EXISTENCE OF INFINITELY MANY SOLUTIONS FOR A FRACTIONAL

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 66, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR A FRACTIONAL DIFFERENTIAL INCLUSION WITH NON-SMOOTH POTENTIAL FARIBA FATTAHI, MOHSEN ALIMOHAMMADY Communicated by Vicentiu Radulescu

Abstract. In this article, we use non-smooth critical point theory and variational methods to study the existence solutions for a fractional boundary-value problem. We provide some intervals for positive parameters in which the problem possess infinitely many solutions.

1. Introduction In this article, we consider the boundary-value problem α α t DT (0 Dt u(t))

∈ λ∂F (u(t)) + µ∂G(u(t))

a.e. t ∈ [0, T ],

u(0) = u(T ) = 0,

(1.1)

where 0 Dtα and t DTα are the left and right Riemann-Liouville fractional derivatives of order α with 0 < α ≤ 1, and where λ > 0 and µ ≥R0 are two parameters. R ω F, G : ω R → R are locally Lipschitz functions, where F (ω) = 0 f (s)ds, G(ω) = 0 g(s)ds, ω ∈ R and f, g : R → R are locally essentially bounded functions. ∂F (u(t)) denotes the generalized Clarke gradient of the function F (u(t)) at u ∈ R. We consider the following problem: Find u ∈ E0α [0, T ], called a weak solution of (1.1), such that for any v ∈ E0α [0, T ], we have Z T Z T Z T [0 Dtα u(t) ·0 Dtα v(t)]dt = λ u∗1 (t)v(t)dt + µ u∗2 (t)v(t)dt, (1.2) 0

0

0

where u∗1 (t) ∈ ∂F (u(t)) and u∗2 (t) ∈ ∂G(u(t)). Fractional differential problems were studied by many authors, see for example [11, 12, 17, 23]. Recently, fractional differential inclusions were considered by many authors: Ahamd et al. [1] studied the existence of solutions for impulsive fractional differential inclusions with antiperiodic boundary conditions. Ntouyas et al. [15] studied the existence of solutions for boundary value problems for nonlinear fractional differential inclusions with mixed type integral boundary conditions. More recently, the study of differential equations by variational method and critical point theory has attracted a lot of attention; see for example [9, 18, 20, 21]. Variational-hemivariational inequalities 2010 Mathematics Subject Classification. 35J87, 26A33, 49J40, 49J52. Key words and phrases. Non-smooth critical point theory; fractional differential inclusion; infinitely many solutions. c

2017 Texas State University. Submitted August 21, 2016. Published March 4, 2017. 1

2

F. FATTAHI, M. ALIMOHAMMADY

EJDE-2017/66

have been extensively studied in recent years via variational methods; see [2, 3, 4, 5, 19]. Here, we investigate the existence of infinitely many solutions for a fractional differential inclusion under some hypotheses on the behavior of the locally Lipschitz functions F and G in theorem 3.1. We prove the existence of infinitely many solutions for a variational-hemivariational inequality depending on two parameters. Also, we list some consequences of theorem 3.1 and give an example. Finally, we consider the uniform convergence of a sequence of solutions to zero in theorem 3.6. 2. Preliminaries In this section, first we recall some basic definitions of fractional calculus and locally Lipschitz functions. Definition 2.1 ([8]). Let f be a function defined on [a, b]. The left and right Riemann-Liouville fractional integrals of order α of f are denoted by a Dt−α f (t) and t Db−α f (t), respectively, and defined by Z t 1 −α (t − s)α−1 f (s)ds, t ∈ [a, b], α > 0, a Dt f (t) = Γ(α) a Z b 1 −α D f (t) = (s − t)α−1 f (s)ds, t ∈ [a, b], α > 0, t b Γ(α) t provided the right-hand sides are pointwise defined on [a, b], where Γ(α) is the gamma function. Definition 2.2 ([8]). Let f be a function defined on [a, b]. For n − 1 ≤ α < n (n ∈ N), the left and right Riemann-Liouville fractional derivatives of order α for function f denoted by a Dtα f (t) and t Dbα f (t), respectively, are defined by Z t dn 1 dn α−n α D f (t) = D f (t) = (t − s)n−α−1 f (s)ds, t ∈ [a, b], a t a t dtn Γ(n − α) dtn a Z t n (−1)n dn α−n α n d D f (t) = (s − t)n−α−1 f (s)ds, t ∈ [a, b]. D f (t) = (−1) t b t b dtn Γ(n − α) dtn a Proposition 2.3 ([8, 22]). We have the following property of fractional integration Z b Z b [a Dt−α f (t)]g(t)dt = [t Db−α g(t)]f (t)dt, α > 0, (2.1) a

a

p

N

q

provided that f ∈ L ([a, b], R ), g ∈ L ([a, b], RN ) and p ≥ 1, q ≥ 1, or p 6= 1, q 6= 1, p1 + 1q = 1 + α.

