EXISTENCE OF SOLUTION FOR A FRACTIONAL DIFFERENTIAL INCLUSION VIA NONSMOOTH CRITICAL POINT THEORY. Bian-Xia Yang and Hong-Rui Sun

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1 Korean J. ath ), No. 4, pp EXISTENCE OF SOLUTION FOR A FRACTIONAL DIFFERENTIAL INCLUSION VIA NONSOOTH CRITICAL POINT THEORY Bian-Xia Yang and Hong-Rui Sun Abstract. This paper is concerned with the existence of solutions to the following fractional differential inclusion { ) d dx p Dx β u x)) + q x D β 1 u x)) F u x, u), x, 1), u) = u1) =, where Dx β and x D β 1 are left and right Riemann-Liouville fractional integrals of order β, 1) respectively, < p = 1 q < 1 and F : [, 1] R R is locally Lipschitz with respect to the second variable. Due to the general assumption on the constants p and q, the problem does not have a variational structure. Despite that, here we study it combining with an iterative technique and nonsmooth critical point theory, we obtain an existence result for the above problem under suitable assumptions. The result extends some corresponding results in the literatures. Received August 24, 215. Revised October 14, 215. Accepted October 28, athematics Subject Classification: 26A33, 34B15, 34B4. Key words and phrases: Fractional differential inclusion, Nonsmooth critical point theory, Existence, Solutions. Corresponding author. This works was supported by the program for New Century Excellent Talents in University NECT ), FRFCUlzujbky-213-k2) and FRFCUlzujbky ). c The Kangwon-Kyungki athematical Society, 215. This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License -nc/3./) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.

2 538 B.X. Yang and H.R. Sun 1. Introduction Fractional differential equations and inclusions have been proved that they are very useful tools in modeling of many phenomena in various fields of science and engineering, such as, viscoelasticity, electrochemistry, electromagnetism, economics, optimal control and so forth. For details and examples, see [1,3,4,6,7,11,16,19] and the references therein. In consequence, more and more attention has been paid to fractional differential equations and inclusions. The study of fractional differential inclusions was initiated by Sayed and Ibrahim, see [21]. Very recently several qualitative results for fractional differential inclusions were obtained in [2, 5, 11, 14, 23] and the references therein. Especially, in [23], Teng et al. considered the fractional differential inclusion 1.1) { d dx ) 1 2 Dx β u x)) x D β 1 u x)) F u x, u), x, 1), u) = u1) =, by using nonsmooth mountain pass theorem and nonsmooth symmetric mountain pass theorem, they derived the existence and multiplicity of solutions, where Dx β and x D β 1 are left and right Riemann-Liouville fractional integrals of order β, 1) respectively, defined by D β x u = 1 Γβ) x x s) β 1 us)ds, xd β 1 u = 1 Γβ) x s x) β 1 us)ds. Obviously, in 1.1), the coefficient 1 is very special. So a natural question 2 is what will happen for the existence with coefficient p and q, which only satisfy p + q = 1? We will give a positive answer in present paper, so we attempt to use nonsmooth mountain pass theorem and iterative technique to study the existence of nontrivial solutions of fractional differential inclusion 1.2) { d dx p Dx β u) = u1) =, ) u x)) + q x D β 1 u x)) F u x, u), x, 1), where < p = 1 q < 1, for s R, F, s) is measurable, and for a.e. x [, 1], F x, ) is locally Lipschitz, F s x, s) denotes the generalized subdifferential in the sense of Clarke [9].

