutions for impulsive fractional boundary value problem by using variational methods and critical point theory, see [22–30]. For example, Heidarkhani et al. [26] and [28] studied the following impulsive nonlinear fractional boundary value problem: ⎧ α c α ⎪ ⎨ DT – ( D0+ u(t)) + a(t)u(t) = λf (t, u(t)) + h(u(t)), t ∈ [0, T], t = tj , c α j = 1, 2, . . . , m, (Dα–1 T – ( D0+ u))(tj ) = μIj (u(tj )), ⎪ ⎩ u(0) = u(T) = 0.

(1.2)

Based on variational methods and critical point theory, they obtained the existence results of infinitely many classical solutions and three solutions for problem (1.2). In particular, Rodríguez-López and Tersian [22] established one and three solutions for problem (1.2) when h(u(t)) ≡ 0. In [25], Heidarkhani and Salari obtained the existence of two and three weak solutions for a class of nonlinear impulsive fractional systems by applying variational methods. Furthermore, the p-Laplacian often occurs in non-Newtonian fluid theory, nonlinear elastic mechanics, and so on. So, the impulsive fractional boundary value problem with pLaplacian is worth considering. For instance, in [31], basing on the mountain pass theorem and minimax methods, the existence of multiple solutions for BVP(1.1) is obtained. To the best of our knowledge, there are fewer results on the existence and multiplicity of solutions for impulsive fractional boundary value problem with p-Laplacian. Inspired by the above references, we apply variant fountain theorems to study the existence of infinitely many small or high energy solutions for BVP (1.1). The main new features presented in this paper are as follows. Firstly, the main results of this paper are different from those in the aforementioned references, and extend the results obtained in [31]. Secondly, the main tool of this paper is variant fountain theorems, which is different from the aforementioned papers. Thirdly, the assumed conditions in this paper are easier to verify than those in [31]. Finally, two examples are worked out to demonstrate the effectiveness of our results. For convenience, we list the following assumptions. u (H1 ) Ij (u) (j = 1, 2, . . . , m) are odd about u and satisfy 0 Ij (s) ds ≥ 0 for all u ∈ R. (H2 ) There exist bj > 0 and γj ∈ (p – 1, +∞) such that Ij (u) ≤ bj |u|γj .

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(H3 ) There exist bj > 0, μ > p, and γj ∈ (p – 1, μ – 1) such that

Ij (u) ≤ bj |u|γj ,

u

Ij (u)u ≤ μ

Ij (s) ds. 0 p

(F1 ) There exist η ∈ (p – 1, p) and b(t) ∈ L p–η [0, T] with b(t) ≥ 0 such that   f (t, u) ≤ b(t) 1 + |u|η–1 ,

∀(t, u) ∈ [0, T] × R.

(F2 ) There exist σ ∈ (p – 1, η) and d > 0 such that

F(t, u) lim > d, |u|→∞ |u|σ (F3 ) (F4 ) (F5 ) (F6 )

u

uniformly for t ∈ [0, T], where F(t, u) =

f (t, s) ds. 0

f (t,u) lim|u|→0 |u| p–1 = 0, uniformly for t ∈ [0, T]. F(t, u) ≥ 0, ∀(t, u) ∈ [0, T] × R. F(t, –u) = F(t, u), ∀(t, u) ∈ [0, T] × R. There exist constants θ1 > 0, θ2 > 0, and q > p such that

f (t, u) ≤ θ1 |u|p–1 + θ2 |u|q–1

for all t ∈ [0, T], u ∈ R.

(F7 ) There exists μ > p such that –μF(t, u) + uf (t, u) ≥ 0

for all t ∈ [0, T], u ∈ R.

Here are our main results. Theorem 1.1 Assume that (H1 )–(H2 ) and (F1 )–(F5 ) hold. Then BVP (1.1) possesses infinitely many small energy solutions uk ∈ E \ {0} satisfying 1 p

0

+

T  c

p p  Dα0+ uk (t) + uk (t) dt

m

 j=1



uk (tj )

T

Ij (s) ds –

0

  F t, uk (t) dt → 0–

as k → ∞.

