Even homoclinic orbits for a class of damped vibration systems

Accepted Manuscript Even homoclinic orbits for a class of damped vibration systems Khelifi Fathi, Mohsen Timoumi

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S0019-3577(17)30083-6 http://dx.doi.org/10.1016/j.indag.2017.08.002 INDAG 502

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Indagationes Mathematicae

Received date : 31 July 2016 Revised date : 9 January 2017 Accepted date : 14 August 2017 Please cite this article as: K. Fathi, M. Timoumi, Even homoclinic orbits for a class of damped vibration systems, Indagationes Mathematicae (2017), http://dx.doi.org/10.1016/j.indag.2017.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS KHELIFI FATHI, MOHSEN TIMOUMI Abstract. In this paper, we study the existence of even homoclinic solutions for a damped vibration system x ¨(t) + q(t)x(t) ˙ + V 0 (t, x(t)) = 0, where q is a continuously differentiable function and V ∈ C 1 (R × RN , R), V (t, x) = −K(t, x) + W (t, x). We use a new kind of superquadratic condition instead of the global Ambrosetti-Rabinowitz condition . For the proof we use the Mountain Pass Theorem.

1. Introduction and main results Consider the following damped vibration system (DV )

x ¨(t) + q(t)x(t) ˙ + V 0 (t, x(t)) = 0,

1 where V ∈ C 1 (R × RN , R), V 0 (t, x) = ∂V ∂x (t, x) and q ∈ C (R, R) satisfying Z t q(s)ds. lim Q(t) = +∞, where Q(t) =

|t|→+∞

0

As usual, we say that a solution x of (DV) is homoclinic (to 0) if x ∈ C 2 (R, RN ), x 6= 0, x(t) → 0 and x(t) ˙ → 0 as t → ±∞. When q(t) = 0 for all t ∈ R, (DV ) is just the following second order Hamiltonian system (HS)

x ¨(t) + ∇V (t, x(t)) = 0.

During the last two decades, the existence of homoclinic solutions for system (HS) has been extensively investigated by many authors see [1, 2, 3, 13, 14, 16, 17, 18, 19, 20, 21] and references therein. After the work of Rabinowitz [16], several authors used the same method to obtain some interesting results on the existence of homoclinic orbits for system (HS) under various suitable assumptions see [3, 8, 18, 19, 20, 21]. In particular, M. Izoderk and J.Janczewska [8] considered the following more general Hamiltonian system x ¨(t) + ∇V (t, x(t)) = f (t),

where V is of the type V (t, x) = −K(t, x)+W (t, x) and is T-periodic with respect to t, T > 0. As far as the case q(t) 6= 0, is concerned, to our best knowledge, there is no research about the existence of such solutions for (DV) when V is of the type V = −K + W. Mathematics Subject Classification: 34C37. Keywords: Damped vibration system; even homoclinic orbits; Mountain pass Theorem. 1

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2

KHELIFI FATHI, MOHSEN TIMOUMI

In the present paper, we shall study the existence of even homoclinic solutions for (DV ) when q(t) 6= 0 and assuming that V (t, x) is even and W (t, x) satisfies a kind of new superquadratic condition which is different from the corresponding condition in the known results. We obtain the existence of homoclinic solution as the limit of solutions of a certain sequence of boundary-value problem which are obtained by the minimax methods. Our basic hypotheses on K and W are the following: (K1 ) There exist constants b > 0 and λ ∈]1, 2[ such that

K 0 (t, x).x ≤ 2K(t, x)

K(t, 0) = 0, K(t, x) ≥ b|x|λ ,

for all (t, x) ∈ R × RN , (W1 ) V (t, 0) = 0, V (−t, x) = V (t, x) for all (t, x) ∈ R × RN and V 0 (t, x) → 0 as |x| → 0 uniformly in t ∈ R. (t,x) < 0 uniformly in t ∈ R. (W2 ) lim sup|x|→0 V|x| 2 2

2−σ (R, R) such that (W3 ) There exist µ > 2, σ ∈]1, λ[ and d ∈ LQ

µW (t, x) − W 0 (t, x).x ≤ d(t)|x|σ .

(W4 ) There exists constant ξ0 > 0 such that

W (t, x) M0 π 2 + M1 > |x|2 2m0 ξ02 |x|→+∞ lim inf

uniformly in t ∈ [−ξ0 , ξ0 ], where M1 = M0 =

sup t∈[−ξ0 ,ξ0 ] Vt0 (t, x)

sup

K(t, x),

t∈[−ξ0 ,ξ0 ],|x|=1

eQ(t) and m0 = min eQ(t) . t∈R

∂V ∂t

(W5 ) = (t, x) ≤ 0 and q(t) ≥ 0, f or all t ≥ 0 and x ∈ RN . (q) q : R → R is an odd, continuously differentiable and a bounded function . Now we state our main results. Theorem 1.1. Assume that (W1 ) - (W5 ), (K1 ) and (q) hold, then the system 1 (R, RN ) such (DV) has at least one nontrivial even homoclinic solution x ∈ HQ that x(t) ˙ → 0 as |t| → +∞. Remark 1.1. There are functions K and W which satisfy Theorem 1.1, but do not satisfy the corresponding results in [1], [4]-[8], [13]-[15],[19] . For example ( 5 1 |x| 4 exp(|x| 4 ) + |x|2 , if |x| ≤ 1, K(t, x) = 3 |x| 2 + exp(1)|x|2 if |x| > 1,

and

5

1

W (t, x) = d(t)|x| 4 exp(|x| 4 ), where 

π cos(t), if t ∈ [ −π 2 , 2 ], −π π 0 if t ∈ R[ 2 , 2 ]. A straightforward computation shows that K and W satisfy Theorem 1.1, but they do not satisfy the corresponding results on the above papers, in particular do not satisfy the Ambrosetti-Rabinowitz condition.

