On variational and topological methods in nonlinear difference equations

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

1. Introduction. Subject. Difference equations are widely used in modeling real life phenomena (cf. [1,23,34,39,43,55,92]). Typically, difference equations display very complex dynamics compared to their continuous counterparts (cf. [76,103,64,113]). A considerable work was done to study various dynamical aspects of difference equations: stability and attractivity (cf. [1,55,77,103]), oscillations (cf. [1,43,90,94]), homoclinic and heteroclinic orbits (cf. [96,111,112,116,119,120,121]) to mention a few. In this paper, we are mainly concerned with the multiple periodic/subharmonic solutions to second order difference equations and discrete Hamiltonian systems. Methods: paradigmatic difficulty. It is well-known for a long time that the existence of solutions to continuous nonlinear equations depends on certain topological characteristics of associated operators rather than on their complicated singularities. In the context relevant to the existence of periodic solutions to periodic non-autonomous systems/solutions to boundary value problems, this idea leads to the Fixed Point Theory based on the usage of Brouwer degree, Leray-Schauder degree, coincidence degree, etc. (cf. [33,38,56,57]). Periodic solutions to autonomous systems appeal to the so-called S 1 -degree with one free parameter (cf. [11,52,58]). Observe, however, that the above invariants do not allow to study multiple solutions. If a system admits an extra variational structure, then the Critical Point Theory based methods can be used to attack the multiplicity problem: (i) Minimax theory utilizing Lusternik-Schnirelman category, Krasnoselskii's genus, index theories (cf. [56,57,75,79]), (ii) Morse theory including Morse inequalities, Conley index, Floer-Witten complex (cf. [24,25,30,75,37]).
Although the research on difference equations has a long history which can be traced back to 2000BC [15], up to the recent years there was a little progress achieved regarding the multiple periodic solutions and solutions to BVPs in such systems. Due to the discrete nature of difference systems, a direct application of the classical methods of Critical Point Theory to this type of systems met a paradigmatic difficulty: essentially, it was not clear how to provide a framework allowing to view a periodic solution/solution to BVPs as a critical point of a smooth functional defined on an appropriate (finite-dimensional) vector space. For the first time, such significant breakthrough was successfully made in [45,46], where the second order difference systems with variational structure/discrete Hamiltonian systems were considered. These pioneering contributions, since 2003, have opened the door for a systematic study of multiple periodic solutions difference systems. Observe, however, that we are not aware of any results on multiple periodic solutions to difference systems without variational structure.
Goal. The goal of this paper is two-fold: (i) to provide a survey of results on multiple periodic solutions to second order difference equations with variational structure and discrete Hamiltonian system initiated in [45,46], and (ii) to suggest a new method to study multiple solutions to difference systems without variational structure based on the usage of the equivariant degree theory (cf. [11]). Several remarks related to (ii) are in order.
Typical applications of the equivariant degree to establish periodic solutions to continuous systems without extra assumption on variational structure and spacial symmetries are related to autonomous systems. For these systems, the S 1 -action is coming from the shift in time giving rise to the S 1 -equivariant degree -the simplest version of the so-called twisted degree with one free parameter related to unknown period (see [11] for details). In the present paper, we consider non-autonomous difference systems assuming the periodicity of the potential with respect to the "discrete time variable". Combining the periodicity with specific properties of the nonlinearity allows one to reformulate the original system as a G-equivariant operator equation F (x) = 0 (with a continuous operator F defined on an appropriate vector space on which the group G := D m × Z 2 naturally acts (here D m stands for the dihedral group)). Applying to F the standard methodology related to the equivariant extension of the Brouwer degree, one obtains the existence of multiple periodic solutions to the original system with a period that is not minimal. One can easily see a strong parallelism between the usage of the S 1 -degree for continuous autonomous systems and equivariant extension of the Brouwer degree to discrete non-autonomous system. Observe also that our method can be easily adapted to systems with more complicated spatial symmetries and nonlinearities than the ones considered in this paper.
However, in this paper, our intention is to provide the basic ideas behind the application of the equivariant degree theory to difference systems rather than to introduce the most complicated setting and/or to formulate the most general results. In fact, the obtained result shows that it is possible to deduce the existence of a large number of different periodic solutions just from the periodicity of the potential.
