Electronic Journal of Qualitative Theory of Differential Equations

2015, No. 11, 1–10; http://www.math.u-szeged.hu/ejqtde/

Variational approach to solutions for a class of

fractional boundary value problems

Ziheng Zhang B 1 and Jing Li2

1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2 School of Management, Qingdao Huanghai University, Qingdao 266427, Shandong, China

Received 11 April 2014, appeared 12 March 2015

Communicated by Gabriele Bonanno

Abstract. In this paper we investigate the existence of infinitely many solutions for the
following fractional boundary value problem
(
t DT (0 Dt u ( t )) = ∇W ( t, u ( t )), t ∈ [0, T ],
α α
(FBVP)
u(0) = u( T ) = 0,

where α ∈ (1/2, 1), u ∈ Rn , W ∈ C1 ([0, T ] × Rn , R) and ∇W (t, u) is the gradient of

W (t, u) at u. The novelty of this paper is that, assuming W (t, u) is of subquadratic
growth as |u| → +∞, we show that (FBVP) possesses infinitely many solutions via the
genus properties in the critical theory. Recent results in the literature are generalized
and significantly improved.
Keywords: fractional Hamiltonian systems, critical point, variational methods, genus.
2010 Mathematics Subject Classification: 34C37, 35A15, 35B38.

1 Introduction
Fractional differential equations, both ordinary and partial ones, are extensively applied in
mathematical modeling of processes in physics, mechanics, control theory, biochemistry, bio-
engineering and economics. Therefore, the theory of fractional differential equations is an
area intensively developed during the last decades [4]. The monographs [7, 9, 12] enclose a
review of methods of solving fractional differential equations.
Recently, equations including both left and right fractional derivatives are discussed. Apart
from their possible applications, equations with left and right derivatives provide an interest-
ing and new field in fractional differential equations theory. In this topic, many results are
obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differ-
ential equations by using techniques of nonlinear analysis, such as fixed point theory (includ-
ing Leray–Schauder nonlinear alternative), topological degree theory (including co-incidence
degree theory) and comparison method (including upper and lower solutions and monotone
iterative method).
B Corresponding author. Email: zhzh@mail.bnu.edu.cn
2 Z. Zhang and J. Li

It should be noted that critical point theory and variational methods have also turned out
to be very effective tools in determining the existence of solutions for integer order differential
equations. The idea behind them is trying to find solutions of a given boundary value problem
by looking for critical points of a suitable energy functional defined on an appropriate function
space. In the last 30 years, the critical point theory has become a wonderful tool in studying the
existence of solutions to differential equations with variational structures, we refer the reader
to the books by Mawhin and Willem [8], Rabinowitz [13], Schechter [16] and the references
listed therein.
Motivated by the above classical works, in the recent paper [6], for the first time, the
authors showed that critical point theory is an effective approach to tackle the existence of
solutions for the following fractional boundary value problem

t D α (0 D α u(t)) = ∇W (t, u(t)), t ∈ [0, T ],
T t
(FBVP)
 u (0) = u ( T ),

where α ∈ (1/2, 1), u ∈ Rn , W ∈ C1 ([0, T ] × Rn , R) and ∇W (t, u) is the gradient of W (t, u)

at u. Explicitly, under the assumption that

(H1 ) |W (t, u)| ≤ ā|u|2 + b̄(t)|u|2−τ + c̄(t) for all t ∈ [0, T ] and u ∈ Rn ,

where ā ∈ [0, Γ2 (α + 1)/2T 2α ), τ ∈ (0, 2), b̄ ∈ L2/τ [0, T ] and c̄ ∈ L1 [0, T ], combining with
some other reasonable hypotheses on W (t, u), the authors showed that (FBVP) has at least
one nontrivial solution. In addition, assuming that the potential W (t, u) satisfies the following
superquadratic condition:

(H2 ) there exists µ > 2 and R > 0 such that

0 < µW (t, u) ≤ (∇W (t, u), u)

for all t ∈ [0, T ] and u ∈ Rn with |u| ≥ R,

and some other assumptions on W (t, u), they also obtained the existence of at least one non-
trivial solution for (FBVP). Inspired by this work, in [18] the author considered the following
fractional boundary value problem
(
t DT (0 Dt u ( t )) = f ( t, u ( t )),
α α t ∈ [0, T ],
(1.1)
u(0) = u( T ) = 0,

with α ∈ (1/2, 1), u ∈ R, f : [0, T ] × R → R satisfying the following hypotheses:

