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Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded -Laplacian
Advances in Difference Equations volume 2014, Article number: 218 (2014)
In this paper, we consider the multiplicity of periodic solutions for a class of difference systems involving the -Laplacian in the cases when the gradient of the nonlinearity has a sublinear growth. By using the variational method, some existence results are obtained. Our results generalize some recent results in (Mawhin in Discrete Contin. Dyn. Syst. 6:1065-1076, 2013).
1 Introduction and main results
Let ℝ denote the real numbers and ℤ the integers. Given in ℤ. Let . Let and N be fixed positive integers.
In this paper, we investigate the multiplicity of periodic solutions for the following nonlinear difference systems:
where and , , satisfy the following condition:
() is a homeomorphism from onto (), such that , , with strictly convex and , .
Remark 1.1 Assumption () is given in , which is used to characterize the classical homeomorphism and the bounded homeomorphism. is called classical when and bounded when . If furthermore is coercive (i.e. as ), there exists such that
where , (see ).
It is well known that the variational method has been an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contributions have been obtained (for example, see [1–13]). However, to the best of our knowledge, few people investigated system (1.1). Recently, in  and , by using the variational approach, Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian:
where , Φ strictly convex, is a homeomorphism of onto the ball or of onto . By using the variational approach, under different conditions, the author found that system (1.3) has at least one or geometrically distinct T-periodic solutions. It is interesting that Mawhin considered three kinds of ϕ: (1) is a classical homeomorphism, for example, for some and all ; (2) () is a bounded homeomorphism, for example, for all ; (3) is a singular homeomorphism, for example, for all .
For a classical and bounded homeomorphism, in , Mawhin obtained the following multiplicity results.
Theorem A (see , Theorem 4.1)
Assume that the following assumptions hold:
(HB) ϕ is a homeomorphism from onto , such that , , with strictly convex and .
(HF) , , and there exist an integer and real numbers such that
for all and .
If there exist and such that
Then, for any such that (the definition of can be seen in ), system (1.3) has at least geometrically distinct T-periodic solutions.
Theorem B (see , Theorem 4.2)
Assume that assumption (HF) and the following condition hold:
(HB)′ ϕ is a homeomorphism from onto (), such that , , with strictly convex and .
If is coercive, such that and , system (1.3) has at least geometrically distinct T-periodic solutions, where is given by (5) in and is such that , .
Obviously, (HF) implies that F is periodic on all variables . Hence, a natural question is that what will occur if F is periodic on some of variables . For differential systems, in  and , the arguments on this question have been given. In , Tang and Wu considered the second order Hamiltonian system
Inspired by [1, 14, 15] and , in this paper, we investigate system (1.1), which is different from (1.3), and consider the case that is periodic on some of the variables and some of the variables , where and . We generalize Theorem A and Theorem B.
Next, in order to present our main results, we consider two decompositions and with
where and are the canonical basis of for , , , and .
In this paper, we make the following assumptions:
() Let , , , and . Assume that there exist positive constants , , , such that
() There exist positive constants , , , with and , , and such that
() There exist constants , , , , , and two nonnegative functions , where , with the properties:
, , ,
, as .
() , is T-periodic in t for all and continuously differentiable in for every , where , .
() is -periodic in , where is a component of vector and , , and -periodic in , where is a component of vector and , .
() There exist , , such that
for all and .(ℰ)
Remark 1.2 A condition similar to () and () was given first in  for the second order Hamiltonian systems
The condition presented some advantages over the following subquadratic condition: there exist and such that
We refer readers to  for more details.
Moreover, assume that and satisfying and . Let
Next, we present our main results.
(I) For classical homeomorphism
Theorem 1.1 Assume that () with , (), (), ()-(), and (ℰ) hold. Assume that F satisfies the following condition:
() For ,
Then system (1.1) has at least geometrically distinct solutions in ℋ, where the definition of ℋ is given in Section 2 below.
Theorem 1.2 Assume that () with , (), (), (), ()-(), and (ℰ) hold. Assume that F satisfies the following condition:
()′ Let and . Assume that there exist positive constants , , , such that
()′ For ,
Then system (1.1) has at least geometrically distinct solutions in ℋ.
(II) For bounded homeomorphism
Theorem 1.3 Assume that () with , are coercive, , (), (), and (ℰ) hold. Assume that F satisfies the following conditions:
() There exists a nonnegative , , such that
for all and ;
() For ,
where , are given in (1.2). Then system (1.1) has at least geometrically distinct solutions in ℋ.
