MULTIPLICITY OF SOLUTIONS FOR NONPERIODIC PERTURBED FRACTIONAL HAMILTONIAN SYSTEMS

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1 Electronic Journal of Differential Equations, Vol (2017), No. 93, pp ISSN: UL: or MULTIPLICITY OF SOLUTIONS FO NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS ABDEAZEK BENHASSINE Communicated by Paul H. abinowitz Abstract. In this article, we prove the existence and multiplicity of nontrivial solutions for the nonperiodic perturbed fractional Hamiltonian systems td α ( Dt α x(t)) λl(t) x(t) + W (t, x(t)) = f(t), x H α (, N ), where α (1/2, 1], λ > 0 is a parameter, t, x N, Dt α and tdα are left and right Liouville-Weyl fractional derivatives of order α on the whole axis respectively, the matrix L(t) is not necessary positive definite for all t nor coercive, W C 1 ( N, ) and f L 2 (, N )\{0} small enough. eplacing the Ambrosetti-abinowitz Condition by general superquadratic assumptions, we establish the existence and multiplicity results for the above system. Some examples are also given to illustrate our results. 1. Introduction Hamiltonian systems form a significant field of nonlinear functional analysis, since they arise in phenomena studied in several fields of applied science such as physics, astronomy, chemistry, biology, engineering and other fields of science. Since Newton wrote the differential equation describing the motion of the planet and derived the Kepler ellipse as its solution, the complex dynamical behavior of the Hamiltonian system has attracted a wide range of mathematicians and physicists. The variational methods to investigate Hamiltonian system were first used by Poincaré, who used the minimal action principle of the Jacobi form to study the closed orbits of a conservative system with two degrees of freedom. Ambrosetti and abinowitz in [1] proved Mountain Pass Theorem, Saddle Point Theorem, Linking Theorem and a series of very important minimax form of critical point theorem. The study of Hamiltonian systems makes a significant breakthrough, due to critical point theory. Critical point theorem was first used by abinowitz [20] to obtain the existence of periodic solutions for first order Hamiltonian systems, while the first multiplicity result is due to Ambrosetti and Zelati [2]. Since then, there is a large number of literatures on the use of critical point theory and variational methods to prove the existence of homoclinic or heteroclinic orbits of Hamiltonian systems see for example [6, 9, 17] and the references therein Mathematics Subject Classification. 34C37, 35A15, 37J45. Key words and phrases. Fractional Hamiltonian systems; critical point; variational methods. c 2017 Texas State University. Submitted December 13, Published March 30,

2 2 A. BENHASSINE EJDE-2017/93 On the other hand, fractional calculus has received increased popularity and importance in the past decades to describe long-memory processes. For more details, we refer the reader to the monographs [7, 13, 15] and the reference therein. ecently, the critical point theory has become an effective tool in studying the existence of solutions to fractional differential equations by constructing fractional variational structures. ecently, in Jiao and Zhou [11] showed that critical point theory is an effective approach to tackle the existence of solutions for the fractional boundary-value problem td α T ( 0 D α t x(t)) = W (t, x(t)), a.e. t [0, T ], x(0) = x(t ), where α (1/2, 1), x N, W C 1 ([0, T ] N, ), W (t, x) is the gradient of W at x, and obtained the existence of at least one nontrivial solution. Inspired by this paper, Torres [22] studied the fractional Hamiltonian system t D α ( D α t x(t)) L(t) x(t) + W (t, x(t)) = 0, x H α (, N ), (1.1) where α ( 1 2, 1), t, x N, Dt α and t D α are left and right Liouville-Weyl fractional derivatives of order α on the whole axis respectively, L(t) C(, N 2 ) is symmetric and positive definite matrix for all t and W C 1 ( N, ). The author showed that (1.1) possesses at least one nontrivial solution via Mountain Pass Theorem, by assuming that L and W satisfy the following hypotheses: (A1) L C(, N 2 ) is a positive definite symmetric matrix for all t ; (A2) the smallest eigenvalue of L(t) + as t ; (A3) W (t, x) = o( x ) as x 0 uniformly in t ; (A4) there is W C( N, ) such that W (t, x) + W (t, x) W (x), (t, x) N. (A5) there exists a constant µ > 2 such that 0 < µw (t, x) W (t, x) x, t, x N \{0}. When α = 1, (1.1) reduces to the standard second-order Hamiltonian systems ẍ(t) L(t)x(t) + W (t, x(t)) = 0. (1.2) When L(t) is a symmetric matrix valued function for all t and W (t, x) satisfies the so-called global Ambrosetti-abinowitz Condition (A5), the existence and multiplicity of homoclinic solutions for Hamiltonian systems (1.2) have been extensively investigated in many recent papers see for example [2, 6, 9, 21] and the references therein. If L(t) and W (t, x) are neither periodic in t, the problem of existence of homoclinic orbits for (1.2) is quite different from the ones just described, because of lack of compactness of Sobolev embedding. In [21] and without periodicity assumptions on both L and W, abinowitz and Tanaka first studied system (1.2) and prove the existence of one nontrivial homoclinic orbit of (1.2) under assumptions (A1) (A5). emark 1.1. Although the technical coercively assumption (A2) plays a key role to guarantee the compactness of the Sobolev embedding, it is somewhat restrictive and eliminates many functions.