( f 1 ) f ∈ C ([0, T ] × R, R);

( f 2 ) there is a constant µ > 2 such that

0 < µF (t, u) ≤ u f (t, u) for all t ∈ [0, T ] and u ∈ R\{0},

the author showed that (1.1) possesses at least one nontrivial solution via the mountain pass
theorem. For the other works related to the solutions of fractional boundary value problems,
we refer the reader to the papers [2, 5, 10, 17, 19] and the references mentioned there.
Note that all the papers mentioned above are concerned with the existence of solutions
for (FBVP). As far as the multiplicity of solutions for (FBVP) is concerned, to the best of
 Solutions for a class of fractional boundary value problems 3

our knowledge, there is no result about this. Therefore, motivated by the above results, in
this paper using the genus properties of critical point theory we establish some new criterion
to guarantee the existence of infinitely many solutions of (FBVP) for the case that W (t, u) is
subquadratic as |u| → +∞. Note that in [11] the same techniques are used to consider the
existence of solutions to nonlocal Kirchhoff equations of elliptic type. For the statement of
our main result in the present paper, the potential W (t, u) is supposed to satisfy the following
conditions:

(W1 ) W (t, 0) = 0 for all t ∈ [0, T ], W (t, u) ≥ a(t)|u|ϑ and |∇W (t, u)| ≤ b(t)|u|ϑ−1 for all
(t, u) ∈ [0, T ] × Rn , where 1 < ϑ < 2 is a constant, a : [0, T ] → R+ is a continuous
function and b : [0, T ] → R+ is a continuous function;

(W2 ) there is a constant 1 < σ ≤ ϑ < 2 such that

(∇W (t, u), u) ≤ σW (t, u) for all t ∈ [0, T ] and u ∈ Rn .

Now, we can state our main result.

Theorem 1.1. Suppose that (W1 ) and (W2 ) are satisfied. Moreover, assume that W (t, u) is even in u,
i.e.,

(W3 ) W (t, u) = W (t, −u) for all t ∈ [0, T ] and u ∈ Rn ,

then (FBVP) has infinitely many nontrivial solutions.

Remark 1.2. From (W1 ), it is easy to check that W (t, u) is subquadratic as |u| → +∞. In fact,
in view of (W1 ), we have
Z 1
b(t) ϑ
W (t, u) = (∇W (t, su), u)ds ≤ |u| , (1.2)
0 ϑ

which implies that W (t, u) is of subquadratic growth as |u| → +∞.

(H2 ) is the so-called Ambrosetti–Rabinowitz condition due to Ambrosetti and Rabinowitz
(see e.g., [1]), which implies that W (t, u) is superquadratic as |u| → +∞. Here we consider the
case that W (t, u) is of subquadratic growth. Therefore, the result in [18] is complemented. In
addition, in view of (1.2), it is obvious that if W (t, u) satisfies (W1 ), then (H1 ) holds. However,
in [6] the authors only obtained the existence of at least one nontrivial solution for (FBVP). In
our Theorem 1.1, we obtain that (FBVP) possesses infinitely many nontrivial solutions.

Example 1.3. Here we give an example to illustrate Theorem 1.1. Take

3
W (t, u) = (2 + sin t)|u| 2 , ∀(t, u) ∈ [0, T ] × R,

then it is easy to check that (W1 ), (W2 ) and (W3 ) are satisfied where a(t) = 2 + sin t, b(t) =
3 3
2 (2 + sin t ) and σ = ϑ = 2 .

The remaining part of this paper is organized as follows. Some preliminary results are
presented in Section 2. In Section 3, we are devoted to accomplishing the proof of Theorem 1.1.
4 Z. Zhang and J. Li

2 Preliminary results
In this section, for the reader’s convenience, firstly we introduce some basic definitions of
fractional calculus which are used further in this paper, see [7].