0

Theorem 1.2 Assume that (H1 ), (H3 ) and (F4 )–(F7 ) hold. Then BVP (1.1) possesses infinitely many high energy solutions uk ∈ E \ {0} satisfying 1 p



T  c

0

+

p p  Dα0+ uk (t) + uk (t) dt

m

 j=1

0



uk (tj )

T

Ij (s) ds –

  F t, uk (t) dt → ∞

as k → ∞.

0

The rest of this paper is organized as follows. Section 2 contains some preliminary results. In Sect. 3, we apply variant fountain theorems to prove the existence of infinitely many small or high energy solutions for BVP (1.1). In Sect. 4, two examples are presented to illustrate the main results.

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2 Preliminaries To obtain multiple solutions for BVP (1.1), it is necessary to introduce several definitions and preliminary lemmas which are used further in this paper. Let AC[a, b] be the space of absolutely continuous functions on [a, b]. Definition 2.1 ([10]) Let f be a function defined on [a, b] and 0 < α ≤ 1. The left and right Riemann–Liouville fractional integrals of order α for the function f are defined by D–α a+ f (t) =

1

(α)

D–α b– f (t) =

1

(α)



t

t ∈ [a, b],

(t – s)α–1 f (s) ds, a



b

t ∈ [a, b],

(s – t)α–1 f (s) ds, t

while the right-hand side is pointwise defined on [a, b]. Definition 2.2 ([10]) Let f ∈ AC[a, b] and 0 < α ≤ 1. The left and right Riemann–Liouville fractional derivatives of order α for the function f are defined by

d α–1 1 d Da+ f (t) = dt

(1 – α) dt

Dαa+ f (t) =

Dαb– f (t) = –

t

(t – s)–α f (s) ds, a

d d α–1 1 Db– f (t) = – dt

(1 – α) dt



t ∈ [a, b],

b

(s – t)–α f (s) ds,

t ∈ [a, b].

t

Definition 2.3 ([10]) Let f ∈ AC[a, b] and 0 < α ≤ 1. The left and right Caputo fractional derivatives of order α for the function f are defined by c

c

Dαa+ f (t) = Dα–1 a+ f (t) =

1

(1 – α)

Dαb– f (t) = –Dα–1 b– f (t) = –

a

1

(1 – α)

t

(t – s)–α f (s) ds,



b

t ∈ [a, b],

(s – t)–α f (s) ds,

t ∈ [a, b].

t

In particular, when α = 1, we have c D1a+ f (t) = f (t) and c D1b– f (t) = –f (t). Lemma 2.4 ([32]) (1) If u ∈ Lp [a, b], v ∈ Lp [a, b], and p ≥ 1, q ≥ 1, 1 + q1 = 1 + α, then p

b

 D–α a+ u(t) v(t) dt =

a



b a

1 p

+

≤ 1 + α or p = 1, q = 1,

1 q

  u(t) D–α b– v(t) dt.

(2) If 0 < α ≤ 1, u ∈ AC[a, b], and v ∈ Lp [a, b] (1 ≤ p < ∞), then

b

a

t=b   u(t) c Dαa+ v(t) dt = Dα–1 b– u(t)v(t) t=a +



b a

Dαb– u(t)v(t) dt.

Denote

u

Lp

= 0

T

u(t) p dt

p1 ,

u∞ = max u(t) . t∈[0,T]

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α,p

Definition 2.5 Let 0 < α ≤ 1, 1 < p < ∞. The fractional derivative space E0 is defined by the closure of C0∞ ([0, T], R) with respect to the norm

uEα,p = 0

T c 0

p Dα0+ u(t) dt +

p1 u(t) p dt ,



T

0

α,p

∀u ∈ E0 .

(2.1)

Remark 2.6 α,p (1) E0 is a reflexive and separable Banach space. α,p (2) For any u ∈ E0 , we have u ∈ Lp ([0, T], R), c Dαa+ u ∈ Lp ([0, T], R), and u(0) = u(T) = 0. α,p

Lemma 2.7 ([32]) Let 0 < α ≤ 1 and 1 < p < ∞. For any u ∈ E0 , we have uLp ≤

 Tα  c Dα+ u p . 0 L

(α + 1)

In addition, for

1 p

< α ≤ 1 and

1 p

1

u∞ ≤

T α– p

(α)(αq – q + 1)

1 q

+

(2.2) 1 q

= 1, we have

c α   D + u p . 0 L

(2.3)

α,p

Remark 2.8 According to Lemma 2.7, it is easy to see that the norm of E0 defined in (2.1) is equivalent to the following norm:

T c

uα,p = 0

p Dα0+ u(t) dt

p1 ,

α,p

∀u ∈ E0 .

