EXISTENCE OF INFINITELY MANY SOLUTIONS FOR A FRACTIONAL DIFFERENTIAL INCLUSION WITH NON-SMOOTH POTENTIAL

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1 Electronic Journal of Differential Equations, Vol. 27 (27), No. 66, pp. 3. ISSN: URL: or EXISENCE OF INFINIELY MANY SOLUIONS FOR A FRACIONAL DIFFERENIAL INCLUSION WIH NON-SMOOH POENIAL FARIBA FAAHI, 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.. Introduction In this article, we consider the boundary-value problem td α ( D α t u(t)) λ F (u(t)) + µ G(u(t)) a.e. t [, ], u() = u( ) =, (.) where Dt α and t D α are the left and right Riemann-Liouville fractional derivatives of order α with < α, and where λ > and µ are two parameters. F, G : R R are locally Lipschitz functions, where = ω f(s)ds, G(ω) = ω 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 E α [, ], called a weak solution of (.), such that for any v E α [, ], we have [ D α t u(t) D α t v(t)]dt = λ u (t)v(t)dt + µ u 2(t)v(t)dt, (.2) where u (t) F (u(t)) and u 2(t) G(u(t)). Fractional differential problems were studied by many authors, see for example [, 2, 7, 23]. Recently, fractional differential inclusions were considered by many authors: Ahamd et al. [] studied the existence of solutions for impulsive fractional differential inclusions with antiperiodic boundary conditions. Ntouyas et al. [5] 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, 8, 2, 2]. Variational-hemivariational inequalities 2 Mathematics Subject Classification. 35J87, 26A33, 49J4, 49J52. Key words and phrases. Non-smooth critical point theory; fractional differential inclusion; infinitely many solutions. c 27 exas State University. Submitted August 2, 26. Published March 4, 27.

2 2 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 have been extensively studied in recent years via variational methods; see [2, 3, 4, 5, 9]. 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.. 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. and give an example. Finally, we consider the uniform convergence of a sequence of solutions to zero in theorem Preliminaries In this section, first we recall some basic definitions of fractional calculus and locally Lipschitz functions. Definition 2. ([8]). Let f be a function defined on [a, b]. he left and right Riemann-Liouville fractional integrals of order α of f are denoted by a Dt α f(t) and t D α b f(t), respectively, and defined by adt α f(t) = Γ(α) td α b f(t) = Γ(α) t a b t (t s) α f(s)ds, t [a, b], α >, (s t) α f(s)ds, t [a, b], α >, 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 α < n (n N), the left and right Riemann-Liouville fractional derivatives of order α for function f denoted by a Dt α f(t) and t Db α f(t), respectively, are defined by ad α t f(t) = dn dt n a D α n t f(t) = Γ(n α) dt n d n tdb α f(t) = ( ) n dn dt n t D α n b f(t) = ( )n Γ(n α) t a d n t dt n (t s) n α f(s)ds, t [a, b], a (s t) n α f(s)ds, t [a, b]. Proposition 2.3 ([8, 22]). We have the following property of fractional integration b a [ a D α t f(t)]g(t)dt = b a [ t D α b g(t)]f(t)dt, α >, (2.) provided that f L p ([a, b], R N ), g L q ([a, b], R N ) and p, q, p + q + α or p, q, p + q = + α. Definition 2.4 ([8, 6]). For n N, n α < n (n N) and a function f AC n ([a, b], R N ), we define adt α f(t) = Γ(n α) td α b f(t) = where t [a, b]. Γ(n α) t a b t f (n) (s) (t s) f (n) (s) (s t) f j (a) ds + Σn α+ n j= Γ(j α + ) (t a)j α, ( ) j f j (b) ds + Σn α+ n j= Γ(j α + ) (b t)j α,