Theorem 1.4 Assume that () with , are coercive, , (), (), (), and (ℰ) hold. If F satisfies the following conditions:
()′ For ,
where , are given in (1.2), then system (1.1) has at least geometrically distinct solutions in ℋ.
First, we present some basic notations. We use to denote the usual Euclidean norm in . Define
ℋ is defined as a subspace of by
Then . For , set
Obviously, we have
For , on , we define
and, on , we define
For , we define
Then ℋ can be decomposed into the direct sum . So, for any , u can be expressed in the form , where and . Obviously, , .
For , let
where , . It is easy to verify that
is also a norm on . Since is finite-dimensional, the norm is equivalent to the norm in ℋ if .
Lemma 2.1 (see )
Let . Then
where , , , and are defined by (1.7)-(1.10).
Lemma 2.2 (see )
Let , , .
If , then ;
if , then there exists such that
Lemma 2.3 For any , the following two equalities hold:
Proof In fact, since and for all , we have
Hence, (2.6) holds. Similarly, it is easy to obtain (2.7). The proof is complete. □
Lemma 2.4 Let , and assume that L is continuously differential in for all . Then the function defined by
is continuously differentiable on ℋ and
Proof Define , by
Since L is continuously differential in for all , is differential in λ and
The proof is complete. □
It follows from (), (), and Lemma 2.4 that
By Lemma 2.3, it is easy to see that the critical points of φ in ℋ are periodic solutions of system (1.1).
Next, we recall a definition. Let G be a discrete subgroup of a Banach space X and let be the canonical surjection. A subset A of X is G-invariant if . A function f defined on X is G-invariant if for every and every (see ).
Definition 2.1 (see , Definition 4.2)
A G-invariant differentiable functional satisfies the (PS) G condition, if for every sequence in X such that is bounded and , the sequence contains a convergent subsequence.
We will use the following two lemmas to obtain the critical points of φ.
Lemma 2.5 (see , Theorem 4.12)
Let be a G-invariant functional satisfying the (PS) G condition. If φ is bounded from below and if the dimension N of the space generated by G is finite, then φ has at least critical orbits.
Lemma 2.6 (see )
Let X be a Banach space and have a decomposition: where Y and Z are two subspaces of X with . Let V be a finite-dimensional, compact -manifold without boundary. Let be a -function and satisfy the (PS) condition. Suppose that f satisfies
where , , R, a, and b are constants. Then the function f has at least critical points.
and , are the unique integers such that
Let , , , and V is the quotient space which is isomorphic to the torus . Now define by
It is easy to verify that Ψ is continuously differentiable and that
Then () implies that
Hence, we have
and, by (ℰ), we have
Hence and .
For the sake of convenience, we denote by and , , below the various positive constants, by and , , below the various positive constants depending on ε and
Proof of Theorem 1.1 It follows from () that there exist and such that
for . It follows from (), (), Lemma 2.1, and Lemma 2.2 that
By (), (3.2), and Lemma 2.1, we have
It follows from (3.1), (3.3), , and that φ is bounded from below. Let G be a discrete subgroup of ℋ defined by (2.10) and let be the canonical surjection. By (2.14)-(2.17), it is easy to verify that φ is G-invariant. In what follows, we show that the functional φ satisfies the (PS) G condition, that is, for every sequence in ℋ such that is bounded and , the sequence has a convergent subsequence. In fact, the boundedness of , (3.1), (3.3), and the facts that and imply that and are bounded. Furthermore, by Lemma 2.1, we know that is also bounded. Hence is bounded in ℋ. Since , we know that has a convergent subsequence. Since , also has a convergent subsequence. Thus, by Lemma 2.5, we know that φ has critical orbits. Hence, system (1.1) has at least geometrically distinct solutions in ℋ. The proof is complete. □
Proof of Theorem 1.2 First, we prove that Ψ defined by (2.11) satisfies the (PS) condition. Assume that is (PS) sequence for Ψ, that is, is bounded and , where , , for . Let
Then it is easy to see that
By (2.12) and (2.13), we find that is bounded and . Then there exists a positive constant such that
By ()′, there exist and such that
It follows from (), Lemma 2.1, and Young’s inequality that, for all ,