d(t) =

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

3

Theorem 1.2. Assume that V and q are T-periodic in t, (T > 0), (W1 )-(W4 ), (K1 ), (q) and the following assumption hold  K(t, x) ≤ K 0 (t, x).x, f or all (t, x) ∈ [0, T ] × RN , (W50 ) W 0 (t, x) = o(|x|) as |x| → 0, unif ormly in t,

then the system (DV) has at least one nontrivial even homoclinic solution x ∈ 1 (R, RN ) such that x(t) HQ ˙ → 0 as |t| → +∞. 2. Proof of the main results.

By the idea of [9], we approximate an even homoclinic solution of (DV) by a solution of the following problem:  x ¨(t) + q(t)x(t) ˙ − K 0 (t, x(t)) + W 0 (t, x(t)) = 0 f or t ∈] − ξ, ξ[ (2.1) x(−t) = x(t) f or t ∈] − ξ, ξ[, x(−ξ) = x(ξ) = 0

where ξ is a positive constant. For 1 ≤ s < ∞, let LsQ (−ξ, ξ; RN ) be the Banach space of measurable functions Z ξ N x defined on [−ξ, ξ] with values in R satisfying eQ(t) |x(t)|s dt < ∞, with the norm

kxkLsQ =

Z

−ξ

ξ

Q(t)

e

−ξ

s

|x(t)| dt

 1s

.

The space L2Q (−ξ, ξ; RN ) provided with the inner product < x, y >=

Z

ξ

−ξ

eQ(t) x(t).y(t)dt, x, y ∈ L2Q (−ξ, ξ; RN )

is a Hilbert space. Let E be the space defined by  E = x ∈ L2Q ([−ξ, ξ], RN ), x˙ ∈ L2Q (−[ξ, ξ], RN ), x(−ξ) = x(ξ) = 0 The space E provided with the inner product

hx, yi = and the associated norm

kxk =

Z

ξ

eQ(t) [x(t).y(t) + x(t). ˙ y(t)]dt, ˙

−ξ

Z

ξ

−ξ

e

Q(t)

2

2

(|x(t)| + |x(t)| ˙ )dt

 21

is a Hilbert space. Consider the functional Iξ : E → R defined by   Z ξ Q(t) 1 2 Iξ (x) = e |x(t)| ˙ + K(t, x(t)) − W (t, x(t)) dt. 2 −ξ

It is easy to check that Iξ ∈ C 1 (E, R) and for all x, y ∈ E, we have Z ξ   0 Iξ (x)y = eQ(t) (x(t). ˙ y(t) ˙ + K 0 (t, x(t)).y(t) − W 0 (t, x(t)).y(t) dt. −ξ

(2.2)

(2.3)

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4

KHELIFI FATHI, MOHSEN TIMOUMI

Moreover, the critical points of Iξ in E are the classical solutions of (DV) in [−ξ, ξ] satisfying x(ξ) = x(−ξ) = 0. We will obtain a critical point of Iξ by using the Mountain Pass Theorem: Lemma 2.1. ([15]) Let H be a real Banach space and I ∈ C 1 (H, R) satisfying the Palais-Smale condition. If I satisfies the following conditions: (i) I(0) = 0, (ii) there exist constants ρ, α > 0 such that I|∂Bρ (0) ≥ α, (iii) there exists e ∈ H\B ρ (0) such that I(e) ≤ 0. Then I possesses a critical value c ≥ α given by c = inf max I(g(s)), g∈Γ s∈[0,1]

where Bρ (0) is the open ball in H centered in 0, with radius ρ, ∂Bρ (0) as its boundary and Γ = {g ∈ C([0, 1], H) : g(0) = 0, g(1) = e}. For a fixed ξ > 0, consider the subspace Eξ of E defined by Eξ = {x ∈ E / x(−t) = x(t), a.e. t ∈] − ξ, ξ[} . We will proceed by successive lemmas. Lemma 2.2. The critical points of Φξ on Eξ are exactly the solutions of problem (2.1), where Φξ is the restriction of Iξ on Eξ . Proof : Let Fξ = {x ∈ E / x(−t) = −x(t), a.e. t ∈] − ξ, ξ[} .