3 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS 3 Definition 2.5 ([8, 6]). Let < α. he fractional derivative space E α [, ] is defined as the closure of C ([, ], R) with respect to the norm ( u α = Dt α u(t) 2 dt + u(t) 2 dt) /2, u E α [, ]. Clearly, the fractional derivative space E α [, ] is the space of functions u L 2 [, ] having an α order fractional derivative D α t u(t) L 2 [, ] and u() = u( ) =. Proposition 2.6 ([6]). Let < α. he fractional derivative space E α [, ] is reflexive and separable Banach space. Proposition 2.7 ([6]). Let < α. hen for all u E α [, ], u α u L 2 Γ(α + ) Dt α u(t) L 2, (2.2) α 2 Γ(α)(2(α ) + ) /2 D α t u(t) L 2. (2.3) According to (2.2), one can consider E α [, ] with the equivalent norm ( /2 u α = Dt α u(t) dt) 2 = Dt α u L 2, u E α [, ]. Definition 2.8. A function u : [, ] R is called a solution for (.) if () t D α ( Dt α u(t)) and Dt α u(t) exist for almost all t [, ]; (2) u satisfies in (.). Definition 2.9. A function u E α [, ] is called a weak solution of (.) if there exist u (x) F (u), u 2(x) G(u), such that u v, u v L [, ] and [ D α t u(t) D α t v(t)]dt = λ for all v E α [, ]. u (x)v(x)dx + µ For α > /2, propositions 2.7 and 2.8, imply that where u 2(x)v(x)dx, (2.4) ( /2 u M Dt α u(t) dt) 2 = M u α, u E α [, ], M = α 2 Γ(α)(2(α ) + ) /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, 3, 4, 8] for more details. Let X be a Banach space and X its topological dual. By we denote the norm in X and by, X 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 V x of x and a constant L x such that h(z) h(w) L x z w, z, w V x.

4 4 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 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 h h(w + tγ) h(w) (u; γ) = lim sup. w u,t t + he generalized gradient of h at u X is h(u) = {x X : x, γ X h (u; γ), γ X}, which is non-empty, convex and w -compact subset of X, where <, X is the duality pairing between X and X. Proposition 2. ([4]). Let h, g : X R be locally Lipschitz functionals. hen, for any u, v X the following hold: () h (u; ) is subadditive, positively homogeneous; (2) h is convex and weak compact; (3) ( h) (u; v) = h (u; v); (4) the set-valued mapping h : X 2 X is weak u.s.c.; (5) h (u; v) = max{< ξ, v >: ξ h(u)}; (6) (λh)(u) = λ h(u) for every λ R; (7) (h + g) (u; v) h (u; v) + g (u; v); (8) the function m(u) = min ν h(u) ν X exists and is lower semicontinuous; i.e., lim inf u u m(u) m(u ); (9) h (u; v) = max u h(u) u, v L v. Definition 2. ([5]). An element u X is called a critical point for functional h if h (u; v u), 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 L λ := Φ λυ. For every r > inf X Φ, we define ( supv Φ (],r[) Υ(v) ) Υ(u) ϕ(r) := inf u Φ (],r[) r Φ(u) 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. heorem 2.2 ([]). Under the assumptions stated for X, Φ and Υ, the following statements hold: (a) For any r > inf X Φ and λ (, ϕ(r) ), the restriction of the functional L λ = Φ λυ to Φ (, r) admits a global minimum which is a critical point (local minimum) of L λ in X. (b) If γ < +, then for each λ (, γ ), the following alternative holds: either (b) L λ possesses a global minimum, or (b2) there is a sequence {u n } of critical points (local minima) of L λ such that lim Φ(u n) = +. n + (c) If δ < +, then for each λ (, δ ), the following alternative holds: either,

