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The existence of periodic solutions for a class of nonautonomous lattices with bounded-coupling
Nonlinear Analysis 71 (2009) 1438–1444
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The existence of periodic solutions for a class of nonautonomous lattices with bounded-couplingI Chao Wang a,∗ , Dingbian Qian b a
School of Mathematical Science, Yancheng Teachers University, Yancheng 224002, PR China
b
School of Mathematical Science, Suzhou University, Suzhou 215006, PR China
article
a b s t r a c t
info
Article history: Received 31 August 2008 Accepted 3 December 2008
In this paper, by using a definition of topological degree and limiting arguments, we study the existence of periodic solutions of a class of one-dimensional nonautonomous chains with bounded-coupling. An approach is developed to deal with the case of lack of the a priori bounds for an infinite lattices. © 2008 Elsevier Ltd. All rights reserved.
MSC: 34C15 34C25 Keywords: Nonautonomous lattices Bounded-coupling Super-linear Periodic solutions A relative priori bounds Topological degree
1. Introduction In this paper, we are concerned with the 2π -periodic boundary value problem of the infinite system of weakly-coupled second-order differential equations u00i + gi (ui ) = pi (t , u, u0 ),
i ∈ Z,
(1.1)
u(2π ) − u(0) = u (2π ) − u (0) = 0, 0
0
(1.2)
where u ∈ R , u ∈ R , gi : R → R are continuous, pi : [0, 2π ] × R × R are continuous functions and 2π -periodic for the first variant and, for each i ∈ Z, pi dependents only on a finite numbers of components of u and u0 and satisfying Z
0
Z
Z
Z
(p0 ) |pi (t , u, u0 )| ≤ Pi ∈ R+ . Clearly, by a solution of (1.1)–(1.2) we mean a solution u of (1.1) such that ui (0) = ui (2π ), u0i (0) = u0i (2π ) for each i ∈ Z. We assume
(g0 ) gi (s) · s > 0, for all s 6= 0, i ∈ Z, gi (s) (g1 ) lim = +∞, i ∈ Z. |s|→+∞
s
I This work is supported by NNSF of China (10571131).
∗
Corresponding author. E-mail addresses: [email protected] (C. Wang), [email protected] (D. Qian).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.011
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
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The problems of the existence of the periodic solutions of infinite systems have been studied in many papers for its applied backgrounds, where different methods are applied in different cases. In [1–4], due to the variational structure of the autonomous conservative one-dimensional lattice of particles, the authors can make use of the classical variational techniques to obtain the existence of the periodic solutions. At the same time, in [5,6], a method of topological degree accompanied with limiting arguments are used to obtain the existence of the periodic solutions to a class of one-dimensional chain of particles periodically perturbed and with nearest-neighbor interaction between particles. Besides [5,6], we note that there are few papers that study the conditions of the existence of periodic solutions by using the method of topological degree to the infinite lattices with coupling. However, the researches on the existence of the periodic solutions and the related problems for the infinite lattices attracts much attentions. We refer the readers to [7,8] and [9] for the classical works about the existence of discrete breathers to some autonomous Hamiltonian lattices where in [7] the methods of Anti-integrable and in [9] the variational method are used. In this paper, because of the appearance of the x0 in the external force term, the classical variational method cannot be used. Meanwhile, under the condition (g1 ), the main difficulty of the problem is that lack of the a priori bounds for periodic solutions makes the methods, which are used to prove the existence of the uniform a priori bounds of the periodic solutions to finite systems in [5], fail. We note that a continuation approach is given in [10] to deal with the PBV problem of lack of the a priori bounds to equations, and by a continuation theorem proved in [10], the existence of the periodic solutions to the finite systems with weak-couple is obtained. However, the continuation theorem is not available to the infinite systems. Motivated by the ideal in [5], we organize the following text as follow. In Section 2, we study the finite system which is related to system (1.1), where, by the arguments developed in [10] and the definition of the topological degree, we obtain the existence of a periodic solution with suitable large numbers of zero points which leads to the fact that the norms of the periodic solution is independent of the dimensions of the finite system. So, as a consequence, in Section 3, we are able to use the same a priori bounds obtained in Section 2 to pass to the limit and get T -periodic solutions for the infinite lattice. Our main result in this paper is: Theorem 1.1. Assume (g0 ), (g1 ) hold. Then there is at least one solution to the periodic boundary value problem (1.1)–(1.2). 2. Finite systems Inspired by the ideals of the proof in [5], we firstly consider the corresponding finite system of d equations u00i + gi (ui ) = pi (t , u˜ , u˜ 0 ),
i = 1, . . . , d,
(2.1)
where u˜ = (. . . , 0, u1 , u2 , . . . , ud , 0, . . .), u = (. . . , 0, u1 , u2 , . . . , ud , 0, . . .). The 2π -periodic boundary value conditions of (2.1) is 0
˜0
0
0
u(2π ) − u(0) = u0 (2π ) − u0 (0) = 0.
(2.2)
Let u0i = yi , i = 1, . . . , d, we can write the equivalent system of (2.1) as u0i = yi , y0i = −gi (ui ) + pi (t , u˜ , y˜ ),
(2.3)
i = 1, . . . , d,
where y˜ = (. . . , 0, y1 , y2 , . . . , yd , 0, . . .), and if we let x = col(xi )i=1,...,2d = (u1 , y1 , . . . , ud , yd ), then (2.3) can be written as x0 = F (t , x),
(2.4)
where x ∈ R , n = 2d, F = col(Fi )i=1,...,2d : [0, 2π ]×R → R is a continuous function satisfying F2i−1 (t , x) = yi , F2i (t , x) = −gi (ui )+ pi (t , u˜ i , y˜ i ), for all i = 1, . . . , d. The condition (2.2) corresponds to 2π -periodic boundary value condition, for (2.4), n
n
n
x(0) = x(2π ).
(2.5)
Take a continuous function E : R → R such that E (s) · s < 0
for all s 6= 0
and |E (s)| ≤ 1
for all s ∈ R.
Now we embed system (2.4) into a one-parameter family of differential equations as following, i.e. we consider the system x0 = f (t , x; λ),
λ ∈ [0, 1],
(2.6)
where f (t , x; λ) is the second term of, for i = 1, . . . , d, u0i = yi , y0i = (1 − λ)E (yi ) − gi (ui ) + λpi (t , u˜ , y˜ ).