Overview. After the Introduction, the paper is organized as follows. In Section 2, we outline the variational methods for discrete systems which were introduced in [45,46]. In Section 3, we focus on the progress related to the existence of multiple periodic solutions in second order difference equations, including second order discrete Hamiltonian systems and second order self-adjoint difference equations. Section 4 mainly deals with solutions to boundary value problems to difference equations. In Section 5, we discuss recent results on discrete Hamiltonian systems. The recent research on discrete p-Laplace equation and higher-order difference equations is presented in Section 6. In Section 7, we present our original results on multiple periodic solutions for a system of second order difference equations without an extra assumption on variational structure using the equivariant degree theory. Finally, in Section 8, we put forward a new direction for further investigations.
For the convenience of the reader, we added two appendices: one containing the main results in the critical point theory, and the other one, providing basic information on the equivariant degree theory.

2.
Variational methods for discrete systems: first results. In this section we follow the work done by Guo and Yu in 2003 (cf. [45, 46]).
Let N, Z and R be the set of natural, integers and real numbers respectively. For any two integers a, b, we denote Z(a) = {a, a + 1, · · · , }, and Z(a, b) = {a, a + 1, · · · , b} when a ≤ b.
Consider the following second order difference equation where f : Z × R → R is continuous in the second variable and satisfies f (n + m, z) = f (n, z) for some fixed positive integer m and all (n, z) ∈ Z × R, ∆x n = x n+1 − x n , and ∆ 2 x n = ∆(∆x n ) for each n ∈ Z. Notice that one can define a continuous extension of f by f (t, x) = f (n, x) + (f (n + 1, x) − f (n, x))(t − n) for t ∈ [n, n + 1], so without any loss of generality we can always assume that f ∈ C(R × R, R).
Equation (1) can be viewed as a discretization of the second order differential equation where f ∈ C(R × R, R) is m-periodic with respect to the first variable, i.e., Recall that a subharmonic solution to (1) is a periodic solution with period mp for some positive integer p.
Let S be the vector space of all sequences x = (x n ) n∈Z , namely, For convenience, an element x ∈ S will be written in the form x = (x n ) = (· · · , x −n , x −n+1 , · · · , x −1 , x 0 , x 1 , x 2 , · · · , x n , · · · ). For any given positive integers p and m, we define the set E pm ⊂ S by

ZALMAN BALANOV, CARLOS GARCÍA-AZPEITIA AND WIESLAW KRAWCEWICZ
Clearly E pm is a linear subspace of S isomorphic to R pm , thus it can be equipped with the inner product x, y Epm := pm j=1 x j y j , ∀x, y ∈ E pm and the norm Notice that (E pm , ·, · Epm ) is a finite dimensional Hilbert space isometrically isomorphic to the Euclidean space R pm . Consider the functional J defined on E pm by where F (t, z) = z 0 f (t, s)ds. Then we have Proposition 2.1 ( [45]). For the functional J given by (4), x ∈ E pm is a critical point of J , i.e.,, ∇J (x) = 0, if and only if The following important observation provides a connection between discrete settings and continuous ones, which has solved the paradigmatic difficulty we mentioned in the introduction. Combining Proposition 2.1 with the Linking Theorem (cf. Theorem A.2) leads to the following result: 45]). Suppose that a function f ∈ C(R×R, R) satisfies the following conditions: (f 1 ) There exists a positive integer m such that There exist constants R > 0, β > 2 such that for any |z| ≥ R, Then for any given positive integer p, the equation (1) possesses at least three periodic solutions with period pm. Remark 2.3. Assumptions (f 2 ) and (f 3 ) imply that the function f (t, z) grows superlinearly both at zero and at infinity. In such a case the potential F (t, z) is said to grow superquadratically, both at zero and at infinity. Consider the following system of second order difference equations where x n ∈ R k , F ∈ C 1 (R × R k , R), and ∇ z F (t, z) stands for the gradient of F (t, z) in z. Then the problem of finding periodic solutions to (5) can be reformulated as a problem of finding critical points of a functional J , defined on the appropriately modified space E pm . Assuming that F (t, z) grows subquadratically at infinity and at zero and combining an analog of Proposition 2.1 with the Saddle Point Theorem (see Theorem A.3), one obtains the following results: 46]). Suppose that F ∈ C 1 (R × R k , R) satisfies the following assumptions: (F 1 ) There exists a positive integer m such that F (t + m, z) = F (t, z), for any Then for any given positive integer p, the equation (5) admits at least one pmperiodic solution.