For every x ∈ E, set

1 1 (x(t) + x(−t)) , z(t) = (x(t) − x(−t)) , 2 2 then y ∈ Eξ , z ∈ Fξ and x = y + z. So E = Eξ + Fξ . Furthermore, for all y ∈ Eξ , z ∈ Fξ we have Z ξ Z −ξ Q(t) hy, zi = e (y(t).z(t)+y(t). ˙ z(t))dt ˙ = eQ(−t) (y(−t).z(−t)+y(−t). ˙ z(−t))d(−t) ˙ y(t) =

−ξ

ξ

=

Z

ξ

−ξ

eQ(t) (y(t).(−z(t)) + (−y(t)). ˙ z(t))dt ˙ = −hy, zi,

which implies that hy, zi = 0 and then Eξ ⊥Fξ . Hence E = Eξ ⊕ Fξ . For all x ∈ Eξ , z ∈ Fξ , we have Iξ0 (x)z =

Z

ξ

−ξ

=

Z

ξ

−ξ

  eQ(t) (x(t). ˙ z(t) ˙ − V 0 (t, x(t)).z(t) dt.

  eQ(−t) (x(−t).(− ˙ z(t)) ˙ − V 0 (−t, x(−t)).z(−t) d(−t).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

=

Z

ξ

−ξ

5

  eQ(t) (−x(t). ˙ z(t) ˙ − V 0 (t, x(t)).(−z(t)) d(t) = −Iξ0 (x)z,

which implies that Iξ0 (x)z = 0. Therefore, a vector x ∈ Eξ is a critical point of Φξ on Eξ if and only if, it is a critical point of Iξ on E. The proof of Lemma 2.2 is complete. Lemma 2.3. Assume that (K1 ) holds. Then, for every t ∈ [−ξ0 , ξ0 ] and x ∈ RN , the following inequality holds: K(t, x) ≤ M1 |x|2 + M2 ,

where M1 is defined in (W4 ) and M2 =

sup

(2.4) K(t, x).

t∈[−ξ0 ,ξ0 ],|x|≤1

Proof : To prove this lemma it suffices to show that for every x ∈ RN and t ∈ [−ξ0 , ξ0 ] the function (0, +∞) → R, s 7→ K(t, s−1 x)s2 is nondecreasing; which is an immediate consequence of (K1 ). The proof of Lemma 2.3 is complete. By Sobolev’s embedding theorem, H 1 (R; RN ) is continuously embedded into L∞ (R, RN ). Thus there exists C > 0 such that kxkL∞ (R,RN ) ≤ CkxkH 1 ,

∀ x ∈ H 1 (R; RN ).

Since x ∈ E can be regarded as belonging to H 1 (R; RN ) if one extends it by zero in R\[−ξ, ξ], then we have kxkL∞ ([−ξ,ξ],RN ) ≤ CkxkH 1 ,

∀ x ∈ H01 ([−ξ, ξ], RN ),

where C is independent of ξ > 0. It follows from the above inequality that kxkL∞ ([−ξ,ξ],RN ) ≤ CkxkH 1 ≤ γ2 kxk, ∀x ∈ E, where γ2 =

(2.5)

√C . m0

Lemma 2.4. Suppose that the conditions (W1 ) - (W4 ) , (K1 ) and (q) are satisfied, then for all ξ ≥ ξ0 , the problem (2.1) possesses a nontrivial solution. Proof : Let ηξ : Eξ → [0, +∞[ be given by ηξ (x) =

Z

ξ

Q(t)

e

−ξ

 21 (|x(t)| + 2K(t, x(t)))dt . 2

(2.6)

By (2.2) and (2.6) we have 1 Φξ (x) = ηξ2 (x) − 2

Z

ξ

eQ(t) W (t, x(t))dt.

(2.7)

−ξ

Moreover, using (K1 ) and (2.6) we obtain Φ0ξ (x)x

≤

ηξ2 (x)

−

Z

ξ

−ξ

eQ(t) W 0 (t, x(t)).x(t)dt.

(2.8)

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6

KHELIFI FATHI, MOHSEN TIMOUMI

It is clear that Φξ (0) = 0. Firstly, we will show that Φξ satisfies the Palais-Smale conditions. Let (xn ) ⊂ Eξ be a sequence such that (Φξ (xn )) is bounded and Φ0ξ (xn ) → 0. Then, there exists a constant M > 0 such that |Φξ (xn )| ≤ M, kΦ0ξ (xn )k ≤ M,

(2.9)

for every n ∈ N. Without loss of generality, we can assume that kxn k = 6 0. Then from (2.5), (2.6) and (K1 ), we obtain for n ∈ N ηξ2 (xn )

= ≥ ≥

Z

ξ

−ξ Z ξ −ξ ξ

Z

−ξ

eQ(t) (|xn (t)|2 + 2K(t, xn (t)))dt eQ(t) (|xn (t)|2 + 2b|xn (t)|λ )dt eQ(t) |xn (t)|2 dt + 2b(γ2 ||xn ||)λ−2 λ−2

≥ min{1, 2b(γ2 ||xn ||) 2

λ−2

= min{kxn k , 2b(γ2 )

}kxn k

2

Z

ξ

−ξ

eQ(t) |xn (t)|2 dt

||xn ||λ }.