(2.7)
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Obviously, f (t , x; 1) = F (t , x) and f (t , x; 0) = f0 (x), where f0 (x) is the second term, i = 1, . . . , d, u0i = yi ,
(2.8)
y0i = E (yi ) − gi (ui ). For the Banach space X = {x ∈ C ([0, 2π ]) → Rn : x(0) = x(2π )}, we define the norm as n X
kxkX := max
t ∈[0,2π]
! 12 x2i
(t )
.
i =1
Define Z = C ([0, 2π ], Rn ) and for z ∈ Z , use the norm kz kZ := maxt ∈[0,2π] ( Let
Pn
L : D(L) ⊂ X → Z ,
i=1
1
zi2 (t )) 2 .
L(x)(t ) = x0 (t ),
with D(L) = {x ∈ X : x ∈ C 1 ([0, 2π ], Rn )}, N : X × [0, 1] → Z ,
N (x, λ)(t ) := f (t , x(t ); λ).
Then problem (2.1)–(2.2) is equivalent to the operator equation Lx = N (x, 1),
x ∈ D(L) ⊂ X .
(2.9)
It is easy to see that L is a linear Fredholm operator of index zero and N is an L-completely continuous mapping [10]. We denote by
Σ = {(x, λ) ∈ D(L) × [0, 1] : x0 = f (t , x; λ)} and by Σλ = {x ∈ D(L) : (x, λ) ∈ Σ }(0 ≤ λ ≤ 1), the section of Σ at λ. Clearly, Σ = arguments in [10], we know that the sets Σ and Σλ are closed and locally compact. Now, we define the subspaces Xi ⊂ X (i = 1, . . . , d; n = 2d) by
S
λ∈[0,1]
Σλ . Moreover, by the
Xi = {x(·) = col(xk ) ∈ X : xk = 0 for k ∈ {2i − 1, 2i}}, with projections Πi : X → Xi (i = 1, . . . , d),
(Πi x)(t ) = x2i−1 (t )e2i−1 + x2i (t )e2i , where {e1 , . . . , en } is the canonical orthonormal basis in Rn . Clearly, Xi (i = 1, . . . , d) are closed linear subspaces of X and Ld e e X = i=1 Xi . We also define Πi : X × [0, 1] → Xi by Πi (x, λ) = Πi (x). Obviously, system (2.8) is equivalent to the uncoupled second-order equation u00i − E (u0i ) + gi (ui ) = 0,
i = 1, . . . , d.
(2.10)
It is easy to prove that the only periodic solutions (of any period) are the constant ones. And if we assume xˆ = (ˆu1 , 0, . . . , uˆ d , 0) ∈ Rn is a periodic solution of (2.8), then gi (ˆui ) = 0, i = 1, . . . , d. We refer the readers to [10] for details of the proof. By (g1 ), there is a R0 > 0 such that gi (s) · s > 0 for |s| ≥ R0 ,
i = 1, . . . , d,
which implies that uˆ i < R0 , i = 1, . . . , d. So, there is a r0 > 0 such that for each periodic solution xˆ of (2.8), we have that kˆxk < r0 . Thus, we have: (i1 ) Σ0 is bounded in X and Σ0 ⊂ B(0, r0 ). Define χ0 := |DL (L − N (·, 0), X )| = |DL (L − N (·, 0), Ω )|, where Ω ⊃ Σ0 is any open bounded subset of X and DL is the coincidence degree. By the arguments in [10] (or see [11] Theorem 1), we have that
|DL (L − N (·, 0), B(0, r ))| = dB (f0 , B(0, r ), 0) for each r ≥ r0 . Now, for each i = 1, . . . , d, we define ϕi : X × [0, 1] → R+ as
Z 2π [x2i−1 (t )f2i (t , x(t ); λ) − x2i (t )f2i−1 (t , x(t ); λ)] · δ(x2i−1 (t ), x2i (t ))dt ϕi (x, λ) = 2π 0 1
for (x, λ) ∈ X × [0, 1], where δ : R2 → [0, 1] is a continuous function such that
δ(a, b) :=
1,
a2
for a2 + b2 < 1
1
+
b2
,
for a2 + b2 ≥ 1.
It is easy to see that ϕi (i = 1, . . . , d) are continuous.
(2.11)
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
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Remark 2.1. If x22i−1 (t ) + x22i (t ) ≥ 1
for all t ∈ [0, 2π ],
(2.12)
then
Z
1 ϕi (x, λ) = 2π
2π
x2i−1 (t )f2i (t , x(t ); λ) − x2i f2i−1 (t , x(t ); λ) x22i−1 (t ) + x22i (t )
0
dt .
(2.13)
Thus the functional ϕi (x, λ) can be regarded as a modification of the classic map which counts the number of rotations around the origin of the solution of (2.6) in the (x2i−1 , x2i )− plane. Arguing similarly as in [10], for each i = 1, . . . , d, we have Lemma 2.1. Assume (x, λ) ∈ Σ and (2.12) holds, then ϕi (x, λ) ∈ Z+ . Lemma 2.2. For each i = 1, . . . , d, we have that for each r1 ≥ 0, there is a r2 ≥ 0, such that 1
1
min (x22i−1 (t ) + x22i (t )) 2 ≤ r1 ⇒ max (x22i−1 (t ) + x22i (t )) 2 ≤ r2
[0,2π ]
[0,2π]
holds for all (x, λ) ∈ Σ . Proof. For each i = 1, . . . , d, by (g1 ), there is a di > 0, such that a
Z
gi (s)ds + di > 0, 0
holds for each a ∈ R. Define Vi : R2 → R+ as follow Vi (a, b) =
1 2
b2 +
a
Z
gi (s)ds + di + 2, 0
then Vi (a, b) ∈ C 1 and grad Vi = (Vi0,1 , Vi0,2 ). By the definition of Vi , for all (a, b) ∈ R2 , we have
|b| ≤ Vi (a, b). And then, for all (x, λ) ∈ R2d × [0, 1] and for all t ∈ [0, 2π ], we have
|Vi0,1 (x2i−1 , x2i )f2i−1 (t , x; λ) + Vi0,2 (x2i−1 , x2i )f2i (t , x; λ)| = |gi (ui )yi + yi ((1 − λ)E (yi ) − gi (t , u) + λpi (t , u˜ , y˜ ))| ≤ |yi ||1 + Pi | ≤ (1 + Pi )Vi (ui , yi ) = (1 + Pi )Vi (x2i−1 , x2i ). Define W : R2 → R, W (z ) = log Vi (z ), z ∈ R2 . Obviously, W ∈ C 1 (R2 , R) and we have lim
|z |→+∞
W (z ) = +∞
(2.14)
grad W (z ) = Vi−1 (z ) · grad Vi (z ).