Periodic solutions obtained in the following two Theorems are non-constant periodic solutions.
Remark 2.8. Assumption (F 4 ) in Theorem 2.5 (resp. (F 7 ) in Theorem 2.6) implies the subquadratic growth of F at infinity, while condition (F 8 ) in Theorem 2.6 (resp. (F 10 ) in Theorem 2.7) implies the subquadratic growth of F at zero. Remark 2.9. In the case that F is independent of the first variable t, i.e., (5) is an autonomous difference equation of the form the results similar to those stated in Theorems 2.2-2.7, were obtained in [45,46]. The next difference system where is m-periodic in the first variable t, can be considered as a discrete analogue of the Hamiltonian system For many interesting results on multiple solutions to system (12), we refer to [24,25,75,79]. Observe that for discrete Hamiltonian system (11), there were no comparable results on the existence of periodic solutions before 2003. The first progress in this direction was achieved by Guo and Yu in [47]. Below we outline a surprising difference scheme introduced in [47] to study (11).
Let S be the set of all sequences x = {x(n)} n∈Z , i.e., Elements x ∈ S can be written as where · T denotes the transposition. The set S admits a natural vector space structure For any given positive integers p, m, we define E pm as the following subspace of S E pm := {x = {x(n)} ∈ S : x(n + pm) = x(n), n ∈ Z}.
The inner product ·, · Epm on E pm given by and the associated with it norm x := x, x Epm 1 2 , lead to the isometric isomorphism ϕ : E pm → R 2pmk defined by We define for x ∈ E pm the functional J : E pm → R by where is the standard 2k × 2k symplectic matrix (here Id stands for the identity k × kmatrix). It was showed in [47] that x = {x(n)} is a critical point of J if and only if x is a pm-periodic solution of (11). Therefore, the problem of finding pm-periodic solutions to (11) was reduced in [47] to finding critical points of the functional J in E pm .
Theorem 2.10 ( [47]). Suppose that H ∈ C 1 (R × R k × R k , R) satisfies the following assumptions: (H 1 ) There exists a positive integer m such that There exist some constants R > 0, β > 2 such that for any |z| ≥ R, Then for any given positive integer p, there exist at least pmk geometrically distinct nontrivial periodic solutions of (11) with period pm.
The assumptions (H 2 )-(H 3 ) imply that H(t, z) grows superquadratically at zero and at infinity. Moreover, without the assumption (H 4 ), one still has . Then for any given positive integer p, there exist at least 2 nontrivial periodic solutions to (11) with the period pm.
3. Variational methods for discrete systems: recent developments. Let us discuss recent developments in the investigation of multiple periodic solutions to variational discrete systems. The results in section 2 ( [45]) provide multiple periodic solutions to (5) when k = 1 and ∇ z F (t, z) is superlinear in the second variable z, and when f (t, z) is sublinear in the second variable z respectivly. One can ask if we can improve or generalize these results for the system (5), especially when ∇ z F (t, z) is neither superlinear nor sublinear?
In 2004, Zhou, Yu and Guo ( [122]) proved the existence of multiple periodic solutions to (5) without assuming that the nonlinearity grows superlinearly or sublinearly.
Then the system (5) admits at least three m-periodic solutions.
It follows that F (t, z) grows no slower than quadratic form both at zero and at infinity. Later, Theorem 3.1 was further improved by Xue and Tang ([97]). As for the case that the potential in (5) grows subquatratically at zero or at infinity, Xue and Tang ([98]) generalized the results from [46].