(2.10)

By (2.7)-(2.9), (W3 ) and H¨ older’s inequality, we have Z ξ   µ 0 2 ( − 1)ηξ (xn ) ≤ µΦξ (xn ) − Φξ (xn )xn + eQ(t) µW (t, xn ) − W 0 (t, xn )xn dt 2 −ξ Z ξ ≤ µM + M kxn k + eQ(t) d(t)|xn |σ dt −ξ ξ

≤ µM + M kxn k + (

Z

−ξ

2

eQ(t) |d(t)| 2−σ )

≤ µM + M kxn k + dkxn kσ ,

2−σ 2

Z (

ξ

−ξ

σ

eQ(t) |xn |2 ) 2 dt (2.11)

Z 2 2−σ where d = ( eQ(t) |d(t)| 2−σ ) 2 . R

Combining (2.10) with (2.11), we obtain min{kxn k2 , 2b(γ2 )λ−2 ||xn ||λ } ≤

2 (µM + M kxn k + dkxn kσ ). µ−2

(2.12)

It follows from(2.12) that kxn k is bounded in Eξ . In a way similar to proposition 4.3 in [12], we can prove that (xn ) has a convergent subsequence in Eξ . Hence, Φξ satisfies the Palais-Smale condition. Now, we show that there exist constants ρ, α > 0 independent of ξ such that Φξ satisfies assumption (ii) of Lemma 2.1. By (W2 ), there exist constants α0 , ρ0 > 0 such that V (t, x) ≤ −α0 |x|2

(2.13)

for all |x| < ρ0 and t ∈ R. Choose ρ = γρ02 and let S = {x ∈ Eξ \ kxk = ρ}. By (2.5), we have kxkL∞ ([−ξ,ξ],RN ) ≤ ρ0 , for all x ∈ S, which together with (2.13)

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EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

implies 1 2

Φξ (x) =

1 2

≥

Z

ξ

−ξ ξ

Z

−ξ

2 eQ(t) |x(t)| ˙ dt −

Z

ξ

eQ(t) V (t, x(t))dt

−ξ

Z

2 eQ(t) |x(t)| ˙ dt + α0

ξ

−ξ

1 ≥ min{ , α0 }ρ2 := α 2

7

eQ(t) |x(t)|2 dt

for every x ∈ S. Finally, it remains to show that Φξ satisfies assumption (iii) of Lemma 2.1. By (W4 ), there exist constants a1 > 0 and R > 0 such that M0 π 2 + a1 W (t, x) ≥ + M1 , |x|2 2m0 ξ02

for all |x| > R and t ∈ [−ξ0 , ξ0 ]. Let M3 = W (t, x) ≥ (

max

t∈[−ξ0 ,ξ0 ],|x|≤R

|W (t, x)|, we have

M0 π 2 + a1 + M1 )(|x|2 − R2 ) − M3 2m0 ξ02

(2.14)

for all x ∈ RN and t ∈ [−ξ0 , ξ0 ]. Let  r| sin(ωt)|e1 , if t ∈ [−ξ0 , ξ0 ], u(t) = 0, if t ∈ [−ξ, ξ]\[−ξ0 , ξ0 ], π where ω = ξ0 , e1 = (1, 0, ..., 0) and r ∈ R \ {0}. By (W1 ), (K1 ), (2.4) and (2.14), we have Z ξ Z ξ Z 1 ξ Q(t) 2 Q(t) e |u(t)| ˙ dt + e K(t, u(t))dt − eQ(t) W (t, u(t))dt Φξ (u) = 2 −ξ −ξ −ξ Z 1 2 2 ξ0 Q(t) = e | cos(ωt)|2 dt r ω 2 −ξ0 Z ξ0 Z ξ0 + eQ(t) K(t, r| sin(ωt)|e1 )dt − eQ(t) W (t, r| sin(ωt)|e1 )dt −ξ0

≤ − + ≤ − +

1 2 2 r ω 2

Z

−ξ0

ξ0

−ξ0

e

Q(t)

2

| cos(ωt)| dt + M1 r

2

Z

ξ0

−ξ0

eQ(t) | sin(ωt)|2 dt + 2ξ0 M2 M0

Z ξ0 M0 π 2 + a1 2 ( + M1 )r eQ(t) | sin(ωt)|2 dt 2m0 ξ02 −ξ0   2 2 M0 π + a 1 2ξ0 M0 R ( + M1 ) + M3 2m0 ξ02 Z ξ0 Z ξ0 1 2 2 2 2 r ω M0 | cos(ωt)| dt + M1 r eQ(t) | sin(ωt)|2 dt + 2ξ0 M2 M0 2 −ξ0 −ξ0 Z ξ0 Z ξ0 M0 π 2 + a1 2 2 m0 ) ( | sin(ωt)| dt − M1 r eQ(t) | sin(ωt)|2 dt 2m0 ξ02 −ξ0 −ξ0   2+a M π 0 1 2ξ0 M0 R2 ( + M1 ) + M3 2m0 ξ02

≤ (

−a1 r2 ) + M4 (ξ0 ) → −∞ as r → ∞, ξ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

8

KHELIFI FATHI, MOHSEN TIMOUMI

where m0 and M0 are defined in (W4 ). All the assumptions of Lemma 2.1 are satisfied, so for all ξ ≥ ξ0 , Φξ possesses a critical value cξ ≥ α > 0 defined by cξ ≡ inf max Φξ (g(s)) g∈Γξ s∈[0,1]

where Γξ = {g(t) ∈ C([0, 1], Eξ )/g(0) = 0, g(1) = u} .