(2.15)
Now, for each r1 ≥ 0, take (x, λ) ∈ Σ such that 1
min (x22i−1 (t ) + x22i (t )) 2 ≤ r1 .
[0,2π ]
Fix a constant number C1 > r1 . Let t0 = t0 (x, λ) such that 1
1
(x22i−1 (t0 ) + x22i (t0 )) 2 = max (x22i−1 (t ) + x22i (t )) 2 := M (x). [0,2π ]
Then, if M (x) > C1 , there is a t1 ∈ [0, 2π ] such that 1
(x22i−1 (t1 ) + x22i (t1 )) 2 = C1 . By the property of period of x, we can choose t0 and t1 such that t0 − t1 > 0 and when t ∈ (t1 , t0 ], we have that 1
(x22i−1 (t ) + x22i (t )) 2 > C1 .
Now, we consider a C 1 function ω(t ) := W (x2i−1 (t ), x2i (t )), t ∈ (t1 , t0 ]. By above arguments and (2.15), considering x2i−1 (t ) = f2i−1 (t , x; λ) and x02i (t ) = f2i (t , x; λ), we have 0
ω(t0 ) = ω(t1 ) +
Z
t0 t1
ω0 (s)ds ≤ ω(t1 ) +
Z
t0
(1 + P (s))ds
t1
≤ max{W (z ) : |z | = C1 } + 2π (1 + Pi ) := C2 .
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It is easy to see that C2 only depends on r1 and i, but not on d. From (2.14), we have a K ≥ 0 such that W (z ) > C2 when 1
|z | > K . It is easy to see that (x22i−1 (t0 ) + x22i (t0 )) 2 ≤ K . Thus, take r2 := max{C1 , K } and we complete the proof.
Remark 2.2. From the proof of Lemma 2.2, we can see that r2 only depends on r1 and i, but not on d. By Lemmas 2.1 and 2.2 and Remark 2.2, for each i = 1, . . . , d, we have: Corollary 2.1. There is a Ri > 0 (only depends on i) such that 1
min (x22i−1 (t ) + x22i (t )) 2 ≥ 1
[0,2π]
for any (x, λ) ∈ Σ with |Πi x|X ≥ Ri . And then ϕi (x, λ) ∈ Z+ . Now, for each i = 1, . . . , d, we define Mi := sup{ϕi (x, λ) : x ∈ X , λ ∈ [0, 1], |Πi x|X ≤ Ri }, where Ri are given in Corollary 2.1. Lemma 2.3. For each i = 1, . . . , d, we have that Mi < +∞. Proof. For each i = 1, . . . , d, and for each r ≥ 0, we have 1
1
(f2i2−1 (t , x; λ) + f2i2 (t , x; λ)) 2 ≤ (r 2 + max gi2 (ui )) 2 + (1 + Pi ) := βr (t ) [0,r ]
1
for all t ∈ [0, 2π ], all λ ∈ [0, 1] and each x ∈ Rn that (x22i−1 + x22i ) 2 ≤ r. Then, from the definitions of ϕi and Mi we have Mi ≤
Ri |βRi |1 2π
and so we complete the proof.
By Corollary 2.1, it is easy to see that Mi only depends on i, but not on d. Combining Corollary 2.1 with Lemma 2.3, we have the following property. Property 2.1. For each i = 1, . . . , d, we have
ϕi (Σ ) ⊂ ([0, Mi ] ∪ N). Now, we define a value
hQ i :=
2π
dθ
2
Z
1
π 0
Q (cos θ , sin θ )
,
where Q : dom Q ⊃ S 1 → R+ is a continuous function, S 1 = {(x1 , x2 ) ∈ R2 , x21 + x22 = 1} is the unit circle in R2 . Then, by Proposition 2 in [10] and the remarks therein, we have Lemma 2.4. Suppose that there are a constant K0 > 0, a continuous function Θ : S 1 → R+ and a measurable function γ ∈ L1 ([0, 2π ], R+ ), such that the following inequalities holds for all t ∈ [0, 2π ], all λ ∈ [0, 1] and each x ∈ Rn such that 1
|(x2i−1 , x2i )| = (x22i−1 + x22i ) 2 ≥ K0 : f2i (t , x; λ) ·
x2i−1
(
x22i−1
+
1 x22i 2
)
− f2i−1 (t , x; λ) · x2i−1
1
≤ −(x22i−1 + x22i ) 2 · θK
(
x22i−1
+
1 x22i 2
)
,
x2i
(
x22i−1
1
+ x22i ) 2 !
x2i
(
x22i−1
1
+ x22i ) 2
+ γK (t ).
Then, for each R0 ≥ 1 a suitable constant (independent of x and λ), we have that
ϕj (x, λ) ≥
1
hΘ i
holds provided that (x, λ) ∈ Σ satisfies x22i−1 (t ) + x22i (t ) ≥ R20 ,
for each t ∈ [0, 2π ].
ei (ϕi−1 (m) ∩ Σ ) is bounded. Lemma 2.5. For each i = 1, . . . , d and for any m ∈ Z+ , we have that Π
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
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Proof. By (g1 ), for each K ∈ R+ , there is a DK ≥ 0 such that gi (s) · s ≥ K 2 s2 − DK ,
∀s ∈ R.
