Periodic solutions of second-order nonautonomous dynamical systems
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We study the existence of periodic solutions for second-order nonautonomous dynamical systems. We give four sets of hypotheses which guarantee the existence of solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a new saddle point theorem using linking methods. Copyright © 2006 Martin Schechter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We consider the following problem. One wishes to solve −x (t) = ∇x V t,x(t) ,
(1.1)
where
x(t) = x1 (t),...,xn (t)
(1.2)
is a map from I = [0,T] to Rn such that each component x j (t) is a periodic function in H 1 with period T, and the function V (t,x) = V (t,x1 ,...,xn ) is continuous from Rn+1 to R with ∇x V (t,x) =
∂V ∂V ,..., ∈ C Rn+1 , Rn . ∂x1 ∂xn
(1.3)
Here H 1 represents the Hilbert space of periodic functions in L2 (I) with generalized derivatives in L2 (I). The scalar product is given by (u,v)H 1 = (u ,v ) + (u,v). For each x ∈ Rn , the function V (t,x) is periodic in t with period T. Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 25104, Pages 1–9 DOI 10.1155/BVP/2006/25104
(1.4)
2
Periodic solutions of second-order nonautonomous dynamical systems We will study this problem under the following assumptions: (1) V (t,x) ≥ 0,
t ∈ I, x ∈ Rn ;
(1.5)
(2) there are constants m > 0, α ≤ 6m2 /T 2 such that |x| ≤ m, t ∈ I, x ∈ Rn ;
V (t,x) ≤ α,
(1.6)
(3) there is a constant μ > 2 such that Hμ (t,x) ≤ W(t) ∈ L1 (I), |x| ≥ C, t ∈ I, x ∈ Rn , |x|2 Hμ (t,x) limsup ≤ 0, |x|2 |x|→∞
(1.7) (1.8)
where Hμ (t,x) = μV (t,x) − ∇x V (t,x) · x;
(1.9)
(4) there is a subset e ⊂ I of positive measure such that liminf |x|→∞
V (t,x) > 0, |x|2
t ∈ e.
(1.10)
We have the following theorem. Theorem 1.1. Under the above hypotheses, the system (1.1) has a solution. As a variant of Theorem 1.1, we have the following one. Theorem 1.2. The conclusion in Theorem 1.1 is the same if Hypothesis (2) is replaced by (2 ) there is a constant q > 2 such that
V (t,x) ≤ C |x|q + 1 ,
t ∈ I, x ∈ Rn ,
(1.11)
and there are constants m > 0, α < 2π 2 /T 2 such that V (t,x) ≤ α|x|2 ,
|x| ≤ m, t ∈ I, x ∈ Rn .
(1.12)
We also have the following theorem. Theorem 1.3. The conclusions of Theorems 1.1 and 1.2 hold if Hypothesis (3) is replaced by (3 ) there is a constant μ < 2 such that Hμ (t,x) ≥ −W(t) ∈ L1 (I), |x| ≥ C, t ∈ I, x ∈ Rn , |x|2 Hμ (t,x) liminf ≥ 0. |x|→∞ |x|2
(1.13)
Martin Schechter 3 And we have the following theorem. Theorem 1.4. The conclusion of Theorem 1.1 holds if Hypothesis (1) is replaced by (1 )
0 ≤ V (t,x) ≤ C |x|2 + 1 ,
t ∈ I, x ∈ Rn
(1.14)
and Hypothesis (3) by (3 ) the function given by H(t,x) = 2V (t,x) − ∇x V (t,x) · x
(1.15)
satisfies H(t,x) ≤ W(t) ∈ L1 (I), H(t,x) −→ −∞,
|x| ≥ C, t ∈ I, x ∈ Rn ,
|x| −→ ∞, t ∈ I, x ∈ Rn .
(1.16)
The periodic
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