1 p

+

1 q

≤ 1+α

Definition 2.4 ([8, 16]). For n ∈ N, n − 1 ≤ α < n (n ∈ N) and a function f ∈ AC n ([a, b], RN ), we define Z t 1 f (n) (s) f j (a) n−1 α ds + Σ (t − a)j−α , a Dt f (t) = j=0 Γ(n − α) a (t − s)α+1−n Γ(j − α + 1) Z b 1 f (n) (s) (−1)j f j (b) α D f (t) = ds + Σn−1 (b − t)j−α , t b j=0 α+1−n Γ(n − α) t (s − t) Γ(j − α + 1) where t ∈ [a, b].

EJDE-2017/66

EXISTENCE OF INFINITELY MANY SOLUTIONS

3

Definition 2.5 ([8, 16]). Let 0 < α ≤ 1. The fractional derivative space E0α [0, T ] is defined as the closure of C0∞ ([0, T ], R) with respect to the norm Z T 1/2 Z T α 2 |u(t)|2 dt , ∀u ∈ E0α [0, T ]. kukα = |0 Dt u(t)| dt + 0

0

Clearly, the fractional derivative space E0α [0, T ] is the space of functions u ∈ L [0, T ] having an α−order fractional derivative 0 Dtα u(t) ∈ L2 [0, T ] and u(0) = u(T ) = 0. 2

Proposition 2.6 ([6]). Let 0 < α ≤ 1. The fractional derivative space E0α [0, T ] is reflexive and separable Banach space. Proposition 2.7 ([6]). Let 0 < α ≤ 1. Then for all u ∈ E0α [0, T ], kukL2 ≤

Tα k0 Dtα u(t)kL2 , Γ(α + 1)

(2.2)

1

kuk∞ ≤

T α− 2 k0 Dtα u(t)kL2 . Γ(α)(2(α − 1) + 1)1/2

(2.3)

According to (2.2), one can consider E0α [0, T ] with the equivalent norm Z T 1/2 = k0 Dtα ukL2 , ∀u ∈ E0α [0, T ]. kukα = |0 Dtα u(t)|2 dt 0

Definition 2.8. A function u : [0, T ] → R is called a solution for (1.1) if (1) t DTα−1 (0 Dtα u(t)) and 0 Dtα−1 u(t) exist for almost all t ∈ [0, T ]; (2) u satisfies in (1.1). Definition 2.9. A function u ∈ E0α [0, T ] is called a weak solution of (1.1) if there exist u∗1 (x) ∈ ∂F (u), u∗2 (x) ∈ ∂G(u), such that u∗1 v, u∗1 v ∈ L1 [0, T ] and Z T Z T Z T α α ∗ [0 Dt u(t) · 0 Dt v(t)]dt = λ u1 (x)v(x)dx + µ u∗2 (x)v(x)dx, (2.4) 0

for all v ∈

0

0

E0α [0, T ].

For α > 1/2, propositions 2.7 and 2.8, imply that Z T 1/2 kuk∞ ≤ M |0 Dtα u(t)|2 dt = Mkukα ,

u ∈ E0α [0, T ],

0

where 1

T α− 2 M= . Γ(α)(2(α − 1) + 1)1/2 Here, we recall some definitions and basic notation of the theory of generalized differentiation for locally Lipschitz functions. We refer the reader to [3, 4, 13, 14, 18] for more details. Let X be a Banach space and X ? its topological dual. By k · k we denote the norm in X and by h·, ·iX the duality brackets for the pair (X, X ? ). A function h : X → R is said to be locally Lipschitz if for any x ∈ X there correspond a neighborhood Vx of x and a constant Lx ≥ 0 such that |h(z) − h(w)| ≤ Lx kz − wk, ∀z, w ∈ Vx .