3 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 3 Motivated by the above works, in this article, when f L 2 (, N )\{0}, L(t) C(, N 2 ) is a symmetric matrix but not necessary positive definite for all t not coercive, W C 1 ( N, ) and replacing the Ambrosetti-abinowitz condition by general superquadratic assumptions, we establish the existence and multiplicity results for the nonperiodic perturbed fractional Hamiltonian system t D α ( D α t x(t)) λl(t) x(t) + W (t, x(t)) = f(t), x H α (, N ), (1.3) where α (1/2, 1], λ > 0 is a parameter. Precisely, we suppose that (A6) min x N, x =1 L(t)x x 0 and there is b > 0 such that meas({l b}) < 1/c 2 α, where meas( ) is the Lebesgue measure, {L b} = {t : L(t) b} and c α defined the Sobolev constant (see section 2); (A7) W (t, 0) = 0 and for any 0 < α 1 < α 2, Cα α1 2 := inf { W (t, x) x 2 ; t, α 1 < x < α 2 } > 0, where W (t, x) := 1 2 W (t, x) x W (t, x); (A8) there exist c 1 > 0, 1 > 1 and β (1, 2) such that W (t, x) x c 1 W (t, x) x 2 β, t, x 1 ; (A9) there exist a constants T 0 > 0 and x 0 N \{0} such that T0 T 0 λl(t)x 0 x 0 W (t, x 0 ) dt < 0. Our main results reads as follows. Theorem 1.2. Assume that f L 2 (, N )\{0} and (A3), (A4), (A6) (A9) hold. Then, there exist constants f 0, λ 0 > 0 such that, for any λ > λ 0 system (1.3) possesses at least two nontrivial solutions whenever f L 2 < f 0. Corollary 1.3. Assume that f L 2 (, N )\{0}, (A3), (W2), (A6) (A8) are satisfied and (A9 ) W (t, x) lim x + x 2 = +, uniformly for a.e. t. Then, there exist constants f 0, λ 0 > 0 such that, for any λ > λ 0 system (1.3) possesses at least two nontrivial solutions whenever f L 2 < f 0. emark 1.4. Assumption (A5) implies (A8), (A9) and (A9 ). In fact assuming (A5) is satisfied, it is clear that (A9) and (A9 ) hold. Choose 1 1 so large that Then, for such x, we have and it follows that 1 µ < x 2 β whenever x 1. W (t, x) ( x 2 β ) W (t, x) x, W (t, x) x x 2 β( 1 2 W (t, x) x W (t, x) ) = x 2 β W (t, x).

4 4 A. BENHASSINE EJDE-2017/93 Here and in the following x y denotes the inner product of x, y N and denotes the associated norm. Throughout this article, we denote by c, c i the various positive constants which may vary from line to line and are not essential to the problem. 2. Preliminaries 2.1. Liouville-Weyl Fractional Calculus. Definition 2.1. The left and right Liouville-Weyl fractional integrals of order 0 < α < 1 on the whole axis are defined by respectively, where x. Ix α u(x) := 1 Γ(α) xi u(x) α := 1 Γ(α) x x (x ξ) α 1 u(ξ)dξ, (ξ x) α 1 u(ξ)dξ, Definition 2.2. The left and right Liouville-Weyl fractional derivatives of order 0 < α < 1 on the whole axis are defined by respectively, where x. D α x u(x) := d dx I 1 α x u(x), (2.1) xd α u(x) := d dx x I 1 α u(x), (2.2) emark 2.3. Definitions (2.1) and (2.2) may be written in the alternative forms: Dx α α u(x) = Γ(1 α) xd u(x) α α = Γ(1 α) 0 0 u(x) u(x ξ) ξ α+1 dξ, u(x) u(x + ξ) ξ α+1 dξ. ecall that the Fourier transform û(w) of u(x) is defined by û(w) = e ix.w u(x)dx. We establish the Fourier transform properties of the fractional integral and fractional operators as follows: I α x u(x)(w) := (iw) α û(w), xi α u(x)(w) := ( iw) α û(w), D α x u(x)(w) := (iw) α û(w), xd α u(x)(w) := ( iw) α û(w).