Definition 2.1 (Left and right Riemann–Liouville fractional integrals). Let u be a function
defined on [ a, b]. The left and right Riemann–Liouville fractional integrals of order α > 0 for
function u are defined by
Z t
1
α
a It u ( t ) = (t − s)α−1 u(s)ds, t ∈ [ a, b]
Γ(α) a

and Z b
1
α
t Ib u ( t ) = (s − t)α−1 u(s)ds, t ∈ [ a, b].
Γ(α) t

Definition 2.2 (Left and right Riemann–Liouville fractional derivatives). Let u be a function
defined on [ a, b]. The left and right Riemann–Liouville fractional derivatives of order α > 0
for function u denoted by a Dtα u(t) and t Dbα u(t), respectively, are defined by

α dn n−α
a Dt u ( t ) = aT u(t)
dtn t
and
α n dn n−α
t Db u ( t ) = (−1) tI u ( t ),
dtn b
where t ∈ [ a, b], n − 1 ≤ α < n and n ∈ N.

In what follows, to establish the variational structure which enables us to reduce the ex-
istence of solutions for (FBVP) to find critical points of the corresponding functional, it is
necessary to construct appropriate function spaces.
We recall some fractional spaces, for more details see [3]. To this end, denote by L p [0, T ]
(1 < p < +∞) the Banach spaces of functions on [0, T ] with values in Rn under the norms
Z T
1/p
p
kuk p = |u(t)| dt ,
0

and L∞ [0, T ] is the Banach space of essentially bounded functions from [0, T ] into Rn equipped
with the norm
kuk∞ = ess sup {|u(t)| : t ∈ [0, T ]} .
For 0 < α ≤ 1 and 1 < p < +∞, the fractional derivative space E0
α,p
is defined by

k·kα,p
= u ∈ L p [0, T ] : 0 Dtα u ∈ L p [0, T ] and u(0) = u( T ) = 0 = C0∞ [0, T ]
α,p 
E0 ,

where k · kα,p is defined as follows

Z T Z T 1/p
kukα,p = |u(t)| p dt + |0 Dtα u(t)| p dt . (2.1)
0 0

α,p
Then E0 is a reflexive and separable Banach space.
 Solutions for a class of fractional boundary value problems 5

Lemma 2.3 ( [6, Proposition 3.3]). Let 0 < α ≤ 1 and 1 < p < +∞. For all u ∈ E0 , if α > 1p , we
α,p

have
α α
0 It (0 Dt u ( t )) = u ( t )

and
Tα
kuk p ≤ k0 Dtα uk p . (2.2)
Γ ( α + 1)
1 1 1
In addition, if α > p and p + q = 1, then

α− 1p
T
kuk∞ ≤ 1 k0 Dtα uk p .
Γ(α)((α − 1)q + 1) q

α,p
Remark 2.4. According to (2.2), we can consider in E0 the following norm

kukα,p = k0 Dtα uk p , (2.3)

which is equivalent to (2.1).

In what follows we denote by Eα = E0α,2 . Then it is a Hilbert space with respect to the
norm kukα = kukα,2 given by (2.3).
The main difficulty in dealing with the existence of infinitely many solutions for (FBVP)
is to verify that the functional corresponding to (FBVP) satisfies (PS)-condition. To overcome
this difficulty, we need the following proposition.

Proposition 2.5 ( [6, Proposition 3.4]). Let 0 < α ≤ 1 and 1 < p < +∞. Assume that α > 1
p and
α,p
uk * u in E0 , then uk → u in C [0, T ], i.e.,

kuk − uk∞ → 0

as k → +∞.

Now we introduce more notations and some necessary definitions. Let B be a real Banach
space, I ∈ C1 (B , R) means that I is a continuously Fréchet differentiable functional defined
on B .

Definition 2.6. I ∈ C1 (B , R) is said to satisfy (PS)-condition if any sequence {uk }k∈N ⊂ B , for
which { I (uk )}k∈N is bounded and I 0 (uk ) → 0 as k → +∞, possesses a convergent subsequence
in B .

In order to find infinitely many solutions of (FBVP) under the assumptions of Theorem 1.1,
we shall use the “genus” properties. Therefore, it is necessary to recall the following defini-
tions and results, see [13, 14].
Let B be a Banach space, I ∈ C1 (B , R) and c ∈ R. We set

Σ = { A ⊂ B − {0} : A is closed in B and symmetric with respect to 0},

Kc = {u ∈ B : I (u) = c, I 0 (u) = 0}, I c = { u ∈ B : I ( u ) ≤ c }.