In 2012, Che and Xie ( [26]) gave a result on the existence of infinitely many periodic solutions to (5) when the potential satisfies some subquadratical growth condition and has a suitable oscillating behavior at infinity. In 2014, Yan, Wu and Tang further extended the results from [98] with the potential partially periodic in state variables (cf. [99]). Very recent generalizations can be found in [53] and [93].
In 2015, Gu and An ( [44]) made use of the Variant Fountain Theorem (cf. [13,14]) to study the existence of periodic solutions of (5) in the superquadratic case, where they assume that and W ∈ C 1 (R × R k , R) satisfies the following conditions: (W 1 ) There exist constants d 1 > 0 and α > 1 such that There exist positive constants c and L such that Note that Condition (W 4 ), which was introduced by Tang and Wu ( [89]), is weaker than the classical Ambrosetti-Rabinowitz condition. It was originally introduced as a modified superquadratic condition to study a class of continuous second order Hamiltonian system by local linking theorem. For more results related to periodic solutions to second order difference equations, we refer to [18,102,108,110] and references therein.
In 2005, Yu, Guo and Zou ( [105]) extended the results from [45,46] to the second order self adjoint difference equation being a discretization of the second order differential system of the type which generalizes the so-called Emden-Fowler equation Notice that the steady state equation of the reaction diffusion equation of the form satisfies the second order self adjoint differential equation of the form (17). As for autonomous second order difference equations (10), by using the geometrical Z p -index theory, Guo and Yu ([48]), under the assumption that the nonlinearity f grows superlinearly or asymptotically linearly both at zero and at infinity, were able to obtain the first results on multiplicity of periodic solutions with a prescribed minimal period. We present these results below.
Then for any prime integer p > 2 the equation (10) admits at least p − 1 distinct Z p -orbits of solutions with minimal period p. Theorem 3.3 is a very interesting result; notice that without any assumptions on symmetric properties of f the authors obtained a result on the number of Z p -orbits of periodic solutions to an autonomous scalar difference equation of second order. We also refer the reader to section 7 where non-symmetric systems are also studied by means of symmetric techniques.
In addition we also have: 48]). Suppose that all the assumptions in Theorem 3.3 hold and that Then for any prime integer p > 2, the equation (10) admits at least 2p − 2 distinct Z p -orbits of solutions with period p. 48]). Suppose that there exist two constants α > β such that f ∈ C(R, R) satisfies the following assumptions: (here · stands for the 'smallest integer' function).
Then (10) admits at least k distinct Z p orbits of periodic solutions with period p, where k stands for the cardinality of the set {j ∈ Z(1, p) : β < λ j < α}.
In [48], the authors also gave two examples to illustrate these results, which also show that the estimate on the number of periodic solutions can not be improved. The existence of periodic solutions with prescribed minimal period in second order difference systems is very difficult, nevertheless it is also a problem that attracts a lot of attention. Therefore, the results from [48] can be very useful to researchers interested in the existence and multiplicity of periodic solutions of difference equations with prescribed minimal period.
Let us remark that for the discrete pendulum equation the existence and multiplicity of periodic solutions with prescribed minimal period has been studied by Yu, Long and Guo in [106]. By using a similar method, Tan and Guo obtained some multiplicity results in [87] when f was also resonant at infinity. Readers who are interested in these topics are referred to [68,70] and references therein.
4. Boundary value problems for second order difference equations. Consider the second order self-adjoint difference equation with boundary value condition where a and b are integers, a ≤ b, A and B are given constants, p(n) is nonzero and real valued for each n ∈ Z(a+1, b+2), q(n) is real valued for each n in Z(a+1, b+1). The condition (21) includes the following four types of boundary conditions as special cases, Without loss of generality, assume that a = 0, b = T − 1 for some positive integer T and put c(n) := q(n)−p(n)−p(n+1). Then the boundary value problems (20)-(21) read Define the functional J on R T for u = (u(1), u(2), · · · , u(T )) ∈ R T by where and It was proved by Yu and Guo (cf. [104] T is a solution of the boundary value problem (22). So the existence of solutions to (22) is reduced to the existence of critical points of J . The value of this finding can help us to provide an effective method to study boundary value problems for discrete systems.
Then the boundary value problem (22) admits at least one solution.