5 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS Fractional derivative spaces. Let us recall for any α > 0, the semi-norm and the norm u I α := D α x u L 2, u I α := ( u 2 L 2 + u 2 I α ) 1/2. Let the space I α () denote the completion of C 0 () with respect to the norm I α, i.e., I α () = C 0 () I α. Next, we define the fractional Sobolev space H α () in terms of the Fourier transform. For 0 < α < 1, define the semi-norm and the norm and let u α = w α û L 2, u α = ( u 2 L 2 + u 2 α) 1/2, H α () := C 0 () α. We note that a function u L 2 () belongs to I α () if and only if w α û L 2 (). In particular, u I α = w α û L 2 (). Therefore H α () and I () α are equivalent, with equivalent semi-norm and norm (see [22]). Analogous to I (), α we introduce I (). α Let us define the semi-norm and norm and let u I α := x D α L 2 (), u I α := ( u 2 L 2 + u 2 I α )1/2, I α () = C 0 () I α. Moreover I α () and I α () are equivalent, with equivalent semi-norm and norm. Lemma 2.4 ([22]). If α > 1/2, then H α () C() and there is a constant C = C α such that u L = sup u(x) C u α (2.3) x where C() denote the space of continuous functions on. emark 2.5. If u H α (), then u L q () for all q [2, ], since u(x) q dx u q 2 L u 2 L. 2 In what follows, we introduce the fractional space in which we will construct the variational framework of (1.3). Let X α = { x H α (, n ) : Dt α x(t) 2 + L(t)x(t) x(t)dt < }. The space X α is a reflexive and separable Hilbert space with the inner product (x, y) X α = ( Dt α x(t). Dt α y(t)) + L(t)x(t) y(t)dt,

6 6 A. BENHASSINE EJDE-2017/93 and the corresponding norm is x X α = (x, x) X α. For λ > 0, we also need the following inner product (x, y) λ = ( Dt α x(t) Dt α y(t) + λl(t)x(t) y(t))dt, and the corresponding norm x λ = (x, x) λ. Set X α λ = (Xα, λ ). Observing x λ x X α for all λ 1. Lemma 2.6. If L satisfies (A6) then, X α is continuously embedded in H α (, n ). Proof. By (A6) and (2.3) we have x(t) 2 dt = x(t) 2 dt + x(t) 2 dt {L<b} {L b} x 2 L meas({l < b}) + 1 L(t)x(t) x(t)dt b {L b} ( ) c 2 α meas({l < b}) ( Dt α x(t) 2 + x(t) 2 )dt + 1 b Therefore, and {L b} L(t)x(t) x(t)dt. x 2 L max{c2 α meas({l < b}), 1 b } 2 1 c 2 x 2 X α meas({l < b}) (2.4) α x 2 α = ( Dt α x(t) 2 + x(t) 2 )dt (1 + max{c2 α meas({l < b}), 1 b } 1 c 2 α meas({l < b}) which yields that the embedding X α H α (, N ) is continuous. ) x 2 X α, (2.5) emark 2.7. Using the same conditions and techniques in (2.4) and (2.5), for all 1 λ bc 2 α meas({l<b}), we also obtain x 2 L c2 α meas({l < b}) 2 1 c 2 α meas({l < b}) x 2 λ, (2.6) ( x 2 α 1 + c2 α meas({l < b}) ) 1 c 2 x 2 α meas({l < b}) λ. (2.7) Furthermore, using (2.3), (2.5) and (2.6), for every p (2, ) and λ 1 bc 2 α meas({l < b}),