Hence, for every ξ > 0, there exists xξ ∈ Eξ such that Φξ (xξ ) = cξ , Φ0ξ (xξ ) = 0.

Since cξ > 0, xξ is nontrivial. The proof of Lemma 2.4 is complete. Remark 2.1. If we define the set of paths Γξ = {g ∈ C([0, 1], Eξ ) : g(0) = 0, g(1) = u},

then there exists a solution xξ of system (2.1) at which inf max Φξ (g(s)) ≡ Rξ

g∈Γ s∈[0,1]

is achieved. Let ξˆ > ξ > ξ0 . Since any function in Eξ can be regarded as belonging ˆ ξ][−ξ, ˆ to Eξˆ if one extends it by zero in [−ξ, ξ], then Γξ ⊂ Γξˆ and Hence we have

Rξˆ ≤ Rξ ≤ Rξ0

1 Φξ (xξ ) = ηξ2 (xξ ) − 2

Z

ξ

−ξ

eQ(t) W (t, xξ (t))dt ≤ Rξ0 ,

(2.15)

uniformly in ξ ≥ ξ0 . Lemma 2.5. There exists a positive constants C1 such that kxξ k ≤ C1 ,

(2.16)

uniformly in ξ ≥ ξ0 . Proof : By (2.15), we have for all ξ ≥ ξ0 . Φξ (xξ ) ≤ Rξ0 , kΦ0ξ (xξ )k = 0. Arguing as in the proof of the Palais-Smale condition in Lemma 2.4, we prove (2.16). Lemma 2.6. Assume that V satisfies (W1 ) − (W4 ), (K1 ) and (q) then xξ (0) 6= 0.

(2.17)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

9

Proof : Since xξ is nontrivial and even, using the Picard’s Existence and Uniqueness Theorem, it is not difficult to show that xξ (0) 6= 0, ∀n ∈ N, (see [3]) Lemma 2.7. Suppose that the conditions (W1 )−(W5 ), (K1 ) and (q) are satisfied, then there exists C2 > 0 such that

for all ξ ≥ ξ0 .

|xξ (0)| > C2

Proof : Let 1 E(t) = |x˙ξ (t)|2 + V (t, xξ (t)). 2 Differentiating E(t), and using (DV ) and (W5 ) we obtain E 0 (t) = −q(t)|x˙ξ (t)|2 + Vt0 (t, xξ (t)) ≤ 0

for all 0 ≤ t ≤ ξ, and hence one has

1 E(0) ≥ E(ξ) = |x˙ξ (t)|2 ≥ 0. 2 Since xξ is even, x˙ ξ (0) = 0, and then V (0, xξ (0)) = E(0) ≥ 0. By (2.17), we have V (0, xξ (0)) ≥ 0. |x(0)|2

(2.18)

V (t, x) < 0, |x|2

(2.19)

|xξ (0)| > C2

(2.20)

By (W2 ), there exists C2 > 0 such that

for all t ∈ R and |x| ≤ C2 . Comparing (2.18) and (2.19), we obtain

for all ξ ≥ ξ0 . The proof of Lemma 2.7 is complete. Proof of Theorem 1.1. Take a sequence (ξn )n∈N with ξ0 ≤ ξ1 ≤ ξ2 ≤ ... → ∞ ˜ ) on Eξ , i.e. and consider problem (DV n  0 ¨(t) + q(t)x(t) ˙ − K (t, x(t)) + W 0 (t, x(t)) = 0 f or t ∈] − ξn , ξn [ ˜ ) x (DV x(−t) = x(t) f or t ∈] − ξn , ξn [, x(−ξn ) = x(ξn ) = 0.

˜ ) possesses a nontrivial solution xn such Then, by Lemmas 2.4 and 2.5, (DV that (kxn k) is bounded uniformly in n. As (xn ) satisfies x ¨n (t) + q(t)x˙ n (t) + V 0 (t, xn (t)) = 0,

(2.21)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

10

KHELIFI FATHI, MOHSEN TIMOUMI

by integrating (2.21) between 0 and t we obtain Z t Z t V 0 (s, xn (s))ds. q(s)x ˙ (s)ds − q(t)x (t) − x˙ n (t) − x˙ n (0) = q(0)xn (0) + n n 0

0

Since (xn ) is even and q is odd by assumption (q), then x˙ n (0) = 0, and q(0) = 0. Hence we get Z t Z t V 0 (s, xn (s))ds, q(s)x ˙ x˙ n (t) = n (s)ds − q(t)xn (t) − 0

0

which by (2.16) and assumption (q) implies that there exists C2 > 0 independent of n such that kx˙ n k∞ ≤ C2 , ∀n ∈ N.