4


EJDE-2017/66

For a locally Lipschitz function h : X → R, the generalized directional derivative of h at u ∈ X in the direction γ ∈ X is defined by h0 (u; γ) = lim sup w→u,t→0+

h(w + tγ) − h(w) . t

The generalized gradient of h at u ∈ X is ∂h(u) = {x? ∈ X ? : hx? , γiX ≤ h0 (u; γ), ∀γ ∈ X}, which is non-empty, convex and w? -compact subset of X ? , where < ·, ·iX is the duality pairing between X ? and X. Proposition 2.10 ([4]). Let h, g : X → R be locally Lipschitz functionals. Then, for any u, v ∈ X the following hold: (1) h0 (u; ·) is subadditive, positively homogeneous; (2) ∂h is convex and weak∗ compact; (3) (−h)0 (u; v) = h0 (u; −v); ∗ (4) the set-valued mapping h : X → 2X is weak∗ u.s.c.; (5) h0 (u; v) = max{< ξ, v >: ξ ∈ ∂h(u)}; (6) ∂(λh)(u) = λ∂h(u) for every λ ∈ R; (7) (h + g)0 (u; v) ≤ h0 (u; v) + g 0 (u; v); (8) the function m(u) = minν∈∂h(u) νX ∗ exists and is lower semicontinuous; i.e., lim inf u→u0 m(u) ≥ m(u0 ); (9) h0 (u; v) = maxu∗ ∈∂h(u) hu∗ , vi ≤ Lkvk. Definition 2.11 ([5]). An element u ∈ X is called a critical point for functional h if h0 (u; v − u) ≥ 0, ∀v ∈ X. Let X be a reflexive real Banach space, Φ : X → R a sequentially weakly lower semicontinuous and coercive, Υ : X → R a sequentially weakly upper semicontinuous, λ a positive real parameter. Moreover, assumeing that Φ and Υ are locally Lipschitz functionals, we set Lλ := Φ − λΥ. For every r > inf X Φ, we define supv∈Φ−1 (]−∞,r[) Υ(v) − Υ(u) ϕ(r) := inf , r − Φ(u) u∈Φ−1 (]−∞,r[) γ := lim inf ϕ(r), δ := lim inf + ϕ(r). r→+∞

r→(inf X Φ)

First, we need to the following non-smooth version of a critical point theorem. Theorem 2.12 ([10]). Under the assumptions stated for X, Φ and Υ, the following statements hold: 1 (a) For any r > inf X Φ and λ ∈ (0, ϕ(r) ), the restriction of the functional −1 Lλ = Φ − λΥ to Φ (−∞, r) admits a global minimum which is a critical point (local minimum) of Lλ in X. (b) If γ < +∞, then for each λ ∈ (0, γ1 ), the following alternative holds: either (b1) Lλ possesses a global minimum, or (b2) there is a sequence {un } of critical points (local minima) of Lλ such that lim Φ(un ) = +∞. n→+∞

(c) If δ < +∞, then for each λ ∈ (0, 1δ ), the following alternative holds: either

EJDE-2017/66


5

(c1) there is a global minimum of Φ which is a local minimum of Lλ , or (c2) there is a sequence {un } of pairwise distinct critical points (local minima) of Lλ that converges weakly to a global minimum of Φ. 3. Main results Set C(T, α) =

Z Z 3T /4 16 T /4 2(1−α) T t dt + (t1−α − (t − )1−α )2 dt T2 0 4 T /4 Z T 3T 1−α 2 T ) ) dt . (t1−α − (t − )1−α − (t − + 3T 4 4 4

and A := lim inf

ω→+∞

max|x|≤ω F (x) , ω2

B := lim sup ω→+∞

Theorem 3.1. Let 12 < α ≤ 1. Assume B (i) that A < M2 C(T,α) and f : R → R is a locally Rt such that F (t) = 0 f (ξ)dξ for all t ∈ R.

F (ω) . ω2

essentially bounded function

then, for each λ ∈ (λ1 , λ2 ), where 1 C(T, α) , λ2 = , BT M2 T A and for any locally essentially bounded function g : R → R whose potential G(t) = Rt g(ξ)dξ for all t ∈ R is a non-negative function satisfying the condition 0 λ1 =

G∞ = lim sup ω→+∞

max|x|≤ω G(x) < +∞ ω2

(3.1)

and for any µ ∈ [0, µG,λ ), where µG,λ =

1 M2 T G∞

(1 − λM2 T A).

Then problem (1.1) has a sequence of weak solutions for every µ ∈ [0, µG,λ ). ¯ ∈ (λ1 , λ2 ) and G satisfying Proof. Our purpose is to apply theorem 2.12(b). Fix λ ¯ < λ2 , it implies that our assumptions. Since λ µG,λ¯ =

1 M2 T G∞

¯ 2 T A) > 0. (1 − λM

Fix µ ¯ ∈ [0, µG,λ¯ ) and define the functionals Φ, Υ : X → R for each u ∈ X as follows: 1 kuk2α (3.2) 2 Z T Z µ ¯ T Υ(u) = [F (u(t))]dt + ¯ [G(u(t))]dt. (3.3) λ 0 0 ¯ Put Lλ¯ (u) := Φ(u)− λΥ. The critical points of the functional Lλ¯ are the weak solutions of problem (1.1). According to [6], Φ is continuous and convex, so it is weakly sequentially lower semicontinuous, also Φ is continuously Gˆateaux differentiable and coercive. By standard argument, Υ is sequentially weakly continuous. Φ(u) =