So, for all (a, b) ∈ R with a2 + b2 ≥ 1 and for all t ∈ [0, 2π], we have 2
gi (a)a + b2 − (1 − λ)E (b)a − λpi (t , u˜ , y˜ )a ≥ K 2 a2 + b2 − |a| − Pi |a| − DK 1
≥ K 2 a2 + b2 − (Pi + 1 + DK )(a2 + b2 ) 2 . 1
1
Let (x, λ) ∈ Σ with min[0,2π] (x22i−1 (t ) + x22i (t )) 2 = min[0,2π] (u2i (t ) + y2i (t )) 2 ≥ 1, then by (2.13), we have
Z 2π gi (ui (t ))ui (t ) + y2i (t ) − (1 − λ)E (yi (t ))ui (t ) − λpi (t , u˜ , y˜ )ui (t ) ϕi (x, λ) = dt . 2π 0 u2i (t ) + y2i (t ) 1
Thus, we have f2i (t , x; λ) ·
x2i−1
(
x22i−1
≤ −(
x22i−1
(t ) +
(t ) +
x22i
x22i
(t ))
1 2
(t )) · θK
1 2
− f2i−1 (t , x; λ) ·
x2i
(
x22i−1
1
(t ) + x22i (t )) 2
x2i−1
x2i
, 1 1 (x22i−1 (t ) + x22i (t )) 2 (x22i−1 (t ) + x22i (t )) 2
! + γK (t ),
where θK (a, b) := K 2 a2 + b2 , γK (t ) := (Pi + 1 + DK ). By Lemma 2.4, for the K given above, there is a CK ≥ 1 such that
ϕi (x, λ) ≥
1
hθK i
=K
ei (ϕi−1 (m) ∩ Σ ) must be bounded, otherwise, we will for all (x, λ) ∈ Σ with (x22i−1 (t ) + x22i (t )) ≥ CK2 for all t ∈ [0, 2π ]. So, Π have a contradiction. The proof is completed. Remark 2.3. From the proof of Proposition 2 in [10], we can easily to see that CK only depends on K . For each i = 1, . . . , d, take fixed numbers ni ∈ N such that ni − 1 > Mi . By the proof of Lemma 2.5, we see that for each fixed number K > ni suitable large, there is a positive number CK , only depends on i and K , such that ei (ϕi−1 (m) ∩ Σ ) ⊂ B(0, CK ), m = 0, 1, . . . , ni . So, considering Corollary 2.1, it is easy to prove the following result. Π
ei (ϕi−1 ([0, ni ]) ∩ Σ ) ⊂ B(0, e Property 2.2. For each i = 1, . . . , d, there is a e Ri (independent of d) such that Π Ri ). In what follows, for each i = 1, . . . , d, we denote by Ai := {(x, λ) ∈ X × [0, 1] : ϕi (x, λ) < ni + 12 },
Ω=
d \
Ai ,
i=1
Σi := {(x, λ) ∈ Σ : ϕi (x, λ) ∈ [0, ni ]} = ϕi−1 ([0, ni ]) ∩ Σ ,
b := Σ
d \
Σi ⊂ Ω .
i=1
b and Property 2.2, we have that, for each (x, λ) ∈ Σ b , |Πi (x)|X < It is easy to that Ai and Ω are open sets. By the definition of Σ e b is bounded. Note that Σ b ⊂ Σ and Ri , i = 1, . . . , d. So |x|X < R0 , where R0 is a certain positive number. And so Σ b is closed, and then compact. Thus, we can choose a bounded open set B ϕi (i = 1, . . . , d) are continuous, we have that Σ such that
b ⊂ B ⊂ clX ×[0,1] B ⊂ Ω . Σ b. Lemma 2.6. Σ ∩ Ω = Σ b . By contradiction, assume that there is a (x, λ) ∈ (Σ ∩ (Ω \ Σ b )), Proof. Obviously, it is enough to prove that Σ ∩ Ω ⊂ Σ then there is a i0 ∈ {i = 1, . . . , d} such that ni 0 +
1 2
> ϕi0 (x, λ) > ni0 > Mi0 ,
which contradicts ϕi0 (Σ ) ⊂ ([0, Mi0 ] ∪ N) by Property 2.1. Thus, we prove the lemma. Now, we claim that Lx 6= N (x, λ)
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C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
for any x ∈ (frX ×[0,1] B )λ , λ ∈ [0, 1], where Bλ := {x ∈ X : (x, λ) ∈ B }, (frX ×[0,1] B )λ := {x ∈ X : (x, λ) ∈ frX ×[0,1] B }. In fact, if there are λ0 ∈ [0, 1] and x0 ∈ (frX ×[0,1] B )λ0 such that L(x0 ) = N (x0 , λ0 ), then, by Lemma 2.6,
b. (x0 , λ0 ) ∈ Σ ∩ clX ×[0,1] B ⊂ Σ ∩ Ω = Σ Then, x ∈ Bλ and we have a contradiction. Using the homotopy invariance of coincidence degree, we have DL (L − N (·, 1), B1 ) = DL (L − N (·, 0), B0 ) = dB (f0 (·), B0 ∩ Rn , 0) = dB (f0 (·), B(0, r0 ), 0), where r0 is defined in (i1 ) above. By standard computations as in [10], we have for any r ≥ r0 , dB (f0 (·), B(0, r0 ), 0) = dB (f0 (·), B(0, r ), 0) = 1.
b satisfying So, there is at least one solution x(t ) of equation Lx = N (x, 1) in B1 and then in Σ ϕi (x, 1) ≤ ni ,
i = 1, . . . , d.
Thus, we obtain at least a solution to the periodic boundary value problem (2.1)–(2.2). By Property 2.2 again, we have |Πi x|X < e Ri , i = 1, . . . , d. 3. Proof of Theorem 1.1 In this section, we are concerned with the existence of solution to periodic BVP (1.1)–(1.2). The idea of the proof is to pass to the limit from a finite system. Consider the finite system of 2n + 1 equations u00i + gi (ui ) = pi (t , u˜ , u˜ 0 ),
i = −n, . . . , 0, . . . , n.