5 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS 5 and Set (c) there is a global minimum of Φ which is a local minimum of L λ, or (c2) there is a sequence {u n } of pairwise distinct critical points (local minima) of L λ that converges weakly to a global minimum of Φ. 3. Main results C(, α) = 6 ( /4 2 t 2( α) dt /4 /4 (t α (t 4 ) α ) 2 dt (t α (t 4 ) α (t 3 ) 4 ) α ) 2 dt. max x ω F (x) A := lim inf ω + ω 2, B := lim sup ω + ω 2. heorem 3.. Let 2 < α. Assume (i) that A < and f : R R is a locally essentially bounded function B M 2 C(,α) such that F (t) = t then, for each λ (λ, λ 2 ), where f(ξ)dξ for all t R. C(, α) λ = B, λ 2 = M 2 A, and for any locally essentially bounded function g : R R whose potential G(t) = t g(ξ)dξ for all t R is a non-negative function satisfying the condition and for any µ [, µ G,λ ), where max x ω G(x) G = lim sup ω + ω 2 < + (3.) µ G,λ = M 2 G ( λm 2 A). hen problem (.) has a sequence of weak solutions for every µ [, µ G,λ ). Proof. Our purpose is to apply theorem 2.2(b). Fix λ (λ, λ 2 ) and G satisfying our assumptions. Since λ < λ 2, it implies that µ G, λ = M 2 G ( λm 2 A) >. Fix µ [, µ G, λ) and define the functionals Φ, Υ : X R for each u X as follows: Υ(u) = Φ(u) = 2 u 2 α (3.2) [F (u(t))]dt + µ λ [G(u(t))]dt. (3.3) Put L λ(u) := Φ(u) λυ. he critical points of the functional L λ are the weak solutions of problem (.). According to [6], Φ is continuous and convex, so it is weakly sequentially lower semicontinuous, also Φ is continuously Gâteaux differentiable and coercive. By standard argument, Υ is sequentially weakly continuous.

6 6 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 First, we claim that λ < /γ. Note that Φ() = Υ() =, then for n large enoughlarge, ( supv Φ ϕ(r) = inf (],r[) Υ(v) ) Υ(u) u Φ (],r[) r Φ(u) sup v Φ (],r[) Υ(v). r Let {ω n } be a sequence of positive numbers in X such that lim n + ω n = + and max x ωn F (x) A = lim n + ωn 2. Set Hence, r n = ω2 n M 2, n N. ϕ(r n ) max x ω n [F (x) + µ λ G(x)] ω 2 n M 2 M 2 max x ω n [F (x) + µ λ G(x)] [ M 2 max x ωn F (x) ωn 2 + µ λ max x ωn G(x) ] ωn 2. Moreover, from assumption (i) and the condition (3.), it follows that max x ωn F (x) ω 2 n + µ λ ω 2 n max x ωn G(x) ω 2 n < +. hen γ lim inf ϕ(r n) M 2 (A + µ λ G ) < +. n + It is clear that, for any µ [, µ G, λ), γ M 2 A + ( λm 2 A) ; λ therefore, λ = M 2 A + ( λm 2 A)/ λ < γ. We claim that the functional L λ is unbounded from below. We can consider a sequence {τ n } of positive numbers such that τ n +. Let {ξ n } be a sequence in X for all n N, defined by 4Γ(2 α)τ n t t [, 4 ] ξ n (t) = Γ(2 α)τ n t [ 4, 3 4 ] 4Γ(2 α)τ n ( t) t [ 3 4, ], (3.4) Clearly, ξ n () = ξ n ( ) = and ξ n L 2 [, ]. A direct calculation shows that 4τ n t α t [, Dt α 4 ) 4τ ξ n (t) = n (t α (t 4 ) α ) t [ 4, 3 4 ] (3.5) 4τ n (t α (t 4 ) α (t 3 4 ) α ) t ( 3 4, ].