6


EJDE-2017/66

¯ < 1/γ. Note that Φ(0) = Υ(0) = 0, then for n large First, we claim that λ enoughlarge, supv∈Φ−1 (]−∞,r[) Υ(v) − Υ(u) ϕ(r) = inf r − Φ(u) u∈Φ−1 (]−∞,r[) supv∈Φ−1 (]−∞,r[) Υ(v) ≤ . r Let {ωn } be a sequence of positive numbers in X such that limn→+∞ ωn = +∞ and max|x|≤ωn F (x) . A = lim n→+∞ ωn2 Set ω2 rn = n2 , n ∈ N. M Hence, ϕ(rn ) ≤

max|x|≤ωn T [F (x) + µλ¯¯ G(x)] 2 ωn M2

max|x|≤ωn [F (x) + µλ¯¯ G(x)] ωn2 h max µ ¯ max|x|≤ωn G(x) i |x|≤ωn F (x) +¯ ≤ M2 T . 2 ωn ωn2 λ

≤ M2 T

Moreover, from assumption (i) and the condition (3.1), it follows that max|x|≤ωn F (x) µ ¯ max|x|≤ωn G(x) +¯ < +∞. ωn2 ωn2 λ Then

µ ¯ γ ≤ lim inf ϕ(rn ) ≤ M2 T (A + ¯ G∞ ) < +∞. n→+∞ λ It is clear that, for any µ ¯ ∈ [0, µG,λ¯ ), γ ≤ M2 T A +

¯ 2 T A) (1 − λM ; ¯ λ

therefore, 1 1 < . 2 ¯ ¯ γ + (1 − λM A)/λ We claim that the functional Lλ¯ is unbounded from below. We can consider a sequence {τn } of positive numbers such that τn → +∞. Let {ξn } be a sequence in X for all n ∈ N, defined by  4Γ(2−α)τ n  t t ∈ [0, T4 ]  T ξn (t) = Γ(2 − α)τn (3.4) t ∈ [ T4 , 3T 4 ]   4Γ(2−α)τn 3T (T − t) t ∈ [ 4 , T ], T ¯= λ

M2 T A

Clearly, ξn (0) = ξn (T ) = 0 and ξn ∈ L2 [0, T ]. A direct calculation shows that  4τn 1−α  t ∈ [0, T4 )  T t α 4τn 1−α T 1−α (3.5) − (t − 4 ) ) t ∈ [ T4 , 3T 0 Dt ξn (t) = T (t 4 ]   4τn 1−α T 1−α 3T 1−α 3T − (t − 4 ) − (t − 4 ) ) t ∈ ( 4 , T ]. T (t

EJDE-2017/66


7

Moreover, Z

T

(0 Dtα ξn (t))2 dt

0

Z

T /4

Z

T

Z

(0 Dtα ξn (t))2 dt

+ T /4

0

=

3T /4

+

=

3T 4

Z 16τn2 T /4 2(1−α) t dt T2 0

Z

3T /4

T 1−α 2 ) ) dt 4 T /4 Z T 3T 1−α 2 T ) ) dt (t1−α − (t − )1−α − (t − + 3T 4 4 4 (t1−α − (t −

+

(3.6)

= C(T, α)τn2 , for each n ∈ N. Thus, ξn ∈ E0α [0, T ]. Since G is non-negative and by the definition of Υ, we have Z T Z T µ ¯ Υ(ξn ) = [F (ξn (t)) + ¯ G(ξn (t)]dt ≥ F (ξn (t))dt λ 0 0 Z 3T /4 Z 3T /4 ≥ F (ξn (t))dt ≥ F (Γ(2 − α)τn ) dt. T /4

(3.7)

T /4

Let B = lim sup ω→+∞

F (ω) . ω2

(3.8)

If B < +∞, set ∈ (0, B − λΓC(T,α) 2 (2−α)T ). Then from 3.8 there exists N1 such that Z 3T /4 T F (Γ(2 − α)τn )dt > (B − )Γ2 (2 − α)τn2 , ∀n > N1 . 2 T /4 According to (3.6) and (3.7), 1 ¯ − )Γ2 (2 − α)τ 2 T C(T, α)τn2 − λ(B n 2 2 T 2 2 1 ¯ − )Γ (2 − α) ), = τn ( C(T, α) − λ(B 2 2 for n > N1 . Choosing a suitable and limn→+∞ τn = +∞, it results that Lλ¯ (ξn ) ≤

