Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator

: We obtain the existence of multiple pairs of periodic solutions for difference equations of type (cid:18) = λ g ( u n )) ( n ∈ Z ) , where g : R → R is a continuous odd function with anticoercive primitive, and λ > 0 is a real parameter. The approach is variational and relies on the critical point theory for convex, lower semicontinuous perturbations of C 1 -functionals.

In recent years, special attention has been paid to the existence and multiplicity of T-periodic solutions for problems with a discrete relativistic operator. Thus, for instance, in [1,2], variational arguments were employed to prove the solvability of systems of difference equations having the form ∆[φ N (∆u(n − 1))] = ∇ u V(n, u(n)) + h(n) (n ∈ Z), (2) under various hypotheses upon V and h (coerciveness, growth restriction, convexity or periodicity conditions); here, φ N is the N-dimensional variant of φ, i.e., φ N (y) = y 1 − |y| 2 (y ∈ R N , |y| < 1).
The existence of at least N + 1 geometrically distinct T-periodic solutions of (2) was proved in [3], under the assumptions that h is T-periodic, ∑ T j=1 h(j) = 0, and the mapping V(n, x) is T-periodic in n and ω i -periodic (ω i > 0) with respect to each x i (i = 1, . . . , N). For the proof, using an idea from the differential case [4], the singular problem (2) was reduced to an equivalent non-singular one to which classical Ljusternik-Schnirelmann category methods can be applied. In addition, under some similar assumptions on V and h, were obtained in [5] using Morse theory, conditions under which system (2) has at least 2 N geometrically distinct T-periodic solutions.
The motivation of the present study mainly comes from paper [6], where for problems involving Fisher-Kolmogorov nonlinearities of type with q > 0 fixed and λ > 0 a real parameter, it was proved that if λ > 8mT for some m ∈ N with 2 ≤ m ≤ T, then problem (3) has at least m distinct pairs of nontrivial solutions. We also refer the interested reader to [6] for a discussion concerning the origin and steps in the study of this type of nonlinearity. In this respect, we shall see in Example 1 below that a sharper result holds true, namely, (ii) If T is even and λ > 8, then (3) has at least T distinct pairs of nontrivial solutions. Moreover, we prove in Theorem 2 that the above statements (i) and (ii) still remain valid for a larger class of periodic problems.
As in [6], our approach to problem (1) is variational and combines a Clark-type abstract result for convex, lower semicontinuous perturbations of C 1 -functionals, based on Krasnoselskii's genus. However, our technique here brings the novelty that it exploits the interference of the geometry of the energy functional with fine spectral properties of the operator −∆ 2 ; recall that It is worth noting that in paper [7] analogous multiplicity results are obtained in the differential case for potential systems involving parametric odd perturbations of the relativistic operator. In addition, we mention the recent paper [8], where the authors obtain the existence and multiplicity of sign-changing solutions for a slightly modified parametric problem of type (1) using bifurcation techniques.
We conclude this introductory part by briefly recalling some topics in the frame of Szulkin's critical point theory [9], which is needed in the sequel. Let (Y, · ) be a real Banach space and I : Y → (−∞, +∞] be a functional having the following structure: where F ∈ C 1 (Y, R) and ψ : Y → (−∞, +∞] is proper, convex and lower semicontinuous. A point u ∈ D(ψ) is said to be a critical point of I if it satisfies the inequality where ε n → 0. The functional I is said to satisfy the (PS) condition if any (PS)-sequence has a convergent subsequence in Y.
Let Σ be the collection of all symmetric subsets of Y \ {0} which are closed in Y. The genus of a nonempty set A ∈ Σ is defined as being the smallest integer k with the property that there exists an odd continuous mapping h : A → R k \ {0}; in this case, we write γ(A) = k. If such an integer does not exist, then γ(A) := +∞. Notice that if A ∈ Σ is homeomorphic to S k−1 (k − 1 dimension unit sphere in the Euclidean space R k ) by an odd homeomorphism, then γ(A) = k ([10], Corollary 5.5). For other properties and more details on the notion of genus, we refer the reader to [10,11]. The following theorem is an immediate consequence of ([9], Theorem 4.3). Theorem 1. Let I be of type (4) with F and ψ even. In addition, suppose that I is bounded from below, satisfies the (PS) condition and I(0) = 0. If there exists a nonempty compact symmetric then the functional I has at least k distinct pairs of nontrivial critical points.