7 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 7 we have x(t) p dt x p 2 L ( c p 2 α x(t) 2 dt ( D α t x(t) 2 + x(t) 2 )dt ( c p 2 α 1 + c2 α meas({l < b}) 1 c 2 α meas({l < b}) = ( 1 + c2 α meas({l < b}) 1 c 2 α meas({l < b}) ) p 2 2 ) p 2 2 c 2 α meas({l < b}) 1 c 2 α meas({l < b}) x 2 λ x p 2 λ c 2 α meas({l < b}) 1 c 2 α meas({l < b}) x 2 λ ) p 2 2 c p α meas({l < b}) 1 c 2 α meas({l < b}) x p λ (2.8) = meas({l < b})( 1 c 2 α meas({l < b}) )p/2 x p λ := δp x p p λ. c 2 α 3. Proof of Theorem 1.2 and Corollary 1.3 For this purpose, we establish the corresponding variational framework to obtain solutions of (1.3). To this end, define the functional I λ : Xλ α by [ 1 I λ (x) = 2 Dt α x(t) 2 + λ ] 2 L(t)x(t) x(t) W (t, x(t)) + f(t) x(t) dt = 1 2 x 2 λ W (t, x(t))dt + f(t) x(t)dt. Under assumptions (A3), (A4), (A6) (A8), we see that I λ is a continuously Fréchetdifferentiable functional defined on Xλ α; i.e., I λ C 1 (Xλ α, ). Moreover, we have I λ(x)y = [( Dt α x(t). Dt α y(t)) + λl(t)x(t) y(t) W (t, x(t)) y(t) + f(t) y(t)]dt, for all x, y Xλ α, which yields I λ(x)x = x 2 λ (3.1) W (t, x(t)) x(t)dt + f(t) x(t)dt. (3.2) We know that to find a solutions of (1.3), it suffices to obtain the critical points of I λ ; see [22]. For this purpose the lemma below is useful. ecall that φ C 1 (E, ) satisfy the Palais-Smale condition (P S) if any sequence (x n ) E, for which (φ(x n )) is bounded and φ (x n ) 0 as n, possesses a convergent subsequence in E. Lemma 3.1 ([20]). Let E be a real Banach space and φ C 1 (E, ) satisfying the Palais-Smale condition. If φ satisfies the following conditions: (i) φ(0) = 0, (ii) there exist constants ρ, γ > 0 such that φ / Bρ(0) γ, (iii) there exist e E\B ρ (0) such that φ(e) 0.

8 8 A. BENHASSINE EJDE-2017/93 Then φ possesses a critical value c γ given by where c = inf g Γ max s [0,1] φ(g(s)), Γ = {g C([0, 1], E) : g(0) = 0, g(1) = e}. To find the critical points of I λ, we shall show that I λ satisfies the (P S) condition. Because of the lack of the compactness of the Sobolev embedding, we require the following convergence result. Lemma 3.2. Assume that x n f L 2. Then x in Xλ α, (A3), (A4), (A7) are satisfied and I λ (x n x) = I λ (x n ) I λ (x) + o(1) as n +, (3.3) I λ(x n x) = I λ(x n ) I λ(x) + o(1) as n +. (3.4) In particular, if (x n ) is a (P S) sequence of I λ such that I λ (x n ) c for some c then after passing to a subsequence. I λ (x n x) c I λ (x) as n +, (3.5) I λ(x n x) 0 as n +, (3.6) Proof. As x n x in X α λ, we have (x n, x) λ (x, x) λ as n. Then Obviously, x n 2 λ = (x n x, x n x) λ + (x, x n ) λ + (x n x, x) λ = x n x 2 λ + x 2 λ + o(1). (x n, z) λ = (x n x, z) λ + (x, z) λ, z X α λ. Hence, to show (3.3) and (3.4) it suffices to prove that (W (t, x n ) W (t, x n x) W (t, x))dt = o(1), (3.7) ( W (t, x n ) W (t, x n x) W (t, x)) ϕdt = o(1). (3.8) sup ϕ X α λ, ϕ λ=1 Here, we only prove (3.8) the proof of (3.7) is similar. Setting y n := x n x, then y n 0 in Xλ α and y n(t) 0 a.e. t. From (A3), for every ε > 0, there exist σ = σ(ε) (0, 1) such that By (A4) and (3.9), we have where W (t, u) ε u, t, u σ. (3.9) W (t, u) ε u + c ε u 2, t, u N 1, (3.10) N 1 := sup{ y n L, y n + x L, x L + 1}, c ε = max W n u [σ,n (u)σ 2. 1] By (3.10) and the Young Inequality, for each ϕ X α λ with ϕ λ = 1, we have ( W (t, y n + x) W (t, y n )).ϕ ε( y n + x + y n ) ϕ + c ε ( y n + x 2 + y n 2 ) ϕ,