Then the boundary value problem (22) admits at least one solution.
In 2007, Faraci and Iannizzotto [35] considered a nonlinear second-order difference equation together with zero end conditions on a finite interval as follows: They established the existence of three solutions for this boundary value problem. The main tool in proofs was a result by B. Ricceri [80], in which the nonconvexity of a superlevel set of a suitable functional is assumed. The paper is concluded with a nice example illustrating the theory presented.
In 2011, Kristály, Mihȃilescu and Rȃdulescu (cf. [61]) studied the existence of solutions to (28) in the case when the nonlinear term f : [0, ∞) → R is independent of n and has an oscillatory behaviour near the origin or at infinity. By a direct application of the variational method, they showed that (28) has a sequence of nonnegative, distinct solutions which converge to 0 (resp. ∞) in the sup-norm whenever f oscillates at the origin (resp. at infinity). The main results in [61] read as follows.
Then system (11) admits at least one nonconstant m-periodic solution.
Observe that there is a parallelism between the assumptions in Theorems 5.1-5.3 and Theorems 2.4-2.6.
In 2007, Yu, Bin and Guo ([101]), using the Morse theory, established the existence of periodic solutions to first order discrete Hamiltonian systems with two types of nonlinearity: asymptotically linear and superlinear at infinity. They assumed that a Hamiltonian H is T -periodic in the first variable for given integer T > 0.
To be more precise, let λ n = 2 sin nπ T , n ∈ [−T + 1, T − 1] and p 1 = T 2 , where · denotes the 'greatest integer' function. Then for any n ∈ Z[1, T ] and z ∈ R k the authors made the following assumptions: (P 1 ) There exists a 2k × 2k symmetric matrix A 0 (n) with T -periodic entries such that where σ(A 0 (n)) is the spectrum of A 0 (n). Assumption (P 1 ) implies that the function ∇ z H grows asymptotically linear at zero, and assumptions (P 1 )-(P 2 ) imply that the trivial solution of system (11)   Theorem 5.4 provides the existence of periodic solution when the gradient of Hamiltonian grows asymptotically linear both at zero and at infinity, and the system (11) is nonresonant both at zero and at infinity.
The same authors ( [100]) in 2006 studied the existence of periodic solutions for discrete convex Hamiltonian systems with forcing terms of the form J∆u(n) + ∇H(n, Lu(n)) = f (n).
The main results were obtained by using the dual least action principle and a perturbation technique.
In 2011, by making use of minimax theory and geometrical index theory, Long ([69]) obtained some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period for discrete Hamiltonian systems. 6. Discrete φ-Laplacian equations and higher order difference equations. Since 2003, the critical point theory has been extensively applied to study the existence and multiplicity of periodic solutions and boundary value problems to difference equations involving a p-Laplacian operator (more generally, ϕ-Laplacian operator) and higher order difference equations (cf. [3,20,21,22,27,32,51,67,72,84,114]. In this section, we only list several results in the literature. In 2007, Chen and Fang ( [27]) studied the nonlinear second order difference equation ∆(ϕ p (∆x n−1 )) + f (n, x n+1 , x n , x n−1 ) = 0, (33) where ϕ p (s) = |s| p−2 s, 1 < p < ∞ and f is m−periodic in the first variable. By using Linking Theorem (Theorem A.2), the authors obtained the following result.
We remark that (33) can be viewed as a discrete analogue of second order functional differential equation which includes the following equation as a special case Equations with similar structure to (35) arise in the study of solitary waves on infinite lattices of particles with nearest neighbor interaction, see for example [83].
We also point out that the condition (f 1 ) in Theorem 6.1 was first introduced by Guo and Xu in [49], where they discussed the existence of periodic solutions to a class of second order neutral differential difference equations. Since then, many variants of the condition (f 1 ) have been frequently used in the study of nonlinear difference equations containing both advances and delays, see for example [28,62,71,85,95,107,117].