(3.1) (n)
This system has been studied in Section 2, and we have a periodic solution {xi }i=−n,...,n of period 2π to Eq. (3.1) satisfying |(x2i−1 (t ), x2i (t ))| < e Ri , i = −n, . . . , n, where e Ri are defined in Property 2.2 and only depend on i but do not depend on n by Property 2.2. Now, the bounds e Ri mentioned above can be used as in [5] with suitable modifications to prove the convergence of this sequence to a solution of (1.1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
G. Arioli, Chabrowski, Periodic motions of a dynamical system consisting of an infinite lattice of particles, Dynam. Systems Appl. 6 (1997) 287–396. G. Arioli, F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal. TMA 26 (1995) 1103–1114. G. Arioli, F. Gazzola, Existence and approximation of periodic motions of an infinite lattice of particles, Z. Angew. Math. Phys. 46 (1995) 898–912. G. Arioli, F. Gazzola, S. Terracini, Multibump periodic motions of an infinite lattice of particles, Math. Z. 223 (1996) 627–642. P.J. Torres, Periodic motions of forced infinite lattices with nearest neighbor interaction, Z. Angew. Math. Phys. 51 (2000) 333–345. P.J. Torres, Necessary and sufficient conditions for existence of periodic motions of forced systems of particles, Z. Angew. Math. Phys. 51 (2001) 535–540. R.S. Mackay, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlineaity 7 (1994) 1623–1643. R.S. Mackay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713. S. Aubry, G. Kopidakis, V. Kadelburg, Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems, Discrete Contin. Dyn. Syst. (Series B) 1 (2001) 271–298. A. Capietto, J. Mawhin, F. Zanolin, A continuation approach to superlinear periodic boundary value problem, J. Differential Equations 88 (1990) 347–395. A. Capietto, J. Mawhin, F. Zanolin, Continuation theorem for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992) 41–72.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
The existence of periodic solutions for a class of nonautonomous lattices with bounded-couplingI Chao Wang a,∗ , Dingbian Qian b a
School of Mathematical Science, Yancheng Teachers University, Yancheng 224002, PR China
b
School of Mathematical Science, Suzhou University, Suzhou 215006, PR China
article
a b s t r a c t
info
Article history: Received 31 August 2008 Accepted 3 December 2008
In this paper, by using a definition of topological degree and limiting arguments, we study the existence of periodic solutions of a class of one-dimensional nonautonomous chains with bounded-coupling. An approach is developed to deal with the case of lack of the a priori bounds for an infinite lattices. © 2008 Elsevier Ltd. All rights reserved.
MSC: 34C15 34C25 Keywords: Nonautonomous lattices Bounded-coupling Super-linear Periodic solutions A relative priori bounds Topological degree
1. Introduction In this paper, we are concerned with the 2π -periodic boundary value problem of the infinite system of weakly-coupled second-order differential equations u00i + gi (ui ) = pi (t , u, u0 ),
i ∈ Z,
(1.1)
u(2π ) − u(0) = u (2π ) − u (0) = 0, 0
0
(1.2)
where u ∈ R , u ∈ R , gi : R → R are continuous, pi : [0, 2π ] × R × R are continuous functions and 2π -periodic for the first variant and, for each i ∈ Z, pi dependents only on a finite numbers of components of u and u0 and satisfying Z
0
Z
Z
Z
(p0 ) |pi (t , u, u0 )| ≤ Pi ∈ R+ . Clearly, by a solution of (1.1)–(1.2) we mean a solution u of (1.1) such that ui (0) = ui (2π ), u0i (0) = u0i (2π ) for each i ∈ Z. We assume
(g0 ) gi (s) · s > 0, for all s 6= 0, i ∈ Z, gi (s) (g1 ) lim = +∞, i ∈ Z. |s|→+∞
s
I This work is supported by NNSF of China (10571131).
∗
Corresponding author. E-mail addresses: [email protected] (C. Wang), [email protected] (D. Qian).
0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.011
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
1439
The problems of the existence of the periodic solutions of infinite systems have been studied in many papers for its applied backgrounds, where different methods are applied in different cases. In [1–4], due to the variational structure of the autonomous conservative one-dimensional lattice of particles, the authors can make use of the classical variational techniques to obtain the existence of the periodic solutions. At the same time, in [5,6], a method of topological degree accompanied with limiting arguments are used to obtain the existence of the periodic solutions to a class of one-dimensional chain of particles periodically perturbed and with nearest-neighbor interaction between particles. Besides [5,6], we note that there are few papers that study the conditions of the existence of periodic solutions by using the method of topological degree to the infinite lattices with coupling. However, the researches on the existence of the periodic solutions and the related problems for the infinite lattices attracts much attentions. We refer the readers to [7,8] and [9] for the classical works about the existence of discrete breathers to some autonomous Hamiltonian lattices where in [7] the methods of Anti-integrable and in [9] the variational method are used. In this paper, because of the appearance of the x0 in the external force term, the classical variational method cannot be used. Meanwhile, under the condition (g1 ), the main difficulty of the problem is that lack of the a priori bounds for periodic solutions makes the methods, which are used to prove the existence of the uniform a priori bounds of the periodic solutions to finite systems in [5], fail. We note that a continuation approach is given in [10] to deal with the PBV problem of lack of the a priori bounds to equations, and by a continuation theorem proved in [10], the existence of the periodic solutions to the finite systems with weak-couple is obtained. However, the continuation theorem is not available to the infinite systems. Motivated by the ideal in [5], we organize the following text as follow. In Section 2, we study the finite system which is related to system (1.1), where, by the arguments developed in [10] and the definition of the topological degree, we obtain the existence of a periodic solution with suitable large numbers of zero points which leads to the fact that the norms of the periodic solution is independent of the dimensions of the finite system. So, as a consequence, in Section 3, we are able to use the same a priori bounds obtained in Section 2 to pass to the limit and get T -periodic solutions for the infinite lattice. Our main result in this paper is: Theorem 1.1. Assume (g0 ), (g1 ) hold. Then there is at least one solution to the periodic boundary value problem (1.1)–(1.2). 2. Finite systems Inspired by the ideals of the proof in [5], we firstly consider the corresponding finite system of d equations u00i + gi (ui ) = pi (t , u˜ , u˜ 0 ),
i = 1, . . . , d,
(2.1)
where u˜ = (. . . , 0, u1 , u2 , . . . , ud , 0, . . .), u = (. . . , 0, u1 , u2 , . . . , ud , 0, . . .). The 2π -periodic boundary value conditions of (2.1) is 0
˜0
0
0
u(2π ) − u(0) = u0 (2π ) − u0 (0) = 0.
(2.2)
Let u0i = yi , i = 1, . . . , d, we can write the equivalent system of (2.1) as u0i = yi , y0i = −gi (ui ) + pi (t , u˜ , y˜ ),
(2.3)
i = 1, . . . , d,
where y˜ = (. . . , 0, y1 , y2 , . . . , yd , 0, . . .), and if we let x = col(xi )i=1,...,2d = (u1 , y1 , . . . , ud , yd ), then (2.3) can be written as x0 = F (t , x),
(2.4)
where x ∈ R , n = 2d, F = col(Fi )i=1,...,2d : [0, 2π ]×R → R is a continuous function satisfying F2i−1 (t , x) = yi , F2i (t , x) = −gi (ui )+ pi (t , u˜ i , y˜ i ), for all i = 1, . . . , d. The condition (2.2) corresponds to 2π -periodic boundary value condition, for (2.4), n
n
n
x(0) = x(2π ).