3 A fractional differential inclusions via nonsmooth critical point theory 539 In order to use variational method, we consider a family fractional differential inclusions with variational structure, that is, for given w H α, 1), we discuss the following problem 1.3) d q dx Dx β u x)) + q x D β 1 u x)) ) +p q) Dx β w x)) F u x, u), x, 1), u) = u1) =, which can be solved by variational method. Then, for each w H α, 1), we find a solution u w H α, 1) with some bounds. Next, by iterative technique, one gets the existence of solutions of 1.2) under suitable assumptions. This paper is organized as follows. In Section 2, we recall some basic knowledge of nonsmooth analysis and abstract results which we are going to apply. Section 3 is devoted to present the preliminaries about fractional calculus to derive our result, and list the assumptions on the problem and state our main result. In the final Section, we give the proof of the main result. The result extends that in [22, 23]. 2. Nonsmooth analysis We collect some basic notions and results of nonsmooth analysis, namely, the calculus for locally Lipschitz functionals developed by Clarke [9], otreanu and Panagiotopoulos [18], Chang [1], Gasinki and Papageorgiou [13]. Let X, X ) be a Banach space, X, X ) be its topological dual, and ϕ : X R be a functional. We recall that ϕ is locally Lipschitz l.l.) if for u X, there exist a neighborhood U of u and a real number K U > such that ϕv) ϕω) K U v ω X for v, ω U. If ϕ is l.l. and u X, the generalized directional derivative of ϕ at u along the direction v X is ϕ u; v) = lim ω u t ϕω + tv) ϕω), t and the generalized gradient of ϕ at u is the set ϕu) = {u X : u, v ϕ u; v) for v X}.

4 54 B.X. Yang and H.R. Sun Then for u X, ϕu) 2 X \{ } is a convex and weakly -compact subset [9, Proposition 1]. Lemma 2.1. [18, Proposition 1.1]) If ϕ C 1 X, R), then ϕ is l.l. and ϕ u; v) = ϕ u), v, ϕu) = {ϕ u)}, u, v X. Lemma 2.2. [18, Proposition 1.3]]) Let ϕ : X R be a l.l. functional. Then for u X, ϕ u; ) is subadditive and positively homogeneous and ϕ u; v) K U v, v X with K U > being a Lipschitz constant for ϕ around u. Assume ϕ is a l.l. functional defined on Banach space X, set λu) = min{ u X : u ϕu)}, u X, then λu) exists and is lower semi-continuous [1, 13]. u X is said to be a critical point of ϕ if ϕu). A l.l. functional ϕ : X R is said to satisfy the non-smooth PS) condition at level c R, if any sequence {u n } X with ϕu n ) c and λu n ) as n, has a strongly convergent subsequence [1, 13]. 3. Fractional calculus and main result For convenience, hereafter, we denote α = 1 β. In view of β, 1), 2 we have α 1, 1). The fractional Sobolev space 2 Hα, 1) is defined as the completion of C, 1) under the norm u α = D α x u L 2. From [12, Theorem 2.13], we know H α, 1) is a reflexive Banach space, and H α, 1) C[, 1] is compact, moreover, if u H α, 1), then u) = u1) =. For the space H α, 1), we have the following results. Lemma 3.1. [22, Lemma 2.2]) If u H α, 1), then u 1 u Γα)2α 1) 1 α, 2 cosπα) u 2 α u L 2 D α x u xd α 1 udx 1 Γα + 1) u α, 1 cosπα) u 2 α,

5 A fractional differential inclusions via nonsmooth critical point theory 541 x D α 1 u 2 dx 1 cosπα) 2 u 2 α. Definition 3.2. By a weak solution of problem 1.2), it is understood an element u H α, 1) for which there corresponds to a mapping [, 1] x u x) with u x) F u x, ux)) for a.e. x [, 1], and having the property that for every v H α, 1), u v L 1 [, 1] and p D α x u xd α 1 vdx q D α x v xd α 1 udx = u x)vx)dx. Similarly, a function ũ H α, 1) is called a weak solution of problem 1.3) if there exists a corresponding mapping [, 1] x ũ x) with ũ x) Fũx, ũx)) for a.e. x [, 1], and having the property that for every v H α, 1), ũ v L 1 [, 1] and q Dx α ũ xd1 α v + Dx α v xd1 α ũ ) dx p q) D α x w xd α 1 vdx = ũ x)vx)dx. Definition 3.3. A function u H α, 1) is called a solution of problem 1.2) if p Dx 2α 1 u q x D1 2α 1 u is derivable with respect to x, 1) and d dx p Dx 2α 1 u q x D1 2α 1 u) = u, a.e. x, 1), where u x) F u x, ux)) for a.e. x [, 1]. Lemma 3.4. If u H α, 1) is a weak solution of problem 1.2), then u is a solution of problem 1.2). Proof. Suppose u H α, 1) is a weak solution of problem 1.2), then there exists u F u x, u) satisfying 3.1) p D α x u xd α 1 vdx q D α x v xd α 1 udx = u x)vx)dx for all v H α, 1). Similar to the argument of [15, Theorem 4.2], we can get x ) p Dx 2α 1 u q x D1 2α 1 u + u s)ds v x)dx =