(2.4) α,p

Lemma 2.9 ([32]) Let p1 < α ≤ 1. If the sequence {uk } converges weakly to u in E0 , i.e., uk  u, then uk → u in C[0, T], i.e., u – uk ∞ → 0 as k → ∞. α,p

In the following, we denote E = E0 , u = uEα,p , up = uLp for convenience. 0

Definition 2.10 A function 

u ∈ u ∈ AC[0, T] :

tj+1  c

tj

p p  Dα0+ u(t) + u(t) dt < ∞, j = 0, 1, . . . , m



is called a classical solution of BVP (1.1) if (1) u satisfies (1.1). + α–1 – c α c α (2) The limits Dα–1 T – p ( D0+ u)(tj ), DT – p ( D0+ u)(tj ) exist. Definition 2.11 A function u ∈ E is a weak solution of BVP (1.1) if

T

   c Dα+ u(t) p–2 c Dα+ u(t) c Dα+ v(t) dt + 0

0

+

m  j=1

0

  Ij u(tj ) v(tj ) –

0

0

T



T

u(t) p–2 u(t)v(t) dt

0

  f t, u(t) v(t) dt = 0,

∀u ∈ E.

(2.5)

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The energy functional J : E → R associated with BVP (1.1) is defined by 1 J(u) = p +



 c Dα+ u(t) p + u(t) p dt

T 

0

0

m





u(tj ) 0

j=1

T

Ij (s) ds –

  F t, u(t) dt,

∀u ∈ E.

(2.6)

0

It is easy to see that J ∈ C 1 (E, R), and 







T

   c Dα+ u(t) p–2 c Dα+ u(t) c Dα+ v(t) dt +

J (u), v =

0

0

+

m 

0

  Ij u(tj ) v(tj ) –

0



T

T

u(t) p–2 u(t)v(t) dt

0

  f t, u(t) v(t) dt

0

j=1

= 0,



∀u ∈ E.

(2.7)

Moreover, the critical points of J correspond to the weak solutions of BVP (1.1). Lemma 2.12 ([31]) If u ∈ E is a weak solution of BVP (1.1), then u is a classical solution of BVP (1.1). To prove our main results, we need the following two variant fountain theorems in [33].  Let X be a Banach space with the norm  ·  and X = j∈N Xj with dim Xj < ∞ for each   j ∈ N. Set Wk = kj=1 Xj , Zk = ∞ j=k Xj , Bk = {u ∈ Wk : u ≤ ρk }, Sk = {u ∈ Zk : u = rk }, where ρk > rk > 0. Consider a family of C 1 functionals Jλ : X → R defined by Jλ (u) = A(u) – λB(u),

λ ∈ [1, 2],

where A, B : X → R are two functions. Lemma 2.13 ([33]) Assume that the functional Jλ defined above satisfies: (B1 ) Jλ maps bounded sets into bounded sets uniformly for λ ∈ [1, 2], and Jλ (–u) = Jλ (u) for all (λ, u) ∈ [1, 2] × X; (B2 ) B(u) ≥ 0 for all u ∈ X, B(u) → ∞ as u → ∞ on any finite dimensional subspace of X; (B3 ) there exist ρk > rk > 0 such that ak (λ) =

inf

Jλ (u) ≥ 0,

inf

Jλ (u) → 0

u∈Zk ,u=ρk

bk (λ) =

max

u∈Wk ,u=rk

Jλ (u) < 0,

∀λ ∈ [1, 2]

and dk (λ) =

u∈Zk ,u≤ρk

as k → ∞ uniformly for λ ∈ [1, 2].