(2.5)
Take a continuous function E : R → R such that E (s) · s < 0
for all s 6= 0
and |E (s)| ≤ 1
for all s ∈ R.
Now we embed system (2.4) into a one-parameter family of differential equations as following, i.e. we consider the system x0 = f (t , x; λ),
λ ∈ [0, 1],
(2.6)
where f (t , x; λ) is the second term of, for i = 1, . . . , d, u0i = yi , y0i = (1 − λ)E (yi ) − gi (ui ) + λpi (t , u˜ , y˜ ).
(2.7)
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C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
Obviously, f (t , x; 1) = F (t , x) and f (t , x; 0) = f0 (x), where f0 (x) is the second term, i = 1, . . . , d, u0i = yi ,
(2.8)
y0i = E (yi ) − gi (ui ). For the Banach space X = {x ∈ C ([0, 2π ]) → Rn : x(0) = x(2π )}, we define the norm as n X
kxkX := max
t ∈[0,2π]
! 12 x2i
(t )
.
i =1
Define Z = C ([0, 2π ], Rn ) and for z ∈ Z , use the norm kz kZ := maxt ∈[0,2π] ( Let
Pn
L : D(L) ⊂ X → Z ,
i=1
1
zi2 (t )) 2 .
L(x)(t ) = x0 (t ),
with D(L) = {x ∈ X : x ∈ C 1 ([0, 2π ], Rn )}, N : X × [0, 1] → Z ,
N (x, λ)(t ) := f (t , x(t ); λ).
Then problem (2.1)–(2.2) is equivalent to the operator equation Lx = N (x, 1),
x ∈ D(L) ⊂ X .
(2.9)
It is easy to see that L is a linear Fredholm operator of index zero and N is an L-completely continuous mapping [10]. We denote by
Σ = {(x, λ) ∈ D(L) × [0, 1] : x0 = f (t , x; λ)} and by Σλ = {x ∈ D(L) : (x, λ) ∈ Σ }(0 ≤ λ ≤ 1), the section of Σ at λ. Clearly, Σ = arguments in [10], we know that the sets Σ and Σλ are closed and locally compact. Now, we define the subspaces Xi ⊂ X (i = 1, . . . , d; n = 2d) by
S
λ∈[0,1]
Σλ . Moreover, by the
Xi = {x(·) = col(xk ) ∈ X : xk = 0 for k ∈ {2i − 1, 2i}}, with projections Πi : X → Xi (i = 1, . . . , d),
(Πi x)(t ) = x2i−1 (t )e2i−1 + x2i (t )e2i , where {e1 , . . . , en } is the canonical orthonormal basis in Rn . Clearly, Xi (i = 1, . . . , d) are closed linear subspaces of X and Ld e e X = i=1 Xi . We also define Πi : X × [0, 1] → Xi by Πi (x, λ) = Πi (x). Obviously, system (2.8) is equivalent to the uncoupled second-order equation u00i − E (u0i ) + gi (ui ) = 0,
i = 1, . . . , d.
(2.10)
It is easy to prove that the only periodic solutions (of any period) are the constant ones. And if we assume xˆ = (ˆu1 , 0, . . . , uˆ d , 0) ∈ Rn is a periodic solution of (2.8), then gi (ˆui ) = 0, i = 1, . . . , d. We refer the readers to [10] for details of the proof. By (g1 ), there is a R0 > 0 such that gi (s) · s > 0 for |s| ≥ R0 ,
i = 1, . . . , d,
which implies that uˆ i < R0 , i = 1, . . . , d. So, there is a r0 > 0 such that for each periodic solution xˆ of (2.8), we have that kˆxk < r0 . Thus, we have: (i1 ) Σ0 is bounded in X and Σ0 ⊂ B(0, r0 ). Define χ0 := |DL (L − N (·, 0), X )| = |DL (L − N (·, 0), Ω )|, where Ω ⊃ Σ0 is any open bounded subset of X and DL is the coincidence degree. By the arguments in [10] (or see [11] Theorem 1), we have that
|DL (L − N (·, 0), B(0, r ))| = dB (f0 , B(0, r ), 0) for each r ≥ r0 . Now, for each i = 1, . . . , d, we define ϕi : X × [0, 1] → R+ as
Z 2π [x2i−1 (t )f2i (t , x(t ); λ) − x2i (t )f2i−1 (t , x(t ); λ)] · δ(x2i−1 (t ), x2i (t ))dt ϕi (x, λ) = 2π 0 1
for (x, λ) ∈ X × [0, 1], where δ : R2 → [0, 1] is a continuous function such that
δ(a, b) :=
1,
a2
for a2 + b2 < 1
1
+
b2
,
for a2 + b2 ≥ 1.
It is easy to see that ϕi (i = 1, . . . , d) are continuous.
(2.11)
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
1441
Remark 2.1. If x22i−1 (t ) + x22i (t ) ≥ 1
for all t ∈ [0, 2π ],
(2.12)
then
Z
1 ϕi (x, λ) = 2π
2π
x2i−1 (t )f2i (t , x(t ); λ) − x2i f2i−1 (t , x(t ); λ) x22i−1 (t ) + x22i (t )
0
dt .
(2.13)
Thus the functional ϕi (x, λ) can be regarded as a modification of the classic map which counts the number of rotations around the origin of the solution of (2.6) in the (x2i−1 , x2i )− plane. Arguing similarly as in [10], for each i = 1, . . . , d, we have Lemma 2.1. Assume (x, λ) ∈ Σ and (2.12) holds, then ϕi (x, λ) ∈ Z+ . Lemma 2.2. For each i = 1, . . . , d, we have that for each r1 ≥ 0, there is a r2 ≥ 0, such that 1
1
min (x22i−1 (t ) + x22i (t )) 2 ≤ r1 ⇒ max (x22i−1 (t ) + x22i (t )) 2 ≤ r2
[0,2π ]
[0,2π]
holds for all (x, λ) ∈ Σ . Proof. For each i = 1, . . . , d, by (g1 ), there is a di > 0, such that a
Z
gi (s)ds + di > 0, 0
holds for each a ∈ R. Define Vi : R2 → R+ as follow Vi (a, b) =
1 2
b2 +
a
Z
gi (s)ds + di + 2, 0
then Vi (a, b) ∈ C 1 and grad Vi = (Vi0,1 , Vi0,2 ). By the definition of Vi , for all (a, b) ∈ R2 , we have
|b| ≤ Vi (a, b). And then, for all (x, λ) ∈ R2d × [0, 1] and for all t ∈ [0, 2π ], we have
|Vi0,1 (x2i−1 , x2i )f2i−1 (t , x; λ) + Vi0,2 (x2i−1 , x2i )f2i (t , x; λ)| = |gi (ui )yi + yi ((1 − λ)E (yi ) − gi (t , u) + λpi (t , u˜ , y˜ ))| ≤ |yi ||1 + Pi | ≤ (1 + Pi )Vi (ui , yi ) = (1 + Pi )Vi (x2i−1 , x2i ). Define W : R2 → R, W (z ) = log Vi (z ), z ∈ R2 . Obviously, W ∈ C 1 (R2 , R) and we have lim
|z |→+∞
W (z ) = +∞
(2.14)
grad W (z ) = Vi−1 (z ) · grad Vi (z ).