6 542 B.X. Yang and H.R. Sun for all v C, 1). So there exists constant C, such that and then p Dx 2α 1 u q x D1 2α 1 u + x u s)ds = C, d dx p Dx 2α 1 u q x D1 2α 1 u) = u a.e. x, 1). According to Definition 3.3, we know that u is a solution of problem 1.2). We impose F the following conditions. F1) For s R, the function x F x, s) is measurable, for a.e. x [, 1], s F x, s) is l.l. and F x, ) = ; F2) there exist a, b L 1 [, 1], R + ) and r [1, ), such that s ax) + bx) s r 1, a.e. x [, 1], s R and s F s x, s); F3) there exist µ, 1 2 ), c > and >, such that c < F x, s) µf x, s; s) for a.e. x [, 1], and s R with s ; F4) for s F s x, s), lim s s s = in a.e. x [, 1]. Remark 3.5. Noting that from conditions F1), F2) and F4), using the Lebourg s mean value theorem, we obtain that for given ɛ >, there exist b L 1 [, 1], R + ), η > 2, such that F x, s) ɛ s 2 + bx) s η, x [, 1], s R. Remark 3.6. From conditions F1), F2) and the Lebourg s mean value theorem, one has F x, s) ax) s +bx) s r, F x, s; s) ax) s +bx) s r, x [, 1]. Lemma 3.7. Assume conditions F1)F3) hold, then ) 1 s µ F x, s) cx), x [, 1] \ N, s, where N is the Lebesgue-null set outside which the hypothesis F3) uniformly holds, and 3.2) cx) = min{f x, ), F x, )}, x [, 1] \ N, clearly, cx) c for a.e. x [, 1].

7 A fractional differential inclusions via nonsmooth critical point theory 543 Proof. For given s R with s, set Fx, λ) = F x, λs), x [, 1] \ N, λ R, then Fx, ) is l.l.. Via Rademarchers theorem, we see that λ Fx, λ) is differentiable a.e. on R, and at a point of differentiability λ R, it gets d Fx, λ) = F dλ λx, λ). oreover, it follows from Chain rule that F λ x, λ) = s F ξ x, ξ) λs, hence λ F λ x, λ) = λs F ξ x, ξ) λs. At a point of differentiability, condition F3) reduces to so one presents i.e. µf s x, s)s F x, s), x [, 1] \ N, s, λdfx, λ) dλ d Fx, λ) dλ Fx, λ) 1 Fx, λ), µ 1 λµ. Integrating from 1 to λ λ 1), it gives ln Fx,λ ) Fx,1) ln λ 1 µ, hence Fx, λ ) λ 1 µ Fx, 1), that is F x, λ s) λ 1 µ F x, s). Thus, for x [, 1] \ N, s, we have F x, s) = F x, s s s ) ) 1 s µ s F x, s ) cx) ) 1 s µ. For problem 1.2), since the symmetric position of the constants p and q lying in, without loss of generality, one can assume that p q. The functional I w : H α, 1) R corresponding to the problem 1.3) is defined by I w u) = q Dx α u xd1 α udx p q) Dx α w xd1 α udx F x, u)dx. Proposition 3.8. Assume that F satisfies the hypotheses F1),F2), then the functional I w : H α, 1) R is l.l., and every critical point u H α, 1) of I w is a solution of the problem 1.3). Proof. Let I w u) = I 1 u) + I 2 u), where I 1 u) = q D α x u xd α 1 udx p q) D α x w xd α 1 udx,