(3.9)

lim Lλ¯ (ξn ) = −∞.

n→+∞

If B = +∞, we fix ν > λΓC(T,α) 2 (2−α)T and from 3.8 there exists Nν such that Z 3T /4 T F (Γ(2 − α)τn )dt > νΓ2 (2 − α)τn2 , ∀n > Nν . 2 T /4 Hence, 1 ¯ (Γ(2 − α)τn Lλ¯ (ξn ) ≤ C(T, α)τn2 − λF 2

Z

3T /4

T /4

1 ¯ 2 (2 − α) T ), dt < τn2 ( C(T, α) − λνΓ 2 2

for all n > Nν . Taking into account the choice of ν, it leads to limn→+∞ Φ(un ) − ¯ λΨ(u ¯ is unbounded from below, and it follows n ) = −∞. Hence, the functional Lλ that Lλ¯ has no global minimum. Therefore, applying theorem (2.12) there exists a

8


EJDE-2017/66

sequence {un } ∈ X of pcritical points of Lλ¯ such that limn→+∞ Φ(un ) = +∞. From (3.2) it follows that 2 Φ(un ) = kun kα such that limn→+∞ kun kα = +∞. Lemma 3.2. Every critical point u ∈ E0α [0, T ] of Lλ is a solution of problem (1.1). Proof. We suppose that u ∈ E0α [0, T ] is a critical point of Lλ . There exist u∗1 ∈ ∂F (u) and u∗2 ∈ ∂G(u) satisfying Z T Z T Z T u∗2 (t)v(t)dt = 0, (3.10) u∗1 (t)v(t)dt − µ (0 Dtα u(t) · 0 Dtα v(t))dt − λ 0

0

0

for all v ∈ ∈ L ([0, T ], R ), it follows that t DT−α u∗1 , −α ∗ t DT u2 ∈ L ([0, T ], R ). Set k1 (t) = t DT−α u∗1 (t) and k2 (t) = t DT−α u∗2 (t), t ∈ [0, T ]. From the definition of left and right Riemann-Liouville fractional derivatives Z T Z T (k1 (t) · 0 Dtα v(t))dt + (k2 (t) · 0 Dtα v(t))dt E0α [0, T ]. Since 1 N

u∗1 v, u∗2 v

0

1

N

0

Z

T

(t DT−α u∗1 (t) · 0 Dtα v(t))dt +

= 0

Z

Z

T

(t DT−α u∗2 (t) · 0 Dtα v(t))dt

0

T

(u∗1 (t)

=

·

−α α 0 Dt ( 0 Dt v(t)))dt

0

Z =

T

Z

(u∗2 (t) · 0 Dt−α ( 0 Dtα v(t)))dt

+ 0

T

(u∗1 (t) · v(t))dt +

0

Z

T

(u∗2 (t) · v(t))dt.

0

From (3.10), T

Z

(0 Dtα u(t) − λk1 (t) − µk2 (t)) ·0 Dtα v(t))dt

0

Z = 0

T

(3.11)

α−1 (0 Dtα u(t) t DT

0

− λk1 (t) − µk2 (t)) · v (t) dt = 0,

for all v ∈ C0∞ ([0, T ], RN ). Using the argument in [7] we obtain α−1 (0 Dtα u(t) t DT

− λk1 (t) − µk2 (t)) = C,

∀t ∈ [0, T ].

In view of u∗1 , u∗2 ∈ L1 ([0, T ], RN ), we identify the equivalence class t DTα−1 (0 Dtα u(t)) and its continuous representative Z T Z T α−1 α ∗ (0 Dt u(t)) = λ u1 (t)dt + µ u∗2 (t)dt + C, ∀t ∈ [0, T ]. (3.12) t DT t

t

By properties of the left and right Riemann-Liouville fractional derivatives, we have t DTα (0 Dtα u(t)) = −(t DTα−1 (0 Dtα u(t)))0 ∈ L1 ([0, T ], RN ). Hence, it follows from (3.5), that α α t DT (0 Dt u(t))

= λu∗1 (t) + µu∗2 (t),

a.e. t ∈ [0, T ].

Moreover, u ∈ E0α [0, T ] implies that u(0) = u(T ) = 0. Remark 3.3. Under the following two conditions lim inf

ω→+∞

max|x|≤ω F (x) = 0, ω2

lim sup ω→+∞

F (ω) = +∞, ω2

EJDE-2017/66


9

according to Theorem 3.1, for each λ > 0 and each µ ∈ [0, M2 T1 G∞ [, problem (1.1) admits infinitely many solutions in E0α [0, T ]. In addition, if G∞ = 0, the result holds for every λ > 0 and µ ≥ 0. Example. Let {an }n∈N and {bn }n∈N be sequences defined by an = nen ,

bn = (n + 1)en .