(2.15)
Now, for each r1 ≥ 0, take (x, λ) ∈ Σ such that 1
min (x22i−1 (t ) + x22i (t )) 2 ≤ r1 .
[0,2π ]
Fix a constant number C1 > r1 . Let t0 = t0 (x, λ) such that 1
1
(x22i−1 (t0 ) + x22i (t0 )) 2 = max (x22i−1 (t ) + x22i (t )) 2 := M (x). [0,2π ]
Then, if M (x) > C1 , there is a t1 ∈ [0, 2π ] such that 1
(x22i−1 (t1 ) + x22i (t1 )) 2 = C1 . By the property of period of x, we can choose t0 and t1 such that t0 − t1 > 0 and when t ∈ (t1 , t0 ], we have that 1
(x22i−1 (t ) + x22i (t )) 2 > C1 .
Now, we consider a C 1 function ω(t ) := W (x2i−1 (t ), x2i (t )), t ∈ (t1 , t0 ]. By above arguments and (2.15), considering x2i−1 (t ) = f2i−1 (t , x; λ) and x02i (t ) = f2i (t , x; λ), we have 0
ω(t0 ) = ω(t1 ) +
Z
t0 t1
ω0 (s)ds ≤ ω(t1 ) +
Z
t0
(1 + P (s))ds
t1
≤ max{W (z ) : |z | = C1 } + 2π (1 + Pi ) := C2 .
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C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
It is easy to see that C2 only depends on r1 and i, but not on d. From (2.14), we have a K ≥ 0 such that W (z ) > C2 when 1
|z | > K . It is easy to see that (x22i−1 (t0 ) + x22i (t0 )) 2 ≤ K . Thus, take r2 := max{C1 , K } and we complete the proof.
Remark 2.2. From the proof of Lemma 2.2, we can see that r2 only depends on r1 and i, but not on d. By Lemmas 2.1 and 2.2 and Remark 2.2, for each i = 1, . . . , d, we have: Corollary 2.1. There is a Ri > 0 (only depends on i) such that 1
min (x22i−1 (t ) + x22i (t )) 2 ≥ 1
[0,2π]
for any (x, λ) ∈ Σ with |Πi x|X ≥ Ri . And then ϕi (x, λ) ∈ Z+ . Now, for each i = 1, . . . , d, we define Mi := sup{ϕi (x, λ) : x ∈ X , λ ∈ [0, 1], |Πi x|X ≤ Ri }, where Ri are given in Corollary 2.1. Lemma 2.3. For each i = 1, . . . , d, we have that Mi < +∞. Proof. For each i = 1, . . . , d, and for each r ≥ 0, we have 1
1
(f2i2−1 (t , x; λ) + f2i2 (t , x; λ)) 2 ≤ (r 2 + max gi2 (ui )) 2 + (1 + Pi ) := βr (t ) [0,r ]
1
for all t ∈ [0, 2π ], all λ ∈ [0, 1] and each x ∈ Rn that (x22i−1 + x22i ) 2 ≤ r. Then, from the definitions of ϕi and Mi we have Mi ≤
Ri |βRi |1 2π
and so we complete the proof.
By Corollary 2.1, it is easy to see that Mi only depends on i, but not on d. Combining Corollary 2.1 with Lemma 2.3, we have the following property. Property 2.1. For each i = 1, . . . , d, we have
ϕi (Σ ) ⊂ ([0, Mi ] ∪ N). Now, we define a value
hQ i :=
2π
dθ
2
Z
1
π 0
Q (cos θ , sin θ )
,
where Q : dom Q ⊃ S 1 → R+ is a continuous function, S 1 = {(x1 , x2 ) ∈ R2 , x21 + x22 = 1} is the unit circle in R2 . Then, by Proposition 2 in [10] and the remarks therein, we have Lemma 2.4. Suppose that there are a constant K0 > 0, a continuous function Θ : S 1 → R+ and a measurable function γ ∈ L1 ([0, 2π ], R+ ), such that the following inequalities holds for all t ∈ [0, 2π ], all λ ∈ [0, 1] and each x ∈ Rn such that 1
|(x2i−1 , x2i )| = (x22i−1 + x22i ) 2 ≥ K0 : f2i (t , x; λ) ·
x2i−1
(
x22i−1
+
1 x22i 2
)
− f2i−1 (t , x; λ) · x2i−1
1
≤ −(x22i−1 + x22i ) 2 · θK
(
x22i−1
+
1 x22i 2
)
,
x2i
(
x22i−1
1
+ x22i ) 2 !
x2i
(
x22i−1
1
+ x22i ) 2
+ γK (t ).
Then, for each R0 ≥ 1 a suitable constant (independent of x and λ), we have that
ϕj (x, λ) ≥
1
hΘ i
holds provided that (x, λ) ∈ Σ satisfies x22i−1 (t ) + x22i (t ) ≥ R20 ,
for each t ∈ [0, 2π ].
ei (ϕi−1 (m) ∩ Σ ) is bounded. Lemma 2.5. For each i = 1, . . . , d and for any m ∈ Z+ , we have that Π
C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
1443
Proof. By (g1 ), for each K ∈ R+ , there is a DK ≥ 0 such that gi (s) · s ≥ K 2 s2 − DK ,
∀s ∈ R.