8 544 B.X. Yang and H.R. Sun I 2 u) = F x, u)dx. Clearly I 1 C 1 H α, 1), R). By Lemma 2.1, I 1 is l.l. on H α, 1). From condition F2), one knows that I 2 is l.l. on L r [, 1]. oreover H α, 1) is compactly embedded into L r [, 1]. So I 2 is also l.l. on H α, 1), see [1, Proposition 2.3 and Theorem 2.2] and [17], furthermore, 3.3) I 2 u) F u x, u)dx. The interpretation of 3.3) is as follows: for every u I 2 u), we have u x) F u x, ux)) for a.e. x [, 1], and for every v H α, 1), the function u v L 1 [, 1] and u, v = u x)vx)dx. Therefore I w is l.l. on H α, 1). Now we shall show that each critical point u of I w is a weak solution of problem 1.3). Let u H α, 1) be a critical point of I w, then 3.4) I w u) = {u H α, 1)) : u, v Iwu; v) for v H α, 1)}. Set 3.5) A w u), v = q D α x u xd α 1 v + D α x v xd α 1 u) dx p q) u, v H α, 1). It follows from Lemma 2.1, 3.4) that A w u) + u = with u I 2 u), D α x w xd α 1 vdx, hence u x) F u x, ux)) a.e. on [, 1] and for every v H α, 1), one obtains q Dx α u xd1 α v + Dx α v xd1 α u ) dx p q) D α x w xd α 1 vdx + u vdx =. By Definition 3.2, u is a weak solution of problem 1.3), similar to the proof of Lemma 3.4, it gets u is a solution of problem 1.3).

9 A fractional differential inclusions via nonsmooth critical point theory 545 For the µ, given in condition F3), denote 3.6) a = q1 2µ) cosπα), b = Assume that a 2 > b 2 + b, we take 3.7) ɛ 1 = a a 2 bb + 1) 21 + b) For ɛ ɛ 1, ɛ 2 ), define [ 2µ 3.8) t = tɛ) = 1 ) 1 µ 3.9) Cɛ) = q t 2 cosπα) 3.1) R 1 = R 1 ɛ) = p q)1 µ). cosπα), ɛ 2 = a + a 2 bb + 1) b) ) q + p q)2 cosπα) 4ε cosπα) 2 ) 1 t µ µ) cx) ϕx) 1 µ dx 4ɛCɛ) 4aɛ 4b + 1)ɛ 2 b ] µ 1 2µ, cx) ϕx) 1 µ dx + t 2 p q) 2 4ɛ cosπα) 2 ax) + r bx))dx, ) 1 2, R2 = R 2 ɛ) = R 1, Γα)2α 1) 1 2 where c is defined in 3.2), ϕ H α, 1) is a fixed function with ϕ α = 1. The main result of this paper is the following. Theorem 3.9. Assume that F satisfies the hypotheses F1)-F4). If there exists ɛ ɛ 1, ɛ 2 ), such that { } s 3.11) L R2 := sup 1 s 2 s 1 s 2, s 1, s 2 R 2, s 1 s 2 exists, 3.12) q cosπα) 2 ) ɛ Γα)2α 1) 1 2 Γα + 1)) 2 1 bx)dx ) η 2p q)r1 > q cosπα) 2 and 3.13) L R2 < Γα + 1)) 2 2q cosπα) p q ) cosπα) ) η 2