We define a sequence of non-negative functions ( (|x| + n)2 exp − | ((|x|−nee −e1 n )2 −(e2n )) | nen < |x| < (n + 2)en Fn (x) = 0 otherwise. (3.13) A direct computation shows that lim

n→+∞

max|x|≤an Fn (x) = 0, a2n

lim

n→+∞

Fn (bn ) < +∞. b2n

Then Theorem 3.1, R ωimplies that for any non-negative function g : R → R, whose potential G(ω) = 0 g(t)dt satisfies the condition (3.1), problem (1.1) possesses a sequence of solutions. An immediate consequence of theorem 3.1 is a special case when µ = 0. Theorem 3.4. Assume that the assumptions in theorem 3.1 hold. Then, for each h i 1 C(T, α) , , λ∈ max|x|≤ω F (x) T lim supω→+∞ Fω(ω) M2 T lim inf ω→+∞ 2 ω2 the problem α α t DT (0 Dt u(t))

∈ λ∂F (u(t))

has an unbounded sequence of solutions in

a.e. t ∈ [0, T ],

E0α [0, T ].

Theorem R x3.5. Let l1 : R → R be a locally essentially bounded function and denote L1 (x) = 0 l1 (s)ds for all s ∈ R. Suppose that < +∞, (i1) lim inf ω→+∞ L1ω(ω) 2 L1 (ω) (i2) lim supω→+∞ ω2 = +∞. Then, for any locally essentially bounded functions li : R → R for 2 ≤ i ≤ n such that (i3) max supω∈R Li (ω); 2 ≤ i ≤ n ≤ 0 and Rx (i4) min lim inf ω→+∞ Liω(ω) > −∞, where Li (x) = 0 li (s)ds, 2 ;2 ≤ i ≤ n x ∈ R, 2 ≤ i ≤ n, for each h i 1 λ ∈ 0, M2 T lim inf ω→+∞ L1ω(ω) 2 and for any R x locally essentially bounded function g : R → R (whose potential G(x) = 0 g(s)ds for every x ∈ R) satisfying (3.1), and for every µ ∈ [0, µG,λ [, where µG,λ =

1 M2 T G∞

(1 − λM2 T lim inf

ω→+∞

L1 (ω) ), ω2

10


EJDE-2017/66

then the problem α α t DT (0 Dt u(t))

∈ λ∂Σni=1 Li (u(t)) + µ∂G(u(t))

a.e. t ∈ [0, T ], u(0) = u(T ) = 0,

admits an unbounded sequence of solutions in E0α [0, T ]. Proof. Set F (x) = Σni=1 Li (x) for all x ∈ R. In view of (i2) and (i4), lim sup ω→+∞

F (ω) Σni=1 Li (ω) = lim sup = +∞. ω2 ω2 ω→+∞

Conditions (i1) and (i3) imply that max|x|≤ω F (x) L1 (ω) ≤ lim inf < +∞. ω→+∞ ω 2 ω2 By using theorem 3.1, we complete the proof. lim inf

ω→+∞

Theorem 3.6. Let f be a locally essentially bounded function and suppose that lim inf + ω→0

F (ω) 1 F (ω) < lim sup . ω2 M2 C(T, α) ω→0+ ω 2

(3.14)

Then for any λ ∈ Λ1 :=]λ3 , λ4 [, where λ3 =

C(T, α) F (ω) ω2

T lim supω→0+

,

λ4 =

1 M2 T lim inf ω→0+

F (ω) ω2

,

and for any g : R → R such that G0 = lim sup ω→0+

max|x|≤ω G(x) < +∞ ω2

(3.15)

and 1

F (ω) ) ω2 ω→0 ( µ1G,λ = +∞, when G0 = 0). Problem 1.1 has a sequence of solutions, which converges strongly to 0 in E0α [0, T ]. µ1G,λ =

M2 T G0

(1 − λM2 T lim inf +

Proof. Fix λ ∈ Λ1 and pick µ ∈ [0, µ1G,λ [. Suppose that Φ, Υ are the functionals defined by (3.2) and (3.3). Let ln be a sequence of positive numbers such that limn→∞ ln = 0 and F (ln ) F (ω) lim = lim inf . n→∞ ln2 ω2 ω→0+ l2