9 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 9 and If we take c(ε y n ϕ + ε x ϕ + c ε y n 2 ϕ + c ε x 2 ϕ ), c(ε y n 2 + ε x 2 + ε ϕ 2 + ε y n 3 + c ε ϕ 3 + c ε x 3 ), ( W (t, y n + x) W (t, y n ) W (t, x)) ϕ c(ε y n 2 + ε x 2 + ε ϕ 2 + ε y n 3 + c ε ϕ 3 + c ε x 3 ). (3.11) ψ n (t) := max{ ( W (t, y n + x) W (t, y n ) W (t, x))ϕ cε( y n 2 + y n 3 ), 0}, we obtain 0 ψ n (t) c(ε x 2 + ε ϕ 2 + c ε ϕ 3 + c ε x 3 ) L 1 (, N ). The Dominated Convergence Theorem implies that ψ n (t) dt 0 as n. (3.12) It follows from the definition of ψ n (t) that and then ( W (t, y n + x) W (t, y n ) W (t, x)).ϕ ψ n (t) + εc( y n 2 + y n 3 ), ( W (t, y n + x) W (t, y n ) W (t, x)) ϕ dt ψ n (t) L 1 + εc( y n 2 L + y n 3 2 L 3), for all n. Because ϕ is arbitrary in Xλ α, we obtain ( W (t, y n + x) W (t, y n ) W (t, x)) ϕ dt sup ϕ X α λ, ϕ λ=1 ψ n (t) L 1 + εc( y n 2 L2 + y n 3 L 3), which, jointly with (2.8) and (3.12) shows that ( W (t, y n + x) W (t, y n ) W (t, x)) ϕ dt εc, sup ϕ X α λ, ϕ λ=1 for n sufficiently large. Therefore, (3.8) holds. If moreover I λ (x n ) c and I λ (x n) 0 as n, equations (3.3) and (3.4) respectively, imply that and I λ (x n x) c I λ (x) + o(1), I λ(x n x) = I λ(x) as n +. We show that I λ (x) = 0. For every ζ C 0 (, N ), we have Consequently, I λ (x) = 0 and (3.6) holds. I λ(x)ζ = lim n I λ(x n )ζ = 0. Lemma 3.3. Suppose that f L 2 and (A3), (A4), (A6), (A8) are satisfied. Then, there exists λ 0 > 0 such that any bounded (P S) sequence of I λ has a convergent subsequence when λ > λ 0.

10 10 A. BENHASSINE EJDE-2017/93 Proof. Let (x n ) be a bounded sequence such that (I λ (x n )) is bounded and I λ (x n) 0 as n. Then, after passing to a subsequence, we have x n x in Xλ α and y n 0 in L 2 ({L(t) < b}) where y n := x n x. Moreover, y n 2 L 1 λl(t)y 2 n y n dt + y n 2 dt 1 λb λb y n 2 λ + o(1). (3.13) {L b} {L<b} Setting N 2 := sup n y n L. By (A4), we obtain W (t, y n ) = 1 2 W (t, y n) y n W (t, y n ) which, jointly with (3.13) and (A8) yields max W (u)(n 2 + 1), u [0,N 2] W (t, y n ) y n dt c 1 W (t, yn ) y n 2 β dt y n 1 y n 1 cc 1 β 1 y n 2 dt y n 1 cc 1 λb y n 2 λ + o(1). Furthermore, using (A4), (3.9) and (3.13), we have W (t, y n )y n dt y n < 1 ε y n 2 dt + W (t, y n ) y n dt y n σ σ< y n < 1 ε y n 2 dt + max W u [σ, (u)σ 1 y n 2 dt 1] y n δ c λb y n 2 λ + o(1). n, (3.14) (3.15) Because f L 2, one has, for any ε > 0, there exists T ε > 0 such that ( ) 1/2 f(t) 2 dt < ε. t T ε Using (2.8) and the Hölder inequality, we have f(t)y n dt ( ) 1/2 ( f(t) 2 dt t T ε t T ε Obviously ( f(t) y n dt t <T ε y n 2 dt) 1/2 cε n. (3.16) ) 1/2 ( ) 1/2 f(t) 2 dt y n 2 dt 0, (3.17) t <T ε as n. By (3.16) and (3.17), we have f(t) y n (t)dt 0, (3.18) as n. Consequently, a combination of (3.4), (3.14), (3.15) and (3.18) implies o(1) = I λ(y n )y n = y n 2 λ W (t, y n ) y n dt + f(t) y n dt (1 cc 1 λb c λb ) y n 2 λ + o(1).