10 546 B.X. Yang and H.R. Sun hold. Then the problem 1.2) has at least one nonzero solution, where s i F si x, s i ), x [, 1], i = 1, 2, ɛ 1, ɛ 2 are defined in 3.7), b, η are defined in Remark Proof of main result In this section, we give the proof of Theorem 3.9 by the nonsmooth mountain pass theorem and iterative technique, the proof idea inspired from [22]. Proof of Theorem 3.9. We proceed by three steps to prove the main result. Step 1: For given w H α, 1) with w α R 1, one shows that I w has a nontrivial critical point in H α, 1) by the nonsmooth mountain pass theorem. Firstly, we check that I w satisfies the nonsmooth PS) condition. Suppose {u n } H α, 1) satisfies 4.1) I w u n ) C and λu n ) as n. For every n 1, since I w u n ) H α, 1)) is a weakly compact set and the norm function is weakly lower semi-continuous in Banach space, we can find u n I w u n ), such that 4.2) λu n ) = u n H α,1)) and u n = A w u n v n with v n x) F un x, u n x)) for a.e. x [, 1]. Hence, by 4.1), 4.2), 3.5), Lemma 3.1, condition F3) and Remark 3.6, it shows C µ u n α I w u n ) µ u n, u n = q + µq + µp q) D α x u n xd α 1 u n dx p q) D α x u n xd α 1 u n dx + µq D α x w xd α 1 u n dx xd α 1 u n D α x u n dx D α x w xd α 1 u n dx µ v n, u n F x, u n )dx

11 A fractional differential inclusions via nonsmooth critical point theory 547 = q2µ 1) Dx α u n xd1 α u n dx + p q)µ 1) Dx α w xd1 α u n dx F x, u n )dx µ v n, u n q1 2µ) cosπα) u n 2 p q)1 µ) α w α u n α cosπα) F x, un ) + µf x, u n ; u n ) ) dx { u n } { u n >} F x, un ) + µf x, u n ; u n ) ) dx a u n 2 α + b w α u n α 1 + µ) ax) + r bx))dx, where a, b are defined in 3.6). So the sequence {u n } is bounded. Thus, by passing to a subsequence if necessary, we can assume that u n u in H α, 1). Via Rellich-Kondrachov compactness theorem, one gets 4.3) u n u in L 2 [, 1], and u n u in C[, 1]. By Lemma 3.1 and 3.5), it gives A w u n A w u, u n u = 2q D α x u n x) ux)), x D α 1 u n x) ux))) dx 2q cosπα) u n u 2 α. Consequently, in order to prove u n u, it suffices to prove the following fact 4.4) lim n A w u n A w u, u n u. Indeed, from 4.1) and 4.2), there holds ɛ n u n u α u n, u n u = A w u n, u n u v n u n u)dx with ɛ n. In view of 4.3) and Hölder s inequality, one has v n u n u)dx as n. So lim n A w u n, u n u. Via u n u in H α, 1), it is easy to get lim n A w u, u n u =. Hence lim A wu n A w u, u n u lim A w u n, u n u lim A w u, u n u. n n n

12 548 B.X. Yang and H.R. Sun that is, 4.4) holds, we obtain u n u in H α, 1). On the other hand, via Lemma 3.1 and Remark 3.5, one derives I w u) = q Dx α u xd1 α udx p q) Dx α w xd1 α udx q cosπα) u 2 α p q) w α x D1 α u L 2 ɛ u 2 L 2 q cosπα) u 2 α p q)r 1 cosπα) u ɛ α Γα + 1)) 2 u 2 α u η q cosπα) u 2 α p q)r 1 cosπα) u α bx)dx Γα)2α 1) 1 2 ) η u η α = q cosπα) 2 q cosπα) + 2 ɛ Γα + 1)) 2 u α p q)r 1 cosπα) ɛ Γα + 1)) 2 u 2 α bx)dx Γα)2α 1) 1 2 ) η u η 2 α ) u α. ) F x, u)dx bx) ux) η dx u 2 α bx)dx by the assumption 3.12), one can choose ρ > 2p q)r 1 q cosπα) 2, such that q cosπα) 2 ɛ Γα + 1)) 2 > bx)dx Γα)2α 1) 1 2 ) η ρη 2. Now, let u H α, 1) with u α = ρ, then there exists β 1 >, such that 4.5) I w u) β 1 uniformly for w H α, 1) with w α R 1. For ϕ H α, 1) with ϕ α = 1, we will prove I w tϕ) as t.