(2.22)

k¨ xn k∞ ≤ C3 , ∀n ∈ N,

(2.23)

From (2.16), (2.21) and (2.22) we have

where C3 > 0 is independent of n. The task now is to show that (xn )n∈N and (x˙ n )n∈N are equicontinuous. Actually, by (2.22) we get Z t2 |xn (t2 ) − xn (t1 )| ≤ |x˙ n (t)|dt t1

≤ C2 |t2 − t1 | ,

for each p ∈ N, n ≥ p, and t1 , t2 ∈ [−ξp , ξp ], which shows that (xn )n≥p is equicontinuous on [−ξp , ξp ]. Analogously, we have by (2.23) Z t2 |x˙ n (t2 ) − x˙ n (t1 )| ≤ |¨ xn (t)|dt t1

≤ C3 |t2 − t1 | .

For each n ∈ N, set Cn1 = C 1 ([−ξn , ξn ], RN ) with the norm defined as follows: kxkCn1 = maxt∈[−ξn ,ξn ] (|x(t)| + |x(t)|), ˙ x ∈ Cn1 .

Now, we will show that (xn )n∈N possesses a convergent subsequence (xnj ) in 1 (R, RN ). Cloc First, let (xn )n∈N be restricted to [−ξ0 , ξ0 ]. It is clear that (xn ) and (x˙ n ) are uniformly bounded and equicontinuous on [−ξ0 , ξ0 ] . By Arzela-Ascoli Theorem, there exist a subsequence (x0n ) of (xn )n∈N , x0 ∈ C([−ξ0 , ξ0 ], RN ) and y 0 ∈ C([−ξ0 , ξ0 ], RN ) such that kx0n − x0 kC([−ξ0 ,ξ0 ],RN ) → 0, kx˙ 0n − y 0 kC([−ξ0 ,ξ0 ],RN ) → 0, as n → +∞. (2.24)

Note that for t ∈ [−ξ0 , ξ0 ] x0n (t)

=

x0n (−ξ0 )

+

Z

ξ0

−ξ0

x0n (s)ds, n ∈ N.

(2.25)

Let n → +∞ in (2.25) and use (2.24) we obtain 0

0

x (t) = x (−ξ0 ) +

Z

ξ0

−ξ0

y 0 (s)ds, f or t ∈ [−ξ0 , ξ0 ],

(2.26)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

11

which shows that y 0 (t) = x˙ 0 (t) for t ∈ [−ξ0 , ξ0 ] and x0 ∈ C01 . Moreover, it follows from (2.24) that kx0n − x0 kC01 → 0, as n → +∞. Secondly, let (x0n ) be restricted to [−ξ1 , ξ1 ]. It is clear that (x0n ) and (x˙ 0n ) are uniformly bounded and equicontinuous. Similarly as above, by Arzela-Ascoli Theorem, there exist a subsequence (x1n ) of (x0n ) and x1 ∈ C11 such that kx1n − x1 kC11 → 0, as n → +∞. j 1 By repeativy this procedure for all n ∈ N, there exist (xjn ) ⊂ (xj−1 n ) and x ∈ Cj such that

kxjn − xj kC 1 → 0, as n → +∞, j = 1, 2, ....

(2.27)

j

Moreover, we have kxj+1 − xj kC 1 j

≤ kxj+1 − xj+1 kC 1 + kxjn − xj kC 1 n j

+

kxj+1 n

−

xjn kC 1 j

j

→ 0, as n → +∞,

which leads to xj+1 (t) = xj (t), f or t ∈ [−ξj , ξj ], j = 1, 2, ....

(2.28)

x(t) = xj (t), f or t ∈ [−ξj , ξj ], j = 1, 2, ....

(2.29)

Let 1 (R, RN ). Now take a Then x ∈ C 1 (R, RN ) and xjn → x as j → +∞ in Cloc 0 1 2 diagonal sequence (xnj ), which consists of x0 , x1 , x2 , ... (see[10]). j For any j ∈ N, (xii )∞ i=j is a subsequence of (xn )n∈N , so it follows from (2.27) and (2.28) that

kxii − xkC 1 = kxii − xj kC 1 → 0, as i → +∞, j = 1, 2, ... j

j

That is 1 xnj → x, as j → +∞, in Cloc (R, RN ).

(2.30)

x ¨nj (t) + q(t)x˙ nj (t) + V 0 (t, xnj (t)) = 0.

(2.31)

x ¨nj (t) = −q(t)x˙ nj (t) − V 0 (t, xnj (t)) = 0, ∀t ∈ [−ξj0 , ξj0 ].

(2.32)

For each j ∈ N, and t ∈ R we have

Take j0 ∈ N, then

Hence, x ¨nj (t) is continuous in [−ξj0 , ξj0 ] for every j ≥ j0 . (xnj ), (x˙ nj ) and (¨ xnj ) converge uniformly on [−ξj0 , ξj0 ], then x ¨(t) + q(t)x(t) ˙ + V 0 (t, x(t)) = 0, ∀t ∈ [−ξj0 , ξj0 ].