7 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS 7 Moreover, = C(, α)τ 2 n, ( D α t ξ n (t)) 2 dt /4 = + 3/4 / = 6τ n 2 ( /4 2 t 2( α) dt ( D α t ξ n (t)) 2 dt 3/4 /4 (t α (t 4 ) α ) 2 dt (t α (t 4 ) α (t 3 ) 4 ) α ) 2 dt for each n N. hus, ξ n E α [, ]. Since G is non-negative and by the definition of Υ, we have Let Υ(ξ n ) = 3/4 /4 If B < +, set ɛ (, B 3/4 /4 According to (3.6) and (3.7), [F (ξ n (t)) + µ λ G(ξ n (t)]dt F (ξ n (t))dt F (Γ(2 α)τ n ) F (ξ n (t))dt 3/4 /4 dt. (3.6) (3.7) B = lim sup ω + ω 2. (3.8) C(,α) λγ 2 (2 α) ). hen from 3.8 there exists N such that F (Γ(2 α)τ n )dt > (B ɛ)γ 2 (2 α)τn 2 2, n > N. L λ(ξ n ) 2 C(, α)τ n 2 λ(b ɛ)γ 2 (2 α)τn 2 2 = τn( 2 2 C(, α) λ(b ɛ)γ 2 (2 α) (3.9) 2 ), for n > N. Choosing a suitable ɛ and lim n + τ n = +, it results that If B = +, we fix ν > Hence, 3/4 /4 C(,α) λγ 2 (2 α) lim L λ(ξ n ) =. n + and from 3.8 there exists N ν such that F (Γ(2 α)τ n )dt > νγ 2 (2 α)τn 2 2, n > N ν. L λ(ξ n ) 3/4 2 C(, α)τ n 2 λf (Γ(2 α)τ n dt < τn( 2 2 C(, α) λνγ 2 (2 α) 2 ), /4 for all n > N ν. aking into account the choice of ν, it leads to lim n + Φ(u n ) λψ(u n ) =. Hence, the functional L λ is unbounded from below, and it follows that L λ has no global minimum. herefore, applying theorem (2.2) there exists a

8 8 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 sequence {u n } X of critical points of L λ such that lim n + Φ(u n ) = +. From (3.2) it follows that 2 Φ(u n ) = u n α such that lim n + u n α = +. Lemma 3.2. Every critical point u E α [, ] of L λ is a solution of problem (.). Proof. We suppose that u E α [, ] is a critical point of L λ. here exist u F (u) and u 2 G(u) satisfying ( Dt α u(t) Dt α v(t))dt λ u (t)v(t)dt µ u 2(t)v(t)dt =, (3.) for all v E α [, ]. Since u v, u 2v L ([, ], R N ), it follows that t D α u, td α u 2 L ([, ], R N ). Set k (t) = t D α u (t) and k 2 (t) = t D α u 2(t), t [, ]. From the definition of left and right Riemann-Liouville fractional derivatives = = = (k (t) D α t v(t))dt + From (3.), ( t D α u (t) D α t v(t))dt + (k 2 (t) D α t v(t))dt (u (t) D α t ( D α t v(t)))dt + (u (t) v(t))dt + = ( t D α u 2(t) Dt α v(t))dt (u 2(t) v(t))dt. ( D α t u(t) λk (t) µk 2 (t)) D α t v(t))dt ( (u 2(t) Dt α ( Dt α v(t)))dt ) td α ( Dt α u(t) λk (t) µk 2 (t)) v (t) dt =, for all v C ([, ], R N ). Using the argument in [7] we obtain td α ( D α t u(t) λk (t) µk 2 (t)) = C, t [, ]. (3.) In view of u, u 2 L ([, ], R N ), we identify the equivalence class t D α ( Dt α u(t)) and its continuous representative td α ( Dt α u(t)) = λ t u (t)dt + µ t u 2(t)dt + C, t [, ]. (3.2) By properties of the left and right Riemann-Liouville fractional derivatives, we have t D α( Dt α u(t)) = ( t D α ( Dt α u(t))) L ([, ], R N ). Hence, it follows from (3.5), that td α ( D α t u(t)) = λu (t) + µu 2(t), a.e. t [, ]. Moreover, u E α [, ] implies that u() = u( ) =. Remark 3.3. Under the following two conditions max x ω F (x) lim inf ω + ω 2 =, lim sup ω + ω 2 = +,