11 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 11 Choosing λ 0 > 0 large enough such the term (1 cc1 λ > λ 0, we obtain y n 0 and then x n x in Xλ α. λb c λb ) is positive. When Lemma 3.4. If f L 2 and (A3), (A4), (A6) (A8) are satisfied, then I λ satisfies the (P S) condition whenever λ > λ 0. Proof. Let (x n ) be a (P S) sequence of I λ. By Lemma 3.3, it suffices to prove that (x n ) is bounded. Indeed, assume that x n λ as n and setting y n := xn x n λ. Then y n λ = 1 and y n L p δ p for p [2, + ]. Moreover, we have as n. We obtain o(1) = I λ (x n)x n x n 2 λ = 1 W (t, x n ) y n y n dt = x n W (t, x n ) x n x n 2 dt + o(1), λ W (t, x n ) x n x n 2 dt 1, (3.19) λ as n. Let 0 α 1 < α 2 and ω α1,α2 n := {t ; α 1 x n (t) < α 2 }. By (2.8) and because (x n ) is a (P S) sequence of I λ, then there exists N 0 > 0 such that for n N 0 we have This implies, for n N 0, that c + x n λ I λ (x n ) 1 2 I λ(x n )x n W (t, x n )dt + 1 f(t) x n dt 2 W (t, x n )dt δ 2 2 f L 2 x n λ. c(1 + x n λ ) W (t, x n )dt = ω 0,α 1 n W (t, x n )dt + W (t, x n )dt + W (t, x n )dt. ω α 1,α 2 n ω α 2, n (3.20) By (A3), for any ε > 0(ε < 1 3 ) there exists κ ε > 0 such that W (t, u) ( ε δ2 2 ) u, u κ ε, t. Thus, W (t, x n ) ω x 0,κε n y n 2 dt ω 0,κε ε δ 2 2 y n 2 dt ε δ2 2 y n 2 L2 ε, n. (3.21)

12 12 A. BENHASSINE EJDE-2017/93 Because β > 1 and by (A8), (2.8) and (3.20) we can choose θ ε 1 large enough such that W (t, x n )x n y n W (t, x n ) ω x θε,+ n 2 dt c 1 λ ω x θε,+ n β 1 dt x n λ W (t, xn ) c 1 y n L ω θε,+ θε β 1 x n λ cc (3.22) 1 y n L (1 + x n λ ) θε β 1 x n λ cδ θε β 1 < ε, n N 0. By (A7), we have W (t, x n (t)) Cκ θε ε x n 2 for t ωn κε,θε. Noting Cκ θε ε > 0 it follows from (3.20) that y n 2 dt = 1 ω x κε,θε n 2 x n 2 dt λ ω κε,θε 1 W (t, xn )dt Cκ θε ε x n 2 (3.23) λ ω κε,θε c(1 + x n λ ) 0, Cκ θε ε x n 2 λ as n +, which yields that W (t, x n ) y n 2 dt τ ε y n ω x κε,θε n ω 2 dt 0, (3.24) κε,θε as n, where τ ε = max u [κε,θ ε] W (u) κ ε. Hence, by (3.21), (3.22) and (3.24), we have W (t, x n ) y n W (t, x n ) y n y n 2 3ε < 1, x n x n for n large enough, a contradiction with (3.19) and then (x n ) is bounded in Xλ α. Lemma 3.5. If (A3) holds and f L 2, then there exist ρ, γ, f 0 > 0 such that I λ (x) / x λ =ρ γ when f L 2 < f 0. Proof. By (A3), for ε := 1 4δ 2 2 there exists σ 1 = σ 1 (ε) such that W (t, x) ε x 2, t, x σ 1. (3.25) Thus, for x λ ρ := σ 1 /δ, by (3.25), we obtain I λ (x) 1 2 x 2 λ ε x 2 (1 ) dt f L 2 x L 2 x λ 4 x λ f L 2δ 2. Let γ := ρ( 1 4δ 2 ρ f L 2δ 2 ). Then, if f L 2 < f 0 := 1 4δ 2 2 ρ, we have I λ (x) / x λ =ρ γ. Lemma 3.6. If f L 2 < f 0 and (A3), (A7) are satisfied, then there exists x 1 Xλ α\{0} such that I λ (x 1) = 0.