Variational Approach and Preliminaries
To introduce the variational formulation for problem (1), let H T be the space of all T-periodic Z-sequences in R, i.e., of mappings u : Z → R, such that u(n) = u(n + T) for all n ∈ Z. On H T , we consider the following inner product and corresponding norm: , which makes it a Hilbert space. In addition, for each u ∈ H T , we set It is not difficult to check that Now, let the closed convex subset K of H T be defined by where |∆u| ∞ := max i=1,...,T |∆u(i)|. Then, from (5), one has for all u ∈ K. We introduce the even functions It is not difficult to see that Ψ is convex and lower semicontinuos, while G λ is of class C 1 , its derivative being given by Then, the functional I λ : H T → (−∞, +∞] associated to (1) is and it is clear that it has the structure required by Szulkin's critical point theory. A solution of problem (1) is an element u ∈ H T such that |∆u(n)| < 1, for all n ∈ Z, which satisfies the equation in (1). The following result reduces the search of solutions of problem (1) to finding critical points of I λ .
Proposition 1. Any critical point of I λ is a solution of problem (1).
Proof. Let e ∈ H T . By virtue of Lemmas 5 and 6 in [1], the problem has a unique solution u e , which is also the unique solution of the variational inequality ( [6], Proposition 3.1). Next, let w ∈ K be a critical point of I λ . Then, for any v ∈ K, one has which can be written as Hence, w is a solution of the variational inequality with e w ∈ H T being given by e w (n) = −λg(w(n)) − w (n ∈ Z).
Therefore, by (8) and the uniqueness of the solution of (7), we obtain that, in fact, w solves problem (1).
then I λ is bounded from below and satisfies the (PS) condition.
Proof. From (9) we have that −G, hence G λ , are bounded from below on R, respectively on H T . This, together with the fact that Ψ is bounded from below, ensure that the same is true for I λ .
To see that I λ satisfies the (PS) condition, let {u n } ⊂ K be a (PS)-sequence. Assuming by contradiction that {|u n |} is not bounded, we may suppose, going, if necessary, to a subsequence, that |u n | → +∞. Then, by virtue of (6) and (9), we deduce that I λ (u n ) → −∞, contradicting the fact that {I λ (u n )} is convergent. Consequently, {|u n |} is bounded. This, together with |ũ n | ≤ T shows that {u n } is bounded in the finite-dimensional space H T ; hence, it contains a convergent subsequence.

Remark 1.
Notice that until here in this section, no parity assumptions on the continuous function g : R → R must be required.
We end this section by reviewing some spectral properties of the operator −∆ 2 , which is needed in the sequel. A real number λ ∈ R is said to be an eigenvalue of −∆ (n ∈ Z) (10) and in this case, u is called eigensequence corresponding to the eigenvalue λ. On account of the periodicity of u, relation (10) is equivalent to the system If we consider the particular circulant matrix then, having in view (11), the eigenvalues of −∆ 2 are precisely the characteristic roots of M T . In addition, if y = (y 1 , . . . , y T ) ∈ R T \ {0 R T } is an eigenvector corresponding to a characteristic root λ, then its extension u y ∈ H T , defined by u y (i) = y i for i = 1, T, is an eigensequence corresponding to the eigenvalue λ. This means that an orthonormal basis of eigensequences u 1 , . . . , u T can be constructed from an orthonormal basis of eigenvectors x 1 , . . . , x T of M T by extending x i in H T (i = 1, T) as above. From ( [12], p. 38), we know that the characteristic roots of M T , hence the eigenvalues of −∆ 2 , are 4 sin 2 iπ/T (i = 0, T − 1). We can label them according to the parity of T as follows: Todd : Teven : In both cases, we consider an orthonormal basis e 0 , . . . , e T−1 in H T , such that e i is an eigensequence corresponding to λ i (i = 0, T − 1). Observe that, by multiplying equality (10) by arbitrary v ∈ H T and using summation by parts formula, one obtains that if u ∈ H T and λ ∈ R satisfy (10), then

This yields
where δ ik stands for the Kronecker delta function.

Main Result
Our main result is given in the following.
Theorem 2. Assume that g : R → R is a continuous odd function and that G satisfies (9) together with Then, the following hold true: then problem (1) has at least 2m + 1 distinct pairs of nontrivial solutions.
(ii) If T is even and then (1) has at least T distinct pairs of nontrivial solutions.

Proof.
We show (i) in the odd case because the even case follows by exactly the same arguments, and under assumption (15), a quite similar strategy works by simply replacing "2m" with "T − 1". Thus, let 0 ≤ m ≤ (T − 1)/2. On account of Theorem 1 and Propositions 1 and 2, we have to prove that there exists a nonempty compact symmetric subset Since λ > 2λ 2m , we can choose ε ∈ (0, 1), so that λ > 2λ 2m /(1 − ε). Then, by virtue of (13), there exists δ > 0 such that Next, we introduce the set where ρ is a positive number, which is chosen ≤ min Then, it is not difficult to see that the odd mapping H : is a homeomorphism between A m and S 2m ; therefore, γ(A m ) = 2m + 1.
Consider the eigenvalue type problem and set H( then the conclusions (i) and (ii) of Theorem 2 remain valid with (21) instead of (1).
Corollary 2. Assume that f : R → R is a continuous odd function and that Then, the following hold true: then problem (22) has at least 2m + 1 distinct pairs of nontrivial solutions.
(ii) If T is even and then (22) has at least T distinct pairs of nontrivial solutions.

Remark 2.
A multiplicity result for odd perturbations of the discrete p-Laplacian operator is obtained in [13] using a Clark-type result in the frame of the classical critical point theory.