As in theorem 3.1, we set rn = Mn2 , n ∈ N. It follows that δ < +∞. First we show that Φ − λΥ does not have a local minimum at zero. (3.16) Let {θn } be a sequence of positive numbers such that θn → 0 in ]0, θ[, θ > 0 and {ξn } be the sequence defined in 3.4. According to the non-negativity of G it leads that 3.7 satisfies. Using 3.14, λ3 < λ4 . Let B1 = lim sup ω→0+

F (ω) . ω2

If B1 < +∞, then 3.9 holds. By the choice of , lim (Φ(ξn ) − λΥ(ξn )) < 0 = Φ(0) − λΥ(0).

n→+∞

EJDE-2017/66


11

Therefore, Φ − λΥ does not have a local minimum at zero, in view of fact that kξn k → 0. An argument similar to the one in the proof of theorem 3.1 ,for the case B1 = 0, imply 3.16. Since minX Φ = Φ(0), in view of theorem 2.12(c) the consequence is obtained. Next we show an application of theorem 3.1 for obtaining infinitely many solutions. Consider α α t DT (0 Dt u(t)) ∈ −λθ(t)∂F (u) − µϑ(t)∂G(u) t ∈ [0, T ] (3.17) u(0) = u(T ) = 0, where λ, µ are real parameters, R ω λ > 0, µ ≥ 0 and R ω F, G : R → R are locally Lipschitz functions given by F (ω) = 0 f (t)dt, G(ω) = 0 g(t)dt, ω ∈ R such that f, g : R → R are measurable (not necessarily continuous) functions. Moreover, θ, ϑ ∈ L1 [0, T ] and θ, ϑ ≥ 0 will given. Our result is stated as follows. Theorem 3.7. Assume the following two conditions: (i1’) lim inf

ω→+∞

max|x|≤ω F (x) θ∗ lim supω→+∞ Fω(ω) 2 , < ω2 M2 C(T, α)

(i2’) λ1 =

C(T, α) T θ∗

lim supω→+∞ Fω(ω) 2

where θ∗ =

RT 0

,

λ2 =

θ(t)dt and ϑ∗ =

RT 0

1 T M2

lim inf ω→+∞

max|x|≤ω F (x) ω2

,

ϑ(t)dt.

Then for any µ ∈ [0, µG,λ ), problem (3.17) has an unbounded sequence of solutions in E0α [0, T ]. Proof. Define the functionals Φ, E : X → R for each u ∈ X as follows: 1 kuk2α , 2 Z T Z µ T Υ(u) = θ(t)F (u(t))dt + ϑ(t)G(u(t))dt, λ 0 0 E(u) = Υ(u) − χ(u), Lλ (u) := Φ(u) − λE(u). Φ(u) =

¯ < 1 . Note that As in theorem 3.1, we show that λ γ ϕ(rn ) ≤

max|x|≤ωn [θ∗ F (x) + µλ¯¯ ϑ∗ G(x)] 2 ωn M2 ∗

max|x|≤ωn [θ F (x) + µλ¯¯ ϑ∗ G(x)] ωn2 h max ∗ µ ¯ max|x|≤ωn ϑ∗ G(x) i |x|≤ωn θ F (x) ≤ M2 + , ¯ ωn2 ωn2 λ

≤ M2

Therefore, µ ¯ γ ≤ lim inf ϕ(rn ) ≤ M2 (θ∗ A + ¯ ϑ∗ G∞ ) < +∞. n→+∞ λ

12


EJDE-2017/66

It is clear that, for every µ ¯ ∈ [0, µG,λ¯ ), γ ≤ M2 θ∗ A + ϑ∗

¯ 2 T A) (1 − λM . ¯ Tλ

Then ¯= λ

1 1 ¯ 2 T A)/T λ ¯ < γ. M2 θ∗ A + ϑ∗ (1 − λM

We claim that the functional Lλ¯ is unbounded from below. Since G is non-negative, from the definition of Υ we have Z T Z µ T θ(t)F (u(t))dt + ϑ(t)G(u(t))dt Υ(u) = λ 0 0 Z 3T /4 θ(t)F (ξn (t))dt ≥

(3.18)

T /4

Z

3T /4

θ(t)dt = F (Γ(2 − α)τn )θ0 ,

≥ F (Γ(2 − α)τn ) T /4

where θ0 =

R 3T /4 T /4

θ(t)dt. Set B = lim sup ω→+∞

If B < +∞, let ∈ (0, B −

C(T,α) 2λθ 0 Γ2 (2−α) ),

F (ω) . ω2

(3.19)

then from 3.19 there exists N1 such that

θ0 F (Γ(2 − α)τn ) > θ0 (B − )Γ2 (2 − α)τn2 ,

∀n > N1 .

