Consider the second order systems
(1.1)
where T > 0 and F : [0, T] × ℝℕ → ℝ satisfies the following assumption:
-
(A)
F (t, x) is measurable in t for every x ∈ ℝℕ and continuously differentiable in x for a.e. t ∈ [0, T], and there exist a ∈ C(ℝ+, ℝ+), b ∈ L1(0, T ; ℝ+) such that
for all x ∈ ℝℕ and a.e. t ∈ [0, T].
The existence of periodic solutions for problem (1.1) has been studied extensively, a lot of existence and multiplicity results have been obtained, we refer the readers to [1–13] and the reference therein. In particular, under the assumptions that the nonlinearity ∇F (t, x) is bounded, that is, there exists p(t) ∈ L1(0, T ; ℝ+) such that
(1.2)
for all x ∈ ℝℕ and a.e. t ∈ [0, T], and that
(1.3)
Mawhin and Willem in [3] have proved that problem (1.1) admitted a periodic solution. After that, when the nonlinearity ∇F (t, x) is sublinear, that is, there exists f(t), g(t) ∈ L1(0, T ; ℝ+) and α ∈ [0, 1) such that
(1.4)
for all x ∈ ℝℕ and a.e. t ∈ [0, T], Tang in [7] have generalized the above results under the hypotheses
(1.5)
Subsequently, Meng and Tang in [13] further improved condition (1.5) with α ∈ (0, 1) by using the following assumptions
(1.6)
(1.7)
Recently, authors in [14] investigated the existence of periodic solutions for the second order nonautonomous Hamiltonian systems with p-Laplacian, here p > 1, it is assumed that the nonlinearity ∇F (t, x) may grow slightly slower than |x|p-1, a typical example with p = 2 is
(1.8)
solutions are found as saddle points to the corresponding action functional. Furthermore, authors in [12] have extended the ideas of [14], replacing in assumptions (1.4) and (1.5) the term |x| with a more general function h(|x|), which generalized the results of [3, 7, 10, 11]. Concretely speaking, it is assumed that there exist f(t), g(t) ∈ L1(0, T; ℝ+) and a nonnegative function h ∈ C([0, +∞), [0, +∞)) such that
for all x ∈ ℝℕ and a.e. t ∈ [0, T], and that
where h be a control function with the properties:
if α = 0, h(t) only need to satisfy conditions (a)-(c), here C*, K1 and K2 are positive constants. Moreover, α ∈ [0, 1) is posed. Under these assumptions, periodic solutions of problem (1.1) are obtained. In addition, if the nonlinearity ∇F (t, x) grows more faster at infinity with the rate like , f(t) satisfies some certain restrictions and α is required in a more wider range, say, α ∈ [0,1], periodic solutions have also been established in [12] by minimax methods.
An interesting question naturally arises: Is it possible to handle both the case such as (1.8) and some cases like (1.4), (1.5), in which only f(t) ∈ L1(0, T ; ℝ+) and α ∈ [0, 1)? In this paper, we will focus on this problem.
We now state our main results.
Theorem 1.1. Suppose that F satisfies assumption (A) and the following conditions:
(S1) There exist constants C ≥ 0, C* > 0 and a positive function h ∈ C(ℝ+, ℝ+) with the properties:
where . Moreover, there exist f ∈ L1(0, T; ℝ+) and g ∈ L1(0, T; ℝ+) such that
for all x ∈ ℝℕ and a.e. t ∈ [0, T];
(S2) There exists a positive function h ∈ C(ℝ+, ℝ+) which satisfies the conditions (i)-(iv) and
Then, problem (1.1) has at least one solution which minimizes the functional φ given by
on the Hilbert space
defined by
with the norm
Theorem 1.2. Suppose that (S1) and assumption (A) hold. Assume that
Then, problem (1.1) has at least one solution in .
Theorem 1.3. Suppose that (S1), (S3) and assumption (A) hold. Assume that there exist δ > 0, ε > 0 and an integer k > 0 such that
(1.9)
for all x ∈ ℝℕ and a.e. t ∈ [0, T], and
(1.10)
for all |x| ≤ δ and a.e. t ∈ [0, T], where . Then, problem(1.1) has at least two distinct solutions in .
Theorem 1.4. Suppose that (S1), (S2) and assumption (A) hold. Assume that there exist δ > 0, ε > 0 and an integer k ≥ 0 such that
(1.11)
for all |x| ≤ δ and a.e. t ∈ [0, T]. Then, problem (1.1) has at least three distinct solutions in .
Remark 1.1.
-
(i)
Let α ∈ [0, 1), in Theorems 1.1-1.4, ∇F(t, x) does not need to be controlled by |x|2αat infinity; in particular, we can not only deal with the case in which ∇F(t, x) grows slightly faster than |x|2αat infinity, such as the example (1.8), but also we can treat the cases like (1.4), (1.5).
-
(ii)
Compared with [12], we remove the restriction on the function f(t) as well as the restriction on the range of α ∈ [0, 1] when we are concerned with the cases like (1.8).
-
(iii)
Here, we point out that introducing the control function h(t) has also been used in [12, 14], however, these control functions are different from ours because of the distinct characters of h(t).
Remark 1.2. From (i) of (S1), we see that, nonincreasing control functions h(t) can be permitted. With respect to the detailed example on this assertion, one can see Example 4.3 of Section 4.
Remark 1.3. There are functions F(t, x) satisfying our theorems and not satisfying the results in [1–14]. For example, consider function
where f(t) ∈ L1(0, T; ℝ+) and f(t) > 0 for a.e. t ∈ [0, T]. It is apparent that
(1.12)
for all x ∈ ℝℕ and t ∈ [0, T]. (1.12) shows that (1.4) does not hold for any α ∈ [0, 1), moreover, note f(t) only belongs to L1(0, T; ℝ+) and no further requirements on the upper bound of are posed, then the approach of [12] cannot be repeated. This example cannot be solved by earlier results, such as [1–13].
On the other hand, take , , C = 0, C* = 1, then by simple computation, one has
and
Hence, (S1) and (S2) are hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in .
What's more, Theorem 1.1 can also deal with some cases which satisfy the conditions (1.4) and (1.5). For instance, consider function
where q(t) ∈ L1(0, T; ℝℕ). It is not difficult to see that
for all x ∈ ℝℕ and a.e. t ∈ [0, T]. Choose , , C = 0, C* = 1, and g(t) = |q(t)|, then (S1) and (S2) hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in . However, we can find that the results of [14] cannot cover this case. More examples are drawn in Section 4.
Our paper is organized as follows. In Section 2, we collect some notations and give a result regrading properties of control function h(t). In Section 3, we are devote to the proofs of main theorems. Finally, we will give some examples to illustrate our results in Section 4.