Moreover, since

Because of the arbitrariness of j0 , we conclude that x ∈ C 2 (R, RN ) and satisfies (DV). As xnj is even for all j, the same is true for the limit x. By (2.20) x is not identical to zero.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

12

KHELIFI FATHI, MOHSEN TIMOUMI

We prove that x(t) → 0 as |t| → +∞. By the argument of Lemma 2.5, for each i ∈ N there is ni ∈ N such that for all n ≥ ni we have Z ξi eQ(t) (|xn (t)|2 + |x˙ n (t)|2 )dt ≤ kxn k2 ≤ C12 . −ξi

Letting n → +∞, we obtain Z ξi 2 eQ(t) (|x(t)|2 + |x(t)| ˙ )dt ≤ C12 . As i → +∞, we have Z Hence, we get

−ξi

+∞

−∞

Z

|t|≥r

2 eQ(t) (|x(t)|2 + |x(t)| ˙ )dt ≤ C12 .

2 eQ(t) (|x(t)|2 + |x(t)| ˙ )dt → 0 as r → +∞.

(2.33)

By Corollary 2.2 in [18], we have Z t+1 2 |x(t)|2 ≤ (|x(s)|2 + |x(s)| ˙ )ds,

(2.34)

t−1

for every t ∈ R. By (2.33) and (2.34) we conclude that x(t) → 0 as |t| → ∞.

We have to show that x(t) ˙ → 0 as |t| → ∞. By Corollary 2.2 in [18] we have Z t+1 Z t+1 2 2 2 |x(t)| ˙ ≤ (|x(s)| + |x(s)| ˙ )ds + |¨ x(s)|2 ds, t−1

t−1

1 (R, RN ) ⊂ H 1 (R, RN ), we get for every t ∈ R. Since x ∈ HQ Z t+1 2 (|x(s)|2 + |x(s)| ˙ )ds → 0 as |t| → ∞. t−1

Hence, it suffices to prove that Z t+1 |¨ x(s)|2 ds → 0 as |t| → ∞.

(2.35)

t−1

By (DV), we have Z t+1 Z 2 |¨ x(s)| ds = t−1

t+1

t−1

≤

kqk2∞

|q(s)x(s) ˙ + V 0 (s, x(s))|2 ds

Z

+ 2kqk∞ Since

Z

t+1

t−1

t+1

t−1

Z

2

|x(s)| ˙ ds + t+1

t−1

2

Z

|x(s)| ˙ ds

t+1

t−1

|V 0 (s, x(s))|2 ds

 21 Z

t+1

t−1

0

2

|V (s, x(s))| ds

 12

.

2 |x(s)| ˙ ds → 0 as |t| → ∞, x(t) → 0 as |t| → ∞ and V 0 (t, x) → 0 as

|x| → 0 uniformly in t ∈ R, then (2.35) follows. The proof of Theorem 1.1 is complete. Proof of Theorem 1.2. Let

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

EVEN HOMOCLINIC ORBITS FOR A CLASS OF DAMPED VIBRATION SYSTEMS

13

n o 1 HQ,nT (R, RN ) = x : R → RN , 2nT − periodic, x, x˙ ∈ L2Q ([−nT, nT ], RN ) and x(nT ) = 0 . Consider the family of functional (Φn )n≥1 defined on EnT by   Z nT 2 Q(t) 1 Φn (x) = e |x(t)| ˙ + K(t, x(t)) − W (t, x(t)) dt, (2.36) 2 −nT where

 1 EnT = x ∈ HQ,nT (R, RN )/x(−t) = x(t), a.e.t ∈ R .

Arguing as in the proof of Theorem 1.1, we prove that assumptions (W1 )-(W4 ), (K1 ) and (q) imply that for every positive integer n, the problem  x ¨(t) + q(t)x(t) ˙ − K 0 (t, x(t)) + W 0 (t, x(t)) = 0, f or t ∈] − nT, nT [ (DVn ) x(−t) = x(t), f or t ∈] − nT, nT [, x(−nT ) = x(nT ) = 0. possesses a solution xn . Moreover, the sequence (xn ) converges uniformly on 1 (R, RN ) satisfying x(t) any bounded interval to a homoclinic solution x ∈ HQ ˙ →0 as |t| → +∞. It remains to prove that x(t) 6≡ 0. For this aim, consider the function Ψ defined by Ψ(0) = 0 and for s > 0 Ψ(s) =

W 0 (t, x).x . |x|2 t∈R,0<|x|≤s max

Then Ψ is a continuous, non decreasing function and Ψ(s) ≥ 0 for s ≥ 0. The definition of Ψ implies that Z ξn eQ(t) W 0 (t, xn (t)).xn (t)dt ≤ Ψ(kxn kL∞ ([−ξn ,ξn ],RN ) )kxn k2 , (2.37) −ξn

for every n ∈ N. Since Φ0ξn (xn ).xn = 0, we have Z

ξn

e

Q(t)

0

W (t, xn (t)).xn (t)dt =

−ξn

+

Z

ξn

−ξn Z ξn

eQ(t) |x˙ n (t)|2 dt eQ(t) K 0 (t, xn (t)).xn (t)dt. (2.38)

−ξn

From (2.5), (2.16),(2.37), (2.38), (K1 ) and (W50 ), we obtain 2

Ψ(kxn kL∞ ([−ξn ,ξn ],RN ) )kxn k

≥ ≥ ≥ ≥ ≥

Z

ξn

−ξn ξn

Z

−ξn ξn

Z

−ξn ξn

Z

−ξn

e

Q(t)