So, for all (a, b) ∈ R with a2 + b2 ≥ 1 and for all t ∈ [0, 2π], we have 2
gi (a)a + b2 − (1 − λ)E (b)a − λpi (t , u˜ , y˜ )a ≥ K 2 a2 + b2 − |a| − Pi |a| − DK 1
≥ K 2 a2 + b2 − (Pi + 1 + DK )(a2 + b2 ) 2 . 1
1
Let (x, λ) ∈ Σ with min[0,2π] (x22i−1 (t ) + x22i (t )) 2 = min[0,2π] (u2i (t ) + y2i (t )) 2 ≥ 1, then by (2.13), we have
Z 2π gi (ui (t ))ui (t ) + y2i (t ) − (1 − λ)E (yi (t ))ui (t ) − λpi (t , u˜ , y˜ )ui (t ) ϕi (x, λ) = dt . 2π 0 u2i (t ) + y2i (t ) 1
Thus, we have f2i (t , x; λ) ·
x2i−1
(
x22i−1
≤ −(
x22i−1
(t ) +
(t ) +
x22i
x22i
(t ))
1 2
(t )) · θK
1 2
− f2i−1 (t , x; λ) ·
x2i
(
x22i−1
1
(t ) + x22i (t )) 2
x2i−1
x2i
, 1 1 (x22i−1 (t ) + x22i (t )) 2 (x22i−1 (t ) + x22i (t )) 2
! + γK (t ),
where θK (a, b) := K 2 a2 + b2 , γK (t ) := (Pi + 1 + DK ). By Lemma 2.4, for the K given above, there is a CK ≥ 1 such that
ϕi (x, λ) ≥
1
hθK i
=K
ei (ϕi−1 (m) ∩ Σ ) must be bounded, otherwise, we will for all (x, λ) ∈ Σ with (x22i−1 (t ) + x22i (t )) ≥ CK2 for all t ∈ [0, 2π ]. So, Π have a contradiction. The proof is completed. Remark 2.3. From the proof of Proposition 2 in [10], we can easily to see that CK only depends on K . For each i = 1, . . . , d, take fixed numbers ni ∈ N such that ni − 1 > Mi . By the proof of Lemma 2.5, we see that for each fixed number K > ni suitable large, there is a positive number CK , only depends on i and K , such that ei (ϕi−1 (m) ∩ Σ ) ⊂ B(0, CK ), m = 0, 1, . . . , ni . So, considering Corollary 2.1, it is easy to prove the following result. Π
ei (ϕi−1 ([0, ni ]) ∩ Σ ) ⊂ B(0, e Property 2.2. For each i = 1, . . . , d, there is a e Ri (independent of d) such that Π Ri ). In what follows, for each i = 1, . . . , d, we denote by Ai := {(x, λ) ∈ X × [0, 1] : ϕi (x, λ) < ni + 12 },
Ω=
d \
Ai ,
i=1
Σi := {(x, λ) ∈ Σ : ϕi (x, λ) ∈ [0, ni ]} = ϕi−1 ([0, ni ]) ∩ Σ ,
b := Σ
d \
Σi ⊂ Ω .
i=1
b and Property 2.2, we have that, for each (x, λ) ∈ Σ b , |Πi (x)|X < It is easy to that Ai and Ω are open sets. By the definition of Σ e b is bounded. Note that Σ b ⊂ Σ and Ri , i = 1, . . . , d. So |x|X < R0 , where R0 is a certain positive number. And so Σ b is closed, and then compact. Thus, we can choose a bounded open set B ϕi (i = 1, . . . , d) are continuous, we have that Σ such that
b ⊂ B ⊂ clX ×[0,1] B ⊂ Ω . Σ b. Lemma 2.6. Σ ∩ Ω = Σ b . By contradiction, assume that there is a (x, λ) ∈ (Σ ∩ (Ω \ Σ b )), Proof. Obviously, it is enough to prove that Σ ∩ Ω ⊂ Σ then there is a i0 ∈ {i = 1, . . . , d} such that ni 0 +
1 2
> ϕi0 (x, λ) > ni0 > Mi0 ,
which contradicts ϕi0 (Σ ) ⊂ ([0, Mi0 ] ∪ N) by Property 2.1. Thus, we prove the lemma. Now, we claim that Lx 6= N (x, λ)
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C. Wang, D. Qian / Nonlinear Analysis 71 (2009) 1438–1444
for any x ∈ (frX ×[0,1] B )λ , λ ∈ [0, 1], where Bλ := {x ∈ X : (x, λ) ∈ B }, (frX ×[0,1] B )λ := {x ∈ X : (x, λ) ∈ frX ×[0,1] B }. In fact, if there are λ0 ∈ [0, 1] and x0 ∈ (frX ×[0,1] B )λ0 such that L(x0 ) = N (x0 , λ0 ), then, by Lemma 2.6,
b. (x0 , λ0 ) ∈ Σ ∩ clX ×[0,1] B ⊂ Σ ∩ Ω = Σ Then, x ∈ Bλ and we have a contradiction. Using the homotopy invariance of coincidence degree, we have DL (L − N (·, 1), B1 ) = DL (L − N (·, 0), B0 ) = dB (f0 (·), B0 ∩ Rn , 0) = dB (f0 (·), B(0, r0 ), 0), where r0 is defined in (i1 ) above. By standard computations as in [10], we have for any r ≥ r0 , dB (f0 (·), B(0, r0 ), 0) = dB (f0 (·), B(0, r ), 0) = 1.
b satisfying So, there is at least one solution x(t ) of equation Lx = N (x, 1) in B1 and then in Σ ϕi (x, 1) ≤ ni ,
i = 1, . . . , d.
Thus, we obtain at least a solution to the periodic boundary value problem (2.1)–(2.2). By Property 2.2 again, we have |Πi x|X < e Ri , i = 1, . . . , d. 3. Proof of Theorem 1.1 In this section, we are concerned with the existence of solution to periodic BVP (1.1)–(1.2). The idea of the proof is to pass to the limit from a finite system. Consider the finite system of 2n + 1 equations u00i + gi (ui ) = pi (t , u˜ , u˜ 0 ),
i = −n, . . . , 0, . . . , n.
(3.1) (n)
This system has been studied in Section 2, and we have a periodic solution {xi }i=−n,...,n of period 2π to Eq. (3.1) satisfying |(x2i−1 (t ), x2i (t ))| < e Ri , i = −n, . . . , n, where e Ri are defined in Property 2.2 and only depend on i but do not depend on n by Property 2.2. Now, the bounds e Ri mentioned above can be used as in [5] with suitable modifications to prove the convergence of this sequence to a solution of (1.1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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