13 A fractional differential inclusions via nonsmooth critical point theory 549 In fact, by Lemma 3.1 and Remark 3.6, as t, it gives I w tϕ) = qt 2 Dx α ϕ xd1 α ϕdx p q)t Dx α w xd1 α ϕdx qt 2 cosπα) ϕ 2 α + tp q) w α x D α 1 ϕ L 2 qt 2 cosπα) + tp q) w α Dx α ϕ L 2 cosπα) qt 2 cosπα) + tp q)r 1 cosπα). t tϕx) cx) ) 1 µ ) 1 t µ 1 cx) ϕx) 1 µ dx F x, tϕ)dx ) 1 µ dx cx) ϕx) 1 µ dx Thus, there exists t > such that t ϕ α > ρ and I w t ϕ) <. Then noting that I w ) =, combining with 4.5) and the nonsmooth mountain pass theorem [11, 13], we obtain that there is u w H α, 1)\{θ} with I w u w ) and I w u w ) = inf max I wu) β 1 >, g Γ u g[,1]) where Γ = {g C[, 1], H α, 1)) g) =, g1) = t ϕ}. By Proposition 3.8, we get u w is a solution of problem 1.3). Step 2: We construct an iterative sequence {u n } and estimate its norm in H α, 1). For u 1, by Step 1, we know I u1 has a nontrivial critical point u 2. If we can prove u 2 α R 1, then by Step 1, one gets I u2 has a critical point u 3. So in order to obtain iterative sequence {u n }, we need prove that if we assume u n 1 α R 1, then u n, the nontrivial critical point of I un 1 obtained by Step 1, satisfies u n α R 1.

14 55 B.X. Yang and H.R. Sun Indeed, by Lemma 3.1, and Cauchy s inequality with the positive constant ɛ, for ϕ H α, 1) satisfying ϕ α = 1, it gives 4.6) max I u n 1 tϕ) t [, ) qt 2 cosπα) qt 2 cosπα) q t 2 cosπα) t t tϕx) cx) ) 1 µ 1 ) 1 µ = Cɛ) + ɛ u n 1 2 α 1 + µ) ) 1 µ dx + tp q) u n 1 α cosπα) cx) ϕx) 1 µ dx + ɛ un 1 2 α + t2 p q) 2 4ɛ cosπα) 2 cx) ϕx) 1 µ dx + ɛ un 1 2 α + t 2 p q) 2 ax) + r bx))dx, 4ɛ cosπα) 2 where ɛ ɛ 1, ɛ 2 ), ɛ 1, ɛ 2 are defined in 3.7), t, Cɛ) are defined in 3.8), 3.9) respectively. On the other hand, since I un 1 u n ), one has 4.7) 2q Dx α u n xd1 α u n dx+p q) Dx α u n 1 xd1 α u n dx+ u nu n dx =, where u nx) F un x, u n x)) for a.e. x [, 1]. Take 4.7), Lemma 3.1, F3), Remark 3.6 into account, it derives = q I un 1 u n ) p q) = q p q) +µp q) D α x u n xd α 1 u n dx D α x u n 1 xd α 1 u n dx D α x u n xd α 1 u n dx F x, u n )dx F x, u n )dx D α x u n 1 xd α 1 u n dx + 2µq D α x u n 1 xd α 1 u n dx + µ D α x u n xd α 1 u n dx u nu n dx