13 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 13 Proof. Since f L 2 \{0}, we can choose ξ Xλ α such that f(t) ξ(t)dt < 0. By (A3) and (A7) we have W 0 and I λ (sξ) s2 2 ξ 2 λ + s f(t) ξ(t)dt < 0, for s small enough. Thus C 1 := inf{i λ (x), x B ρ (0)} < 0, where ρ is the constants given by Lemma 3.5. From Ekeland s variational principle there exists a sequence (x n ) B ρ such that C 1 I λ (x n ) < C n. Then, by a standard procedure, we can show that (x n ) Xλ α is bounded (P S) sequence. Consequently, Lemma 3.3 implies that, there exist x 1 Xλ α such that x n x 1 Xλ α, I λ (x 1) = 0 and I λ (x 1 ) = C 1 < 0 when λ > λ Proof of Theorem 1.2. Let h(s) = s 2 W (t, sx 0 ) for t, s > 0. Then, by (A7), h (s) = s 3 [ 2W (t, sx 0 ) + W (t, sx 0 ).sx 0 ] > 0, for t, s > 0. Integrating the above from 1 to η, we obtain W (t, ηx 0 ) η 2 W (t, x 0 ), for t, η > 1. (3.26) From (3.26), we have for s > 1, I λ (sx 0 ) = (λs 2 L(t)x 0 x 0 W (t, sx 0 ))dt + s f(t) x 0 dt s 2 ( λl(t)x 0 x 0 W (t, x 0 )dt) + s f(t) x 0 dt. Let e(t) = { sx 0, if t [ T 0, T 0 ] 0, if t \[ T 0, T 0 ], By (A9) there exists s 0 1 such that e λ > ρ and I λ (e) < 0. Since I λ (0) = 0 and all the assumptions of Lemma 3.1 are satisfied, so I λ possesses a critical point x 2 X α λ with I λ (x 2) = 0 and I λ (x 2 ) = C 2 > 0 whenever λ > λ Proof of Corollary 1.3. If (A9 ) holds, let e C0 ()\{0}. Then, by Fatou s Lemma and by W 0 we have I λ (se) s 2 [ 1 W (t, se) 2 e 2 λ e 0 (se) 2 e 2 dt] + s f(t).e(t)dt as s +, which implies that I λ (se) < 0 for s > 0 large. Combining this with Lemmas 3.4 and 3.5, all the assumptions of Lemma 3.1 are satisfied, so I λ possesses a critical point x 3 X α λ with I λ (x 3) = 0 and I λ (x 3 ) > 0 whenever λ > λ An example Let L(t) = h(t)i N where 0, if t < 1, h(t) = 2n 2 t n, if t 1, t n 1 2n (n Z, n 1), 2 1, elsewhere, W (t, x) = k(t) x 2 ln(1 + x 2 ),

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15 EJDE-2017/93 NONPEIODIC PETUBED FACTIONAL HAMILTONIAN SYSTEMS 15 [24] G. M. Zaslavsky; Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, (2005). [25] S. Q. Zhang; Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl., (2011), [26] X. Wu, Z. Zhang; Solutions for perturbed fractional Hamiltonian systems without coercive conditions, BVP, (2015), 1 12 [27] J. Xu, D. O egan, K. Zhang; Multiple solutions for a calss of fractional Hamiltonian systems, Fract. Calc. Appl. Anal., (2015), [28] Z. Zhang,. Yuan; Existence of solutions to fractional Hamiltonian systems with combined nonlinearities, Electron. J. Differential Equations, (2016), Abderrazek Benhassine Dept. of Mathematics, High Institut of Informatics and Mathematics, 5000, Monastir, Tunisia address: ab.hassine@yahoo.com