According to (3.18), 1 ¯ − )θ0 Γ2 (2 − α)τ 2 C(T, α)τn2 − λ(β n 2 (3.20) 1 ¯ − )θ0 Γ2 (2 − α)), = τn2 ( C(T, α) − λ(β 2 for n > N1 . Choosing a suitable and using that limn→+∞ τn = +∞, it results that lim Lλ¯ (ξn ) = −∞. Lλ¯ (ξn ) ≤

n→+∞

If B = +∞, we fix ν >

C(T,α) ¯ 0 Γ2 (2−α) 2λθ

and from 3.19, there exists Nν such that

θ0 F (Γ(2 − α)τn ) > θ0 νΓ2 (2 − α)τn2 ,

∀n > Nν .

Hence, Lλ¯ (ξn ) ≤

1 ¯ 0 F (Γ(2 − α)τn ) < τ 2 ( 1 C(T, α) − λθ ¯ 0 νΓ2 (2 − α)), C(T, α)τn2 − λθ n 2 2

for all n > Nν . Taking into account the choice of ν, implies that limn→+∞ Φ(un ) − ¯ λΨ(u n ) = −∞. From theorem 2.12, there is a sequence {un } ∈ X of critical points of Lλ¯ such that limn→+∞ Φ(un ) = +∞. Acknowledgements. The authors are very grateful to the anonymous referee for the valuable suggestions.

EJDE-2017/66


13

References [1] B. Ahmad, J. J. Nieto; A study of impulsive fractional differential inclusions with antiperiodic boundary conditions, Fractional Differential Calculus, 1 (2012), 1–15. [2] M. Alimohammady, F. Fattahi; Existence of solutions to hemivaritional inequalities involving the p(x)-biharmonic operator, Electron. J. Diff. Equ., Vol. 2015 (2015), 1–12. [3] S. Carl, V. K. Le, D. Motreanu; Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, 2007. [4] F. H. Clarke; Optimization and nonsmooth analysis, Wiley, 1983. [5] A. Iannizzotto; Three critical points for perturbed nonsmooth functionals and applications, Nonlinear Analysis, Theory, Methods & Applications , 72 (2010), 1319–1338. [6] F. Jiao, Y. Zhou; Existence results for fractional boundary value problem via critical point theory, International Journal of Bifurcation and Chaos, 22 (2012), 1–17. [7] F. Jiao, Y. Zhou; Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), 1181–1199. [8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [9] L. Li, V. R˘ adulescu, D. Repovs; Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness, Int. J. Nonlinear Sci. Numer. Simul., 17 (2016), 325332. [10] S. A. Marano, D. Motreanu; Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differential Equations, 182 (1) (2002), 108–120. [11] G. Molica Bisci, V. R˘ adulescu; Ground state solutions of scalar field fractional Schr¨ odinger equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 2985–3008. [12] G. Molica Bisci, D. Repovs; On doubly nonlocal fractional elliptic equations, Rend. Lincei Mat. Appl., 26 (2015), 161-176. [13] D. Motreanu, P. D. Panagiotopoulos; Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Kluwer Academic Publishers, Dordrecht, 1999. [14] D. Motreanu, V. R˘ adulescu; Variational and non-variational methods in nonlinear analysis and boundary value problems, Kluwer Academic Publishers, Boston-Dordrecht-London, 2003. [15] S. K. Ntouyas, S. Etemad, J, Tariboon; Existence of solutions for fractional differential inclusions with integral boundary conditions, Boundary Value Problems, (2015), 1–14. DOI:10.1186/s13661-015-0356-y. [16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [17] P. Pucci, M. Xiang, B. Zhang; Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), no. 1, 27–55. [18] V. R˘ adulescu; Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics and Its Applications, Vol. 6. Hindawi Publ Corp., New York, 2008. [19] V. R˘ adulescu, B. Alleche; Equilibrium problems techniques in the qualitative analysis of quasi-hemivariational inequalities, Optimization 64 (2015), 1855–1868. [20] V. R˘ adulescu, D. Repovˇs; Partial differential equations with variable exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. [21] V. R˘ adulescu; Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Analysis: Theory, Methods and Applications, 121(2015), 336–369. [22] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional integral and derivatives: theory and applications, Gordon and Breach, Longhorne, PA, 1993. [23] X. Zhang, B. Zhang, M. Xiang, Ground states for fractional Schr¨ odinger equations involving a critical nonlinearity, Adv. Nonlinear Anal., 5(2016), no. 3, 293-314. Fariba Fattahi Department of Mathematics, University of Mazandaran, Babolsar, Iran E-mail address: [email protected] Mohsen Alimohammady Department of Mathematics, University of Mazandaran, Babolsar, Iran E-mail address: [email protected]