9 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS 9 according to heorem 3., for each λ > and each µ [, M 2 G [, problem (.) admits infinitely many solutions in E α [, ]. In addition, if G =, the result holds for every λ > and µ. Example. Let {a n } n N and {b n } n N be sequences defined by a n = ne n, b n = (n + )e n. We define a sequence of non-negative functions { ( x + n) 2 exp ( F n (x) = (( x ne e e n ) 2 (e 2n )) ) otherwise. A direct computation shows that ne n < x < (n + 2)e n max x an F n (x) F n (b n ) lim n + a 2 =, lim n n + b 2 < +. n (3.3) hen heorem 3., implies that for any non-negative function g : R R, whose potential G(ω) = ω g(t)dt satisfies the condition (3.), problem (.) possesses a sequence of solutions. An immediate consequence of theorem 3. is a special case when µ =. heorem 3.4. Assume that the assumptions in theorem 3. hold. hen, for each ] C(, α) [ λ,, lim sup ω + ω M 2 max lim inf x ω F (x) 2 ω + ω 2 the problem td α ( D α t u(t)) λ F (u(t)) a.e. t [, ], has an unbounded sequence of solutions in E α [, ]. heorem 3.5. Let l : R R be a locally essentially bounded function and denote L (x) = x l (s)ds for all s R. Suppose that L (i) lim inf (ω) ω + ω < +, 2 L (i2) lim sup (ω) ω + ω = +. 2 hen, for any locally essentially bounded functions l i : R R for 2 i n such that (i3) max { sup ω R L i (ω); 2 i n } and (i4) min { L lim inf i(ω) ω + ω ; 2 i n } >, where L 2 i (x) = x l i(s)ds, x R, 2 i n, for each ] [ λ, M 2 L lim inf (ω) ω + ω 2 and for any locally essentially bounded function g : R R (whose potential G(x) = x g(s)ds for every x R) satisfying (3.), and for every µ [, µ G,λ [, where µ G,λ = M 2 ( λm 2 L (ω) lim inf G ω + ω 2 ),

10 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 then the problem td α ( D α t u(t)) λ Σ n i=l i (u(t)) + µ G(u(t)) a.e. t [, ], u() = u( ) =, admits an unbounded sequence of solutions in E α [, ]. Proof. Set F (x) = Σ n i= L i(x) for all x R. In view of (i2) and (i4), Conditions (i) and (i3) imply that Σ n i= lim sup ω + ω 2 = lim sup L i(ω) ω + ω 2 = +. max x ω F (x) lim inf ω + ω 2 By using theorem 3., we complete the proof. L (ω) lim inf ω + ω 2 < +. heorem 3.6. Let f be a locally essentially bounded function and suppose that lim inf ω + hen for any λ Λ :=]λ 3, λ 4 [, where C(, α) λ 3 = lim sup ω + and for any g : R R such that ω 2 < M 2 lim sup C(, α) ω ω + 2. (3.4) G = lim sup ω + ω 2, λ 4 = M 2 lim inf ω +, ω 2 max x ω G(x) ω 2 < + (3.5) and µ G,λ = M 2 ( λm 2 lim inf G ω + ω 2 ) ( µ G,λ = +, when G = ). Problem. has a sequence of solutions, which converges strongly to in E α [, ]. Proof. Fix λ Λ and pick µ [, µ G,λ [. Suppose that Φ, Υ are the functionals defined by (3.2) and (3.3). Let l n be a sequence of positive numbers such that lim n l n = and F (l n ) lim n ln 2 = lim inf ω + ω 2. As in theorem 3., we set r n = l2 n M 2, n N. It follows that δ < +. First we show that Φ λυ does not have a local minimum at zero. (3.6) Let {θ n } be a sequence of positive numbers such that θ n in ], θ[, θ > and {ξ n } be the sequence defined in 3.4. According to the non-negativity of G it leads that 3.7 satisfies. Using 3.4, λ 3 < λ 4. Let B = lim sup ω + If B < +, then 3.9 holds. By the choice of ɛ, ω 2. lim (Φ(ξ n) λυ(ξ n )) < = Φ() λυ(). n +