The corresponding result about the existence of periodic solutions of a quasilinear difference equation involving a p-Laplacian type operator, namely ∆(ϕ p (∆x(n − 1))) + ∇F x (n, x(n)) = 0 (37) was obtained by He and Chen ([51]). For related results, the readers may refer to [66,67,114] and the references therein. The parameter dependence on the p-Laplacian equation has been considered by Bonanno and Candito ([12]). To be more precise, they studied the following Dirichlet boundary value problem: The authors found that for λ belonging to specific real intervals, problem (38) has at least three distinct positive solutions. To obtain these multiplicity results they impose some growth condition on the function f at 0 and/or at ∞. Different results in this direction have been also given by Candito and D'Agui ( [20]) and Candito and Giovannelli ( [22]). For other results in this direction, we refer to [5,35,109].

Theorem 6.2 ([16]
). Let r, s ∈ R satisfy 0 < r < s. Assume that the following conditions hold: where c 1 and c 2 denote the embedding constants of the solution space (equipped with the Euclidean norm) into R T (equipped with the max-norm). Then, there exists λ * > 0 such that for λ = λ * , system (38) admits at least three solutions.
In [16], it was pointed out that while the first condition on Theorem 6.2 is a standard coercivity assumption, the next two ones are unusual, however such assumptions cannot be, in general, removed or weakened. For more recent results, we refer to [4,21,32].
In 2012, Mawhin ([72]) studied the existence of T -periodic solutions of second order nonlinear difference systems of the type ∆(ϕ(∆u(n − 1))) = ∇ u F [n, u(n)] + h(n), n ∈ Z, in the more general situation where ϕ : Both these two cases are also considered in [72]. The respective motivations are ϕ c (v) = v √ 1+|v| 2 , in which case ∆ϕ c [∆u(n − 1)] may be seen as a discretization of the curvature operator, and ϕ R (v) = v √ 1−|v| 2 , in which case ∆ϕ R [∆u(n − 1)] may be seen as a discretization of the acceleration in special relativity.
The author introduced the action functional whose critical points on a suitable subset of the space of T -periodic sequences should correspond to T -periodic solutions of (39). The author showed that the classical or bounded cases, and the cases of singular homeomorphisms ϕ are essentially different, and gives sufficient conditions for the existence of a minimum. The author shows that the minima of l provide indeed T -periodic solutions of system (39) for singular ϕ. To this aim, an approach is adopted which is based upon elementary auxiliary existence and uniqueness results for some linear systems. The existence of T -periodic solutions for (39), for all classes of ϕ, when the potential F is coercive, is also provided. Next, the author shows that when the nonlinearity satisfies some growth conditions (distinct for the various classes of ϕ), the coercivity condition upon can be weakened to some averaged ones of the Ahmad-Lazer-Paul type. Using Rabinowitz's saddle point theorem, previously obtained results are extended to the case of an averaged anticoercivity condition.
Mawhin further studied T -periodic solutions of second order nonlinear difference systems (39) when F (n, u) is periodic in the u j ( [54,73,74]).
For higher order difference equations, a progress was achieved by Cai, Yu and Guo in 2005 ( [19]) who considered periodic solutions to the following fourth-order difference equation ∆ 2 (r n−2 ∆ 2 x(n − 2)) + f (n, x(n)) = 0. (40) By Linking Theorem (Theorem A.2) they obtained the following result. (A 2 ) There exist R 2 > 0 and β > 2 such that, for every t ∈ Z and every z ∈ R with |z| ≥ R 2 , we have Then, (40) admits at least two nontrivial T -periodic solutions.
In 2010, equation (41) was intensively studied by Zhou,Yu and Chen ([118]). The authors presented a sufficient condition for the nonexistence of nontrivial periodic solutions. They also provided various sufficient conditions for the existence of periodic solutions when f is superlinear, sublinear and asymptotically linear. In particular, they obtained the existence of multiple periodic solutions in the case where f is superlinear, which improved the results of Cai and Yu ([17]).
In recent years, Shi and his collaborators have contributed a lot on higher order difference equations (we refer to [84,86] and references therein for details).