2

|x˙ n (t)| dt +

Z

eQ(t) |x˙ n (t)|2 dt + b e

Q(t)

2

ξn

eQ(t) K 0 (t, xn (t)).xn (t)dt

−ξn Z ξn

−ξn

eQ(t) |xn (t)|λ dt λ−2

|x˙ n (t)| dt + b(γ2 kxn k)

eQ(t) |x˙ n (t)|2 dt + b(γ2 C1 )λ−2

b kxn k2 . b + (γ2 C1 )2−λ

Z

Z

ξn

eQ(t) |xn (t)|2 dt

−ξn ξn Q(t)

−ξn

e

|xn (t)|2 dt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

14

KHELIFI FATHI, MOHSEN TIMOUMI

Since kxn k > 0, it follows that Ψ(kxn kL∞ ([−ξn ,ξn ],RN ) ) ≥

b > 0. b + (γ2 C1 )2−λ

If kxn kL∞ ([−ξn ,ξn ],RN ) → 0 as n → ∞, we would have b Ψ(0) ≥ > 0, a contradiction. Passing to a subsequence if necesb + (γ2 C1 )2−λ sary, there is a constant C4 > 0 such that kxn kL∞ ([−nT,nT ],RN ) ≥ C4

(2.39)

for every n ∈ N. Moreover, for all k ∈ N, t 7→ xkn (t) = xn (t + kT ) is also a 2nT-periodic solution of problem (DVn ). Hence, if the maximum of |xn | occurs in θn ∈ [−nT, nT ] then the maximum of |xkn | occurs in τnk = θn − kT. Then there exists a kn ∈ Z such that τnkn ∈ [−T, T ]. Consequently, kxknn kL∞ ([−nT,nT ],RN ) = max |xknn (t)|. t∈[−T,T ]

Suppose contrary to our claim, that x ≡ 0. Then

kxknn kL∞ ([−nT,nT ],RN ) = max |xknn (t)| → 0, t∈[−T,T ]

which contradicts (2.39) . Then the proof of Theorem 1.2 is complete.

References [1] Alves, C.O.,Carriao, P.C. and Miyagaki, O.H. Existence of homoclinic orbits for asymptotically periodic system involving Duffing-like equation. Appl. Math. Lett. 16 (5) (2003) 639642. 1, 1.1 [2] P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity, Nonlinear Anal.7(9)(1983), 981-1012. 1 [3] A. Benhassine, M. Timoumi, Even homoclinic solutions for a class of second order Hamiltonian systems. Mathematics in Engineering,Science and Aerospace, Vol. 1, No. 3, pp. 279290, (2010). [4] Y.Ding, M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl. 2(1)(1993), 131-145. 1, 2 [5] Y.H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25 (11) (1995), 1095-1113. 1.1 [6] G.H. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potential changing sign, Chinese Ann. Math. Ser. A 17 (4) (1996), 651(a Chinese summary); Chinese Ann. Math. Ser. B 4(1996) 403-410. [7] P.L. Felmer, E.A. De, B.E. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Sc. Norm. Super. Pisca Cl. Sci. (4) 26 (2) (1998)285-301. [8] M. Izydorek, J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems. J. Differential Equations 219 (2) (2005), 375-389. 1, 1.1 [9] P.Korman, A.C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations 1994 (1)(1994), 1-10. [10] M.Ma,Z.Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal.67(2007)1737-1745. [11] Y.Lv, C.L. Tang, Existence of even homoclinic orbits for second order Hamiltonian systems, Nonlinear Anal. 67 (2007), 2189-2198. 2 2 [12] J.Mawhin, M. Willem, Critical point theory and Hamiltonian systems, in: Applied Mathematical Sciences, Vol. 74 Spring-Verlag, New Yorek. 1989. [13] W. Omana, M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5(5)(1992), 1115-1120. 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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[14] Z.Q. Qu, C.L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl. 291(1)(2004), 203-213. 1, 1.1 [15] P.H.Rabinowitz, Minimax Methods in critical point theory with applications to differential equations, CBMS 65, American Mathematical Society, Providence, RI, 1986. 1 [16] P.H.Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A 114(1-2)(1990), 33-38. 1.1, 2.1 [17] P.H.Rabinowitz, K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z.206 (3)(1991), 473-499. 1 [18] X.H. Tang, L. Xiao, Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal. 71 (2009), 1140-1152. 1 [19] X.H. Tang, X. Lin, Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl. 354(2009), 539-549. 1, 2, 2 [20] R.Yuan, Z.Zhang, Homoclinic solutions for a class of second order non-autonomous systems, Electron. J. Diff. Equ., Vol. 2009 (2009), No. 128, pp. 1-9. 1, 1.1 [21] Z.Zhang, Existence of homoclinic solutions for second order Hamiltonian systems with general potentials, Journal of Applied Mathematics and Computing (2013) 44: 263-272. 1 1 Khelifi Fathi, Faculty of Sciences of Monastir, Department of Mathematics, 5000 Monastir, Tunisia E-mail address: [email protected]