11 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS herefore, Φ λυ does not have a local minimum at zero, in view of fact that ξ n. An argument similar to the one in the proof of theorem 3.,for the case B =, imply 3.6. Since min X Φ = Φ(), in view of theorem 2.2(c) the consequence is obtained. Next we show an application of theorem 3. for obtaining infinitely many solutions. Consider td α ( D α t u(t)) λθ(t) F (u) µϑ(t) G(u) t [, ] u() = u( ) =, (3.7) where λ, µ are real parameters, λ >, µ and F, G : R R are locally Lipschitz functions given by = ω f(t)dt, G(ω) = ω g(t)dt, ω R such that f, g : R R are measurable (not necessarily continuous) functions. Moreover, θ, ϑ L [, ] and θ, ϑ will given. Our result is stated as follows. heorem 3.7. Assume the following two conditions: (i ) (i2 ) λ = max x ω F (x) lim inf ω + ω 2 < θ lim sup ω + M 2 C(, α) ω 2 C(, α), λ θ 2 =, lim sup ω + ω M 2 max lim inf x ω F (x) 2 ω + ω 2 where θ = θ(t)dt and ϑ = ϑ(t)dt. hen for any µ [, µ G,λ ), problem (3.7) has an unbounded sequence of solutions in E α [, ]. Proof. Define the functionals Φ, E : X R for each u X as follows: Φ(u) = 2 u 2 α, Υ(u) = θ(t)f (u(t))dt + µ ϑ(t)g(u(t))dt, λ E(u) = Υ(u) χ(u), L λ (u) := Φ(u) λe(u). As in theorem 3., we show that λ < γ. Note that, herefore, ϕ(r n ) max x ω n [θ F (x) + µ λ ϑ G(x)] ω 2 n M 2 M 2 max x ω n [θ F (x) + µ λ ϑ G(x)] ωn 2 M 2[ max x ωn θ F (x) ωn 2 + µ λ max x ωn ϑ G(x) ] ωn 2, γ lim inf n + ϕ(r n) M 2 (θ A + µ λ ϑ G ) < +.

12 2 F. FAAHI, M. ALIMOHAMMADY EJDE-27/66 It is clear that, for every µ [, µ G, λ), hen λ = γ M 2 θ A + ϑ ( λm 2 A) λ. M 2 θ A + ϑ ( λm 2 A)/ λ < γ. We claim that the functional L λ is unbounded from below. Since G is non-negative, from the definition of Υ we have Υ(u) = 3/4 /4 θ(t)f (u(t))dt + µ λ θ(t)f (ξ n (t))dt F (Γ(2 α)τ n ) 3/4 /4 ϑ(t)g(u(t))dt θ(t)dt = F (Γ(2 α)τ n )θ, (3.8) where θ = 3/4 /4 θ(t)dt. Set If B < +, let ɛ (, B According to (3.8), B = lim sup ω + ω 2. (3.9) C(,α) 2λθ Γ 2 (2 α) ), then from 3.9 there exists N such that θ F (Γ(2 α)τ n ) > θ (B ɛ)γ 2 (2 α)τ 2 n, n > N. L λ(ξ n ) 2 C(, α)τ 2 n λ(β ɛ)θ Γ 2 (2 α)τ 2 n = τ 2 n( 2 C(, α) λ(β ɛ)θ Γ 2 (2 α)), (3.2) for n > N. Choosing a suitable ɛ and using that lim n + τ n = +, it results that If B = +, we fix ν > Hence, lim L λ(ξ n ) =. n + C(,α) 2 λθ Γ 2 (2 α) and from 3.9, there exists N ν such that θ F (Γ(2 α)τ n ) > θ νγ 2 (2 α)τ 2 n, n > N ν. L λ(ξ n ) 2 C(, α)τ 2 n λθ F (Γ(2 α)τ n ) < τ 2 n( 2 C(, α) λθ νγ 2 (2 α)), for all n > N ν. aking into account the choice of ν, implies that lim n + Φ(u n ) λψ(u n ) =. From theorem 2.2, there is a sequence {u n } X of critical points of L λ such that lim n + Φ(u n ) = +. Acknowledgements. he authors are very grateful to the anonymous referee for the valuable suggestions.