7. Second order difference systems without an extra assumption on variational structure. In this section, we propose a new topological method, based on the usage of the equivariant version of the Brouwer degree (see [11,10] and Appendix B for the related background), allowing one to establish multiple periodic and subharmonic solutions to several classes of second order difference systems without any extra condition on their variational structure. Consider the difference equation with f : Z × R k → R k satisfying the following conditions: for a certain prime number p > 2, f is p-periodic with respect to the first variable, i.e.,, f (n + p, z) = f (n, z) for all n ∈ Z and z ∈ R k ; (f D 2 ) f is even with respect to the first variable, i.e.,, f (−n, z) = f (n, z) for all n ∈ Z and z ∈ R k ; (f D 3 ) f is odd with respect to the second variable, i.e f (n, −z) = −f (n, z) for all n ∈ Z and z ∈ R k .  (44) equip E k pm with the structure of an orthogonal G := D m × Z 2 -representation. By direct verification, one has the following statement.
x is a pm-periodic solution to (42) if and only if x satisfies the equation With Proposition 7.1 in hands, below we outline the framework allowing one to use the G-equivariant Brouwer degree theory to study multiple periodic solutions to (42). To begin with, observe that the G-equivariance of F implies that F (0) = 0. Suppose that 0 is an isolated solution to (45), i.e.,, there exists ε > 0 such that On the other hand, assume that solutions to (45) admit an priori estimate, i.e.,, there exists a sufficiently large R > 0 such that F (x) = 0 for x ≥ R, meaning that all the non-zero solutions x ∈ E k pm to the equation (45) are located in the set Ω := B R (0) \ B ε (0). Then, combining Proposition 7.1(i) with the additivity property of the G-equivariant degree, one has: The following statement (its proof is an immediate consequence of Proposition 7.1(ii) and the existence property of the G-equivariant Brouwer degree) is crucial for our discussion. (46) is given by where (H j ) ∈ Φ(G; E k pm \ {0}) and n j = 0 for every j = 1, 2, . . . , r (cf. Appendix B). Then: (i) for every j = 1, 2, . . . , r, there exists a pm-periodic solution x j ∈ Ω to (42) such , then the periodic solution x j has exactly the symmetries (H j ), i.e.,, (G x j ) = (H j ). Proposition 7.2 reduces the studying of multiple periodic solutions to (42) to two problems: (a) computation of the invariant (47), and (b) classification of maximal orbit types in E k pm \ {0}. Problem (a) will be discussed in the next two subsections (spectral properties of the linearization of f at the origin will be linked to G-deg(F , B ε (0)) while the so-called Nagumo-type condition on f combined with the standard homotopy argument will give G-deg(F , B R (0)). At the same time, problem (b) is settled below. To simplify our exposition, we introduce the following condition: Condition (α): the numbers k in (42) and m in (43) are odd and m ≥ 3.
Take the primary decomposition of m: For any j = 1, 2, . . . , l, put Keeping in mind that m is odd and applying the same argument as that in the proof of Theorem 5.2 from [8], one can obtain the following characterization of maximal orbit types in E k pm \ {0}. (ii) Suppose a non-zero x ∈ E k pm is such that H := G x and let π 1 : G = D m ×Z 2 → D m , π 2 : G = D m × Z 2 → Z 2 be the natural projections. It is easy to see that x is p-periodic if and only if (π 1 (H)) ≥ (Z m ). On the other hand, if π 2 (H) = Z 2 , then x is not p-periodic. Keeping in mind the structure of groups D z mj (see Appendix B), all the solutions x with the orbit type (G x ) = (D z mj ), j = 1, 2, . . . , l, provided by Propositions 7.1-7.3, cannot have p as their minimal period. 7.2. G-degree of linear operators associated with (45). Following the scheme outlined in the previous subsection, here we obtain an effective computational formula for the equivariant degree of a linear operator associated to F at infinity. Take a map A : Z × R k → R k which is p-periodic and even in the first variable and, in addition, A n := A(n, ·) : R k → R k is a linear operator for any n ∈ Z. Similarly to the case of equation (42), define the linear operator A : where A n+p = A n and A −n = A n for all n ∈ Z. Clearly, A is G-equivariant. Also, G-deg(A , B 1 (0)) is correctly defined if and only if A is non-singular, i.e.,, the spectrum σ(A ) does not contain zero. To provide this condition, for each n = 1, 2, . . . , p, denote the real