Then there exist λn → 1, un (λn ) ∈ Wn such that   Jλ n |Wn u(λn ) = 0,

  Jλn u(λn ) → ck

as n → ∞,

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where ck ∈ [dk (2), bk (1)]. In particular, if {u(λn )} has a convergent subsequence for every k, then J1 has infinitely many nontrivial critical points {uk } ∈ X \ {0} satisfying J1 (uk ) → 0– as k → ∞. Lemma 2.14 ([33]) Assume that the functional Jλ defined above satisfies (A1 ) Jλ maps bounded sets into bounded sets uniformly for λ ∈ [1, 2], and Jλ (–u) = Jλ (u) for all (λ, u) ∈ [1, 2] × X; (A2 ) B(u) ≥ 0 for all u ∈ X, A(u) → ∞ or B(u) → ∞ as u → ∞; or (A3 ) B(u) ≤ 0 for all u ∈ X, B(u) → –∞ as u → ∞; (A4 ) there exist ρk > rk > 0 such that bk (λ) =

inf

u∈Zk ,u=rk

Jλ (u) > ak (λ) =

max

u∈Wk ,u=ρk

Jλ (u),

∀λ ∈ [1, 2].

Then   bk (λ) ≤ ck (λ) = inf max Jλ γ (u) , γ ∈ k u∈Bk

∀λ ∈ [1, 2],

where k = {γ ∈ C(Bk , X) : γ is odd, γ |∂Bk = id}. Moreover, for almost every λ ∈ [1, 2], there exists a sequence {ukn (λ)} such that   supukn (λ) < ∞,

  Jλ ukn (λ) → 0

n

  and Jλ ukn (λ) → ck (λ) as n → ∞.

∗ ∞ As E is a separable and reflexive Banach space, then there exist {ej }∞ j=1 ⊂ E and {ej }j=1 ⊂ E such that ∗

E = span{ej },  ∗  ei , ei = 1,

  E∗ = span e∗j ,  ∗  ej , ei = 0 (i = j).

  Define Xj = span{ej }, Wk = kj=1 Xj , Zk = ∞ j=k Xj . In order to apply Lemma 2.13 and Lemma 2.14 to prove the existence of infinitely many solutions of BVP (1.1), we define A, B, and Jλ on a fractional derivative space E by  1 A(u) = up + p j=1 m





u(tj )

Ij (s) ds,

B(u) =

0

T

  F t, u(t) dt,

0

and Jλ (u) = A(u) – λB(u)  1 = up + p j=1 m





u(tj )

Ij (s) ds – λ 0

T

  F t, u(t) dt,

∀u ∈ E, λ ∈ [1, 2].

0

3 Proof of the main results In order to complete the proof of our main results, it is necessary to give the following two lemmas. Because of using similar arguments to the proofs of Lemma 3.2 and Lemma 3.5 in [15], we omit the proving processes for convenience.

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Lemma 3.1 Let H be any finite dimensional subspace of E. Then there exists a constant ε0 > 0 such that   meas t ∈ [0, T] : u(t) ≥ ε0 u ≥ ε0 ,

∀u ∈ H \ {0}.

(3.1)

Lemma 3.2 Let αr (k) = supu∈Zk ,u=1 ur with r ≥ p. Then αr (k) → 0 as k → ∞. Now we are ready to prove Theorem 1.1 and Theorem 1.2. Proof of Theorem 1.1 By (F1 ) and (F3 ), for ∀ε > 0, there exists δε > 0 such that F(t, u) ≤ ε|u|p + δε b(t)|u|η .

(3.2)

Combining (3.2), (H2 ), and Lemma 2.7, it is easily seen that Jλ maps bounded sets into bounded sets uniformly for λ ∈ [1, 2]. It follows from (H1 ) and (F5 ) that Jλ (–u) = Jλ (u) for all (λ, u) ∈ [1, 2] × E. Thus, condition (B1 ) holds. Next, we verify condition (B2 ). According to (F4 ), B(u) ≥ 0 is obvious. By (F2 ), there exists M > 0 such that F(t, u) ≥ d|u|σ

for all |u| > M.

(3.3)

Assume that H is a finite dimensional subspace of E. According to Lemma 3.1, there exists ε0 > 0 such that (3.1) holds. Then meas(Du ) ≥ ε0 ,

∀u ∈ H \ {0},

where Du = {t ∈ [0, T] : |u(t)| ≥ ε0 u}. Hence, for any u ∈ H with u ≥ get

T

B(u) =

M , ε0

by (3.3), we

  F t, u(t) dt

0

≥

σ d u(t) dt

Du

≥ dε0σ uσ meas(Du ) ≥ dε01+σ uσ . This means that B(u) → ∞ as u → ∞ on any finite dimensional subspace. Hence, condition (B2 ) holds. In the end, we claim that condition (B3 ) holds. For u ∈ Zk , by (3.2), Hölder’s inequality, and (H1 ), we have 1 Jλ (u) ≥ up – p ≥



T

 ε|u|p + δε b(t)|u|η dt

0

  1 up – δ b(t) p uηp , p–η 2p

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where we choose ε =

1 . 2p

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According to the definition of αr (k) in Lemma 3.2, we have

uηp ≤ αpη (k)uη ,

∀u ∈ Zk .

Hence, Jλ (u) ≥

  1 up – δ b(t) p αpη (k)uη . p–η 2p

Choose ρk = (4pδb(t)

ak (λ) =

inf

u∈Zk ,u=ρk

1

η

αp (k)) p–η . Then ρk → 0+ as k → ∞. Therefore,

p p–η

Jλ (u) ≥

1 p ρ > 0. 4p k

In addition, for λ ∈ [1, 2] and u ∈ Zk with u ≤ ρk , we have   Jλ (u) ≥ –δ b(t)

p p–η

η

αpη (k)ρk → 0+ ,

k → ∞.

So, dk (λ) =

inf

u∈Zk ,u≤ρk

Jλ (u) → 0 as k → ∞.

By (F1 )–(F3 ), we have F(t, u) ≥ d|u|σ – ε|u|p – δε b(t)|u|η .

(3.4)

If u ∈ Wk , by the equivalence of any norm in a finite dimensional space, (3.4), (H2 ), Lemma 2.7, and Hölder’s inequality, we get  1 Jλ (u) = up + p j=1 m





u(tj )

T

Ij (s) ds – λ 0

  F t, u(t) dt

0

 bj 1 u(tj ) γj +1 – d ≤ up + p γ +1 j=1 j m



T

|u| dt + ε

0

  ≤ up + Muγj +1 – δ1 uσ + δ b(t)

T

|u| dt + δε p

σ

0 p p–η

T

b(t)|u|η dt 0

uη .

Choose rk > 0 small enough and rk < ρk such that bk (λ) =

max

u∈Wk ,u=rk

Jλ (u) < 0.

This guarantees that condition (B3 ) holds. Consequently, by Lemma 2.13, for every k ∈ N, there exist λn → 1, un (λn ) ∈ Wn such that   Jλ n |Wn u(λn ) = 0,

    Jλn u(λn ) → ck ∈ dk (2), bk (1) as n → ∞.

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For simplicity, denote u(λn ) by un . Now we show that {un } has a strong convergent subsequence for every k ∈ N. In fact, by (3.2), (H1 ), Hölder’s inequality, and Lemma 2.7, we get

un  = pJλn (un ) – p p

m

 j=1



un (tj )

Ij (s) ds + pλn

0

T

F(t, un ) dt 0

  ≤ pck + pλn δ2 un p + pλn δ b(t)

p p–η

un η .

This means that {un } is bounded in E. Without loss of generality, we may assume un  u in E. Since {ej } is a completely orthonormal basis of E, Wn = L(e1 , e2 , . . . , en ),  u= ∞ j=1 (ej , u)ej . Let Pn : E → Wn be the orthogonal projection operator. We know that  Pn u = nj=1 (ej , u)ej and Pn u → u in E as n → ∞. Therefore un – Pn u  0 in E as n → ∞. Moreover, it follows from J1 (u) ∈ E∗ that 

 J1 (u), un – Pn u → 0,

n → ∞.

(3.5)

Also, since J1 ∈ C(E → E∗ ) and Pn u → u in E, we have 

 J1 (Pn u) – J1 (u), un – Pn u → 0,

n → ∞.

(3.6)

Therefore, by (3.5) and (3.6), we get 

 J1 (Pn u), un – Pn u → 0,

n → ∞.

(3.7)

Note that Pn un = un and (Jλ n (Pn un ), Pn (un – u)) = 0 since un ∈ Wn and Jλ n |Wn (un ) = 0. By the continuity of f , Ij , and Lemma 2.9, it is easily seen that

T

λn

f (t, un )(un – Pn u) dt → 0,

n → ∞.

0



T

f (t, Pn u)(un – Pn u) dt → 0,

n → ∞.