13 EJDE-27/66 EXISENCE OF INFINIELY MANY SOLUIONS 3 References [] B. Ahmad, J. J. Nieto; A study of impulsive fractional differential inclusions with antiperiodic boundary conditions, Fractional Differential Calculus, (22), 5. [2] M. Alimohammady, F. Fattahi; Existence of solutions to hemivaritional inequalities involving the p(x)-biharmonic operator, Electron. J. Diff. Equ., Vol. 25 (25), 2. [3] S. Carl, V. K. Le, D. Motreanu; Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, 27. [4] F. H. Clarke; Optimization and nonsmooth analysis, Wiley, 983. [5] A. Iannizzotto; hree critical points for perturbed nonsmooth functionals and applications, Nonlinear Analysis, heory, Methods & Applications, 72 (2), [6] F. Jiao, Y. Zhou; Existence results for fractional boundary value problem via critical point theory, International Journal of Bifurcation and Chaos, 22 (22), 7. [7] F. Jiao, Y. Zhou; Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2), [8] A. A. Kilbas, H. M. Srivastava, J. J. rujillo; heory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 26. [9] L. Li, V. Rădulescu, D. Repovs; Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness, Int. J. Nonlinear Sci. Numer. Simul., 7 (26), [] 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, 82 () (22), 8 2. [] G. Molica Bisci, V. Rădulescu; Ground state solutions of scalar field fractional Schrödinger equations, Calculus of Variations and Partial Differential Equations, 54 (25), [2] G. Molica Bisci, D. Repovs; On doubly nonlocal fractional elliptic equations, Rend. Lincei Mat. Appl., 26 (25), [3] D. Motreanu, P. D. Panagiotopoulos; Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Kluwer Academic Publishers, Dordrecht, 999. [4] D. Motreanu, V. Rădulescu; Variational and non-variational methods in nonlinear analysis and boundary value problems, Kluwer Academic Publishers, Boston-Dordrecht-London, 23. [5] S. K. Ntouyas, S. Etemad, J, ariboon; Existence of solutions for fractional differential inclusions with integral boundary conditions, Boundary Value Problems, (25), 4. DOI:.86/s y. [6] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 999. [7] P. Pucci, M. Xiang, B. Zhang; Existence and multiplicity of entire solutions for fractional p-kirchhoff equations, Adv. Nonlinear Anal., 5 (26), no., [8] V. Rădulescu; Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics and Its Applications, Vol. 6. Hindawi Publ Corp., New York, 28. [9] V. Rădulescu, B. Alleche; Equilibrium problems techniques in the qualitative analysis of quasi-hemivariational inequalities, Optimization 64 (25), [2] V. Rădulescu, D. Repovš; Partial differential equations with variable exponents: Variational Methods and Qualitative Analysis, CRC Press, aylor & Francis Group, Boca Raton FL, 25. [2] V. Rădulescu; Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Analysis: heory, Methods and Applications, 2(25), [22] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional integral and derivatives: theory and applications, Gordon and Breach, Longhorne, PA, 993. [23] X. Zhang, B. Zhang, M. Xiang, Ground states for fractional Schrödinger equations involving a critical nonlinearity, Adv. Nonlinear Anal., 5(26), no. 3, Fariba Fattahi Department of Mathematics, University of Mazandaran, Babolsar, Iran address: F.Fattahi@stu.umz.ac.ir Mohsen Alimohammady Department of Mathematics, University of Mazandaran, Babolsar, Iran address: Amohsen@umz.ac.ir