15 A fractional differential inclusions via nonsmooth critical point theory 551 = 2µ 1)q +p q)µ 1) D α x u n xd α 1 u n dx D α x u n 1 xd α 1 u n dx F x, u n ) + µu n u n )) dx q1 2µ) cosπα) u n 2 α + µ 1)p q) u n 1 α x D1 α u n L 2 F x, un ) + µf x, u n ; u n ) ) dx u n u n > F x, un ) + µf x, u n ; u n ) ) dx a u n 2 α b u n 1 α u n α 1 + µ) a u n 2 α b ɛ u n 2 α + 1 ) 4ɛ u n 1 2 α ax) + r bx))dx 4.8) 1 + µ) ax) + r bx))dx, where ɛ ɛ 1, ɛ 2 ), ɛ 1, ɛ 2 are defined in 3.7). According to the nonsmooth mountain pass characterization of the critical level, we have 4.9) max t [, ) I u n 1 tϕ) I un 1 u n ). So combining with 4.6), 4.8) and 4.9), it concludes a u n 2 α ɛ u n 1 2 α + b ɛ u n 2 α + 1 ) 4ɛ u n 1 2 α + Cɛ), i.e. a bɛ) u n 2 α ɛ + b 4ɛ ) u n 1 2 α + Cɛ).

16 552 B.X. Yang and H.R. Sun Since ɛ ɛ 1, ɛ 2 ), it holds a bɛ > ɛ + b 4ɛ. Then u n 2 α ɛ + b 4ɛ a bɛ u n 1 2 α + Cɛ) a bɛ ɛ + b ) n 1 4ɛ u 1 2 α + Cɛ) a bɛ a bɛ u 1 2 α + 4ɛCɛ) 4aɛ 4b + 1)ɛ 2 b. n 2 k= ɛ + b ) k 4ɛ a bɛ Consequently, if we take u 1 and let u n be a critical point of I un 1 for n = 2, 3,..., then from the above argument, one knows u n α R 1 and I un 1 u n ) β 1 > for n = 2, 3,.... Step 3: We show the iterative sequence {u n } constructed in Step 2 is convergent to a nontrivial solution of the problem 1.2). We intend to prove {u n } is a Cauchy sequence in H α, 1). Indeed, since u n α R 1, in view of Lemma 3.1 and the definition of R 2, one derives u n R 2. By I un 1 u n )u n+1 u n ), I un u n+1 )u n+1 u n ), we get 4.1) q and 4.11) q p q) p q) D α x u n xd α 1 u n+1 u n ) + x D α 1 u n D α x u n+1 u n )) dx D α x u n 1 xd α 1 u n+1 u n )dx = u nu n+1 u n )dx D α x u n+1 xd α 1 u n+1 u n ) + x D α 1 u n+1 D α x u n+1 u n )) dx D α x u n xd α 1 u n+1 u n )dx = u n+1u n+1 u n )dx, where u nx) F un x, u n x)), u n+1x) F un+1 x, u n+1 x)) for a.e. x [, 1]. 4.11) subtracting 4.1), combining lemma 3.1 and 3.11), one

17 concludes A fractional differential inclusions via nonsmooth critical point theory 553 2q cosπα) u n+1 u n 2 α 2q = p q) + D α x u n+1 u n ) xd α 1 u n+1 u n )dx D α x u n u n 1 ) xd α 1 u n+1 u n )dx u n+1 u n)u n+1 u n )dx p q cosπα) u n u n 1 α u n+1 u n α + u n+1 u n)u n+1 u n )dx p q cosπα) u n u n 1 α u n+1 u n α + L R2 u n+1 u n 2 L 2 p q cosπα) u L R2 n u n 1 α u n+1 u n α + Γα + 1)) u 2 n+1 u n 2 α ) p q = cosπα) u L R2 n u n 1 α + Γα + 1)) u 2 n+1 u n α u n+1 u n α. Hence, 4.12) 2q cosπα) L R2 Γα + 1)) 2 ) u n+1 u n α p q) cosπα) u n u n 1 α. By the assumptions 3.13) and 4.12), we know {u n } is a Cauchy sequence in H α, 1). So we suppose that u n u in H α, 1). In view of the definition of {u n }, we know u is a weak solution and then by Lemma 3.4, it is a solution of problem 1.2). Note that I un 1 u n ) β 1 and β 1 > does not depend on n, we derive that u is a nontrivial solution of problem 1.2). Acknowledgments The authors are very grateful to the anonymous referee for valuable suggestions.

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