Definition 2.7. For A ∈ Σ, we say the genus of A is j (denoted by γ( A) = j) if there is an odd

map ψ ∈ C ( A, R j \{0}) and j is the smallest integer with this property.
6 Z. Zhang and J. Li

Lemma 2.8. Let I be an even C1 functional on B and satisfy (PS)-condition. For any j ∈ N, set

Σ j = { A ∈ Σ : γ ( A ) ≥ j }, c j = inf sup I (u).

A∈Σ j u∈ A

(i) If Σ j 6= ∅ and c j ∈ R, then c j is a critical value of I;

(ii) if there exists r ∈ N such that

c j = c j+1 = · · · = c j+r = c ∈ R,

and c 6= I (0), then γ(Kc ) ≥ r + 1.

Remark 2.9. From Remark 7.3 in [13], we know that if Kc ⊂ Σ and γ(Kc ) > 1, then Kc contains
infinitely many distinct points, i.e., I has infinitely many distinct critical points in B .

3 Proof of Theorem 1.1

The aim of this section is to give the proof of Theorem 1.1. To do this, we are going to establish
the corresponding variational framework of (FBVP). Define the functional I : B = Eα → R by
Z T 
1 2
I (u) = |0 Dt u(t)| − W (t, u(t)) dt.
α
(3.1)
0 2
Lemma 3.1 ( [6, Corollary 3.1]). Under the conditions of Theorem 1.1, I is a continuously Fréchet-
differentiable functional defined on Eα , i.e., I ∈ C1 ( Eα , R). Moreover, we have
Z T
0

I (u)v = (0 Dtα u(t), 0 Dtα v(t)) − (∇W (t, u(t)), v(t)) dt
0

for all u, v ∈ Eα , which yields that

Z T Z T
I 0 (u)u = |0 Dtα u(t)|2 dt − (∇W (t, u(t)), u(t))dt. (3.2)
0 0

Lemma 3.2. If (W1 ) and (W2 ) hold, then I satisfies (PS)-condition.

Proof. Assume that {uk }k∈N ⊂ Eα is a sequence such that { I (uk )}k∈N is bounded and
I 0 (uk ) → 0 as k → +∞. Then there exists a constant M > 0 such that

| I (uk )| ≤ M and k I 0 (uk )k(Eα )∗ ≤ M (3.3)

for every k ∈ N, where ( Eα )∗ is the dual space of Eα .

We firstly prove that {uk }k∈N is bounded in Eα . From (3.1) and (3.2), we obtain that
 σ
1− kuk k2α = I 0 (uk )uk − σI (uk )
2
Z T
+ [(∇W (t, uk (t)), uk (t)) − σW (t, uk (t))]dt (3.4)
0
≤ Mkuk kα + σM.

Since 1 < σ < 2, the inequality (3.4) shows that {uk }k∈N is bounded in Eα . Then the sequence
{uk }k∈N has a subsequence, again denoted by {uk }k∈N , and there exists u ∈ Eα such that

uk * u weakly in Eα ,
 Solutions for a class of fractional boundary value problems 7

which yields that

( I 0 (uk ) − I 0 (u))(uk − u) → 0. (3.5)
Moreover, according to Proposition 2.5, we have
Z T

∇W (t, uk (t)) − ∇W (t, u(t)), uk (t) − u(t) dt → 0 (3.6)
0

as k → +∞. Consequently, combining (3.5), (3.6) with the following equality

Z T
( I 0 (uk ) − I 0 (u))(uk − u) = kuk − uk2α −


∇W (t, uk (t)) − ∇W (t, u(t)), uk (t) − u(t) dt,
0

we deduce that kuk − ukα → 0 as k → +∞. That is, I satisfies the (PS)-condition.

Now we are in the position to complete the proof of Theorem 1.1.