0 m        Ij un (tj ) – Ij Pn u(tj ) un (tj ) – Pn u(tj ) → 0,

n → ∞.

j=1

Set

T  c

ψ1 = 0

p–2 c α p–2 c α   Dα0+ un (t) D0+ un (t) – c Dα0+ Pn u(t) D0+ Pn u(t)

  × c Dα0+ un (t) – c Dα0+ Pn u(t) dt,

T    un (t) p–2 un – Pn u(t) p–2 Pn u(t) un (t) – Pn u(t) dt. ψ2 = 0

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Therefore,     ψ1 + ψ2 = Jλ n (un ), un – Pn u – J1 (Pn u), un – Pn u –

m        Ij un (tj ) – Ij Pn u(tj ) un (tj ) – Pn u(tj ) j=1

+ λn



T

T

f (t, un )(un – Pn u) dt –

f (t, Pn u)(un – Pn u) dt

0

0

→ 0. In what follows, we prove un – Pn u → 0 in two cases. Case 1: p ≥ 2. According to the following inequality (see [34], Lemma 4.2) 

 |x|p–2 x – |y|p–2 y (x – y) ≥ ω|x – y|p ,

there exist ω1 > 0, ω2 > 0 such that

ψ1 ≥ ω1

T c

0

ψ2 ≥ ω2

p Dα0+ un (t) – c Dα0+ Pn u(t) dt,

(3.8)

T

un (t) – Pn u(t) p dt.

(3.9)

0

Combining (3.8) and (3.9), we get ψ1 + ψ2 ≥ M1 un – Pn up , where M1 = min{ω1 , ω2 }. Thus, un – Pn u → 0. Case 2: 1 < p < 2. According to the following inequality (see [34], Lemma 4.2)  2–p  p–2  p  |x| x – |y|p–2 y (x – y) 2 |x|p + |y|p 2 ≥ ω|x – y|p , there exist positive numbers ω3 and ω4 such that

ψ1 ≥ ω3

T

0

ψ2 ≥ ω4

0

T

|c Dα0+ un (t) – c Dα0+ Pn u(t)|2 c (| Dα0+ un (t)| + |c Dα0+ Pn u(t)|)2–p

(3.10)

dt,

|un (t) – Pn u(t)|2 dt. (|un (t)| + |Pn u(t)|)2–p

(3.11)

By Hölder’s inequality, we get

un (t) – Pn u(t) p dt

T 0



|un (t) – Pn u(t)|2 dt ≤ 2–p 0 (|un (t)| + |Pn u(t)|) T   2–p ≤ M2 un pp + Pn upp 2 T

0

p2

 un (t) + Pn u(t) p dt

T  0

|un (t) – Pn u(t)|2 dt (|un (t)| + |Pn u(t)|)2–p

p2 ,

2–p 2

Wang et al. Boundary Value Problems (2018) 2018:94

where M2 = 2

(p–1)(2–p) 2

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. Therefore,

|un (t) – Pn u(t)|2 dt 2–p 0 (|un (t)| + |Pn u(t)|) T

p2   p–2 – p2 un (t) – Pn u(t) p dt un pp + Pn upp p . ≥ M2 T

(3.12)

0

Similarly, we have

0

T

|c Dα0+ un (t) – c Dα0+ Pn u(t)|2 c (| Dα0+ un (t)| + |c Dα0+ Pn u(t)|)2–p

– p2 ≥ M2



T c

0

dt

p Dα0+ un (t) – c Dα0+ Pn u(t) dt

p2

p  p–2 p   · c Dα+ un  + c Dα+ Pn u p . 0

p

0

(3.13)

p

Combining (3.10)–(3.13), we get ψ1 + ψ2 ≥ M3n un – Pn u2 , – 2

p

p p–2

p

p p–2

where M3n = M2 p–1 min{ω3 (c Dα0+ un p + c Dα0+ Pn up ) p , ω4 (un p + Pn up ) p }. If {un } has a subsequence (still relabeled {un } for convenience) such that un  → 0 as n → ∞, it is easy to see that un – Pn u → 0. On the other hand, if infn≥1 un  > 0, by the boundedness of {un } in E, there exists M3 > 0 such that M3n ≥ M3 > 0. Then un – Pn u → 0. So, un – Pn u → 0 in E as n → ∞, which means that un → u in E as n → ∞. By Lemma 2.13, we know that J = J1 has infinitely many nontrivial critical points uk . Consequently, BVP (1.1) has infinitely many small energy solutions.  Proof of Theorem 1.2 For any ε > 0, it follows from (F6 ) that there exist positive numbers θ3 and θ4 such that F(t, u) ≤ θ3 |u|p + θ4 |u|q .

(3.14)

Combining (3.14), (H3 ), and Lemma 2.7, it is easily seen that Jλ maps bounded sets into bounded sets uniformly for λ ∈ [1, 2]. By (H1 ) and (F5 ), Jλ (–u) = Jλ (u) for all (λ, u) ∈ [1, 2] × E. Thus, condition (A1 ) holds. Assumption (F4 ) means that B(u) ≥ 0. Condition (A2 ) holds for the fact that A(u) ≥ p1 up → ∞ as n → ∞ and B(u) ≥ 0. In what follows, we verify condition (A4 ). For this sake, we need to prove that there exist two sequences ρk > rk > 0 such that bk (λ) = ak (λ) =

inf

u∈Zk ,u=rk

max

Jλ (u) > 0,

u∈Wk ,u=ρk

Jλ (u) < 0,

∀λ ∈ [1, 2], ∀λ ∈ [1, 2].

First, we prove that (3.15) is true.

(3.15) (3.16)

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For u ∈ Zk , by (3.14), (H1 ), and the definition of αr (k) in Lemma 3.2, we have 1 Jλ (u) ≥ up – 2θ3 upp – 2θ4 uqq p 1 ≥ up – 2θ3 αpp (k)up – 2θ4 αqq (k)uq . p Choose rk =

Then rk → ∞ as k → ∞. For any u ∈ Zk with u = rk , we know

1 . αp (k)+αq (k)

p

q

αp (k) αq (k) 1 – 2θ4 Jλ (u) ≥ up – 2θ3 p |αp (k) + αq (k)|p |αp (k) + αq (k)|q 1 p ≥ rk – 2θ3 – 2θ4 p > 0. Therefore, bk (λ) =

inf

u∈Zk ,u=rk

∀λ ∈ [1, 2].

Jλ (u) > 0,

Next, we prove that (3.16) is true. By (F7 ), there exists δ3 > 0 such that F(t, u) ≥ δ3 |u|μ ,

for all t ∈ [0, T], u ∈ R.

(3.17)

According to (3.17), (H3 ), and Lemma 2.7, we have  1 Jλ (u) = up + p j=1 m





u(tj )

T

Ij (s) ds – λ 0

0

 bj 1 u(tj ) γj +1 – δ3 ≤ up + p γ +1 j=1 j m

  F t, u(t) dt

T

|u|μ dt

0

1 ≤ up + δ4 uγj +1 – δ5 uμ . p Hence, one can take ρk > γk large enough such that ak (λ) =

max

u∈Wk ,u=ρk

Jλ (u) < 0.

Until now, all the conditions of Lemma 2.14 hold. Hence, for λ ∈ [1, 2], there exists a sequence {ukn (λ)}∞ n=1 such that   supukn (λ) < ∞, n

Jλ



 ukn (λ) → 0,

    Jλ ukn (λ) → ck (λ) = inf max Jλ γ (u) , γ ∈ k u∈Bk

n → ∞.

(3.18)

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Furthermore, 1 p ck (λ) ≥ bk (λ) ≥ rk – 2θ3 – 2θ4 := bk → ∞, p

k → ∞,

ck (λ) ≤ max J1 (u) := ck . u∈Bk

Thus, bk ≤ ck (λ) ≤ ck ,

λ ∈ [1, 2].

(3.19)

Choose a sequence λm → 1 such that (3.18) holds. Using similar arguments of the proof of Theorem 1.1, we can show that {ukn (λm )}∞ n=1 possesses a strong convergent subsequence. Thus, we suppose that ukn (λm ) → uk (λm ) in E as n → ∞. By (3.18) and (3.19), we can get   Jλ m uk (λm ) = 0,

  Jλm uk (λm ) ∈ [bk , ck ] for k ≥ k1 .

In the following we prove that {uk (λm )}∞ m=1 is bounded. From (H3 ) and (F6 ), we have       μJλm uk (λm ) – Jλ m uk (λm ) , uk (λm ) =

m uk (λm )(tj )  k p   k  k μ   μ – 1 u (λm ) + Ij (s) ds – Ij u (λm )(tj ) u (λm )(tj ) p 0 j=1

+ λm ≥

    f t, uk (λm ) uk (λm ) – μF t, uk (λm ) dt

T

0

 p μ – 1 uk (λm ) . p

Therefore, {uk (λm )}∞ m=1 is bounded in E. Similar arguments of the proof of Theorem 1.1 k show that u (λm ) → uk in E as m → ∞ (k ≥ k1 ). Then uk is a critical point of J = J1 with I(uk ) ∈ [bk , ck ]. According to bk → ∞ as k → ∞, we know that BVP (1.1) has infinitely many nontrivial high energy solutions. 

4 Examples In this section, two examples are given to illustrate our results. Example 4.1 Consider the following nonlinear impulsive fractional boundary value problem: ⎧ 0.8 c 0.8 ⎪ ⎨ DT – 4 ( D0+ u(t)) + |u(t)|u(t) = f (t, u), –0.2 5 (DT – 4 (c D0.8 0+ u))(t1 ) = u (t1 ), ⎪ ⎩ u(0) = u(T) = 0, where  f (t, u) =

7 t e |u|4 , 2 5 7 t e |u| 2 , 2

|u| ≤ 1, |u| > 1.

t ∈ [0, T], t = t1 , (4.1)

Wang et al. Boundary Value Problems (2018) 2018:94

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Choose p = 4, α = 0.8 ∈ ( 14 , 1], and I1 (u) = u5 (t1 ). It is easy to show that assumption (H1 ) holds. Take b1 = 2, γ1 = 5 ∈ (3, +∞). From this we can see that assumption (H2 ) holds. Moreover,   f (t, u) ≤ 4et 1 + |u| 52 . Choose η = 72 ∈ (3, 4) and b(t) = 4et . This means that assumption (F1 ) is satisfied. 7 Take σ = 13 ∈ (3, 72 ). By a simple calculation, one has F(t, u) = et |u| 2 and 4 7

lim|u|→∞

et |u| 2 13 |u| 4

→ ∞. Therefore, assumption (F2 ) holds. 7 et |u|4

In addition, lim|u|→0 2 |u|3 = 0 implies that assumption (F3 ) holds. Finally, it is easy to see that (F4 ) and (F5 ) hold. Consequently, BVP (4.1) has infinitely many small energy solutions by Theorem 1.1. Example 4.2 Consider the following nonlinear fractional impulsive boundary value problem: ⎧ 1 |u|5 0.8 c 0.8 4 ⎪ ⎪ ⎨ DT –  5 ( D0+ u(t)) + |u(t)| 2 u(t) = 5|u| ln(|u| + 1) + |u|+1 , t ∈ [0, T], t = t1 , 2

c 0.8 3 (D–0.2 T –  52 ( D0+ u))(t1 ) = u (t1 ), ⎪ ⎪ ⎩ u(0) = u(T) = 0.

(4.2)

First, choose p = 52 , α = 0.8 ∈ ( 25 , 1], and I1 (u) = u3 (t1 ). From this one can see that assumption (H1 ) holds. Taking b1 = 2, μ = 5 > 52 , and γ1 = 3 ∈ ( 32 , 4) means that assumption (H3 ) holds. Next, a simple calculation shows that   F(t, u) = |u|5 ln |u| + 1 ,

–5F(t, u) + uf (t, u) =

|u|5 ≥ 0. |u| + 1

Hence, assumption (F7 ) holds. Finally, it is easy to show that (F4 )–(F6 ) hold. Consequently, BVP (4.2) has infinitely many high energy solutions by Theorem 1.2.

Acknowledgements The authors wish to thank anonymous referees for their valuable suggestions. Funding This work was supported by the National Natural Science Foundation of China (No. 11671237), and by Shandong Provincial Natural Science Foundation of China (No. ZR2013AM005). List of abbreviations BVP, boundary value problem; AC[a, b], the space of absolutely continuous functions on [a, b]. Availability of data and materials Not applicable. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China. 2 Department of Mathematics, Shandong University of Science and Technology, Qingdao, P.R. China. 1

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