Proof of Theorem 1.1. According to (W1 ) and (W3 ), it is obvious that I is even and I (0) = 0. In
order to apply Lemma 2.8, we prove that

for any j ∈ N there exists ε > 0 such that γ( I −ε ) ≥ j. (3.7)

To do this, let {e j }∞
j=1 be the standard orthogonal basis of E , i.e.,
α

kei kα = 1 and hei , ek i Eα = 0, 1 ≤ i 6= k. (3.8)

For any j ∈ N, define

Eαj = span{e1 , e2 , . . . , e j }, S j = {u ∈ Eαj : kukα = 1},

then, for any u ∈ Eαj , there exist λi ∈ R, i = 1, 2, . . . , j, such that

j
u(t) = ∑ λ i ei ( t ) for t ∈ [0, T ], (3.9)
i =1

which indicates that Z T

kuk2α = |0 Dtα u(t)|2 dt
0
j Z T
= ∑ λ2i 0
|0 Dtα ei (t)|2 dt (3.10)
i =1
j j
= ∑ λ2i kei k2α = ∑ λ2i .
i =1 i =1

On the other hand, in view of (W1 ), for any bounded open set D ⊂ [0, T ], there exists
η > 0 (dependent on D) such that

W (t, u) ≥ a(t)|u|ϑ ≥ η |u|ϑ , (t, u) ∈ D × Rn . (3.11)

As a result, for any u ∈ S j , we can take some D0 ⊂ [0, T ] such that

Z T Z T  j  Z  j ϑ
W t, ∑ λi ei (t) dt ≥ η  ∑ λi ei (t) dt =: $ > 0.
 
W (t, u(t)) dt = (3.12)
0 0 D0
 
i =1 i =1
8 Z. Zhang and J. Li

Indeed, if not, for any bounded open set D ⊂ [0, T ], there exists {un }n∈N ∈ S j such that

Z Z  j ϑ
∑

|un (t)| dt =
ϑ  λ in e i ( t )  dt → 0
D D
 
i =1

j j j
as n → +∞, where un = ∑i=1 λin ei such that ∑i=1 λ2in = 1. Since ∑i=1 λ2in = 1, we have

j
lim λin =: λi0 ∈ [−1, 1]
n→+∞
and ∑ λ2i0 = 1.
i =1

Hence, for any bounded open set D ⊂ [0, T ], it follows that

Z  j ϑ
 ∑ λi0 ei (t) dt = 0.

D
 
i =1

j
The fact that D is arbitrary yields that u0 = ∑i=1 λi0 ei (t) = 0 a.e. on [0, T ], which contradicts
the fact that ku0 kα =1. Hence, (3.12) holds.
Consequently, according to (W1 ) and (3.9)–(3.12), we have

s2
Z T
I (su) = kuk2α − W (t, su(t)) dt
2 0
j
s2
Z T  
= kuk2α − W t, s ∑ λi ei (t) dt
2 0 i =1
 j ϑ
s2
Z T
kuk2α − sϑ a(t) ∑ λi ei (t) dt
 
≤ 
2 0 i =1
 j ϑ
s2
Z
kuk2α − ηsϑ  ∑ λi ei (t) dt
 
≤
2 D0

i =1


s2
≤ kuk2α − $sϑ
2
s2
= − $sϑ , u ∈ Sj ,
2
which implies that there exist ε > 0 and δ > 0 such that

I (δu) < −ε for u ∈ S j . (3.13)

Let
 j 
Sδj = {δu : u ∈ S j } and Ω = ( λ1 , λ2 , . . . , λ j ) ∈ R : j
∑ λ2i <δ 2
.
i =1

Then it follows from (3.13) that

I (u) < −ε, ∀u ∈ Sδj ,

which, together with the fact that I ∈ C1 ( Eα , R) is even, yields that

Sδj ⊂ I −ε ∈ Σ.
 Solutions for a class of fractional boundary value problems 9

On the other hand, it follows from (3.9) and (3.10) that there exists an odd homeomorphism
mapping ψ ∈ C (Sδj , ∂Ω). By some properties of the genus (see 3◦ of Proposition 7.5 and 7.7
in [13]), we obtain
γ( I −ε ) ≥ γ(Sδj ) = j, (3.14)

so (3.7) follows. Set

c j = inf sup I (u),
A∈Σ j u∈ A

then, from (3.14) and the fact that I is bounded from below on Eα , we have −∞ < c j ≤ −ε < 0,
that is, for any j ∈ N, c j is a real negative number. By Lemma 2.8 and Remark 2.9, I has
infinitely many nontrivial critical points, and consequently (FBVP) possesses infinitely many
nontrivial solutions.

Acknowledgements
The authors would like to express their appreciation to the referee for valuable suggestions.
This work is supported by National Natural Science Foundation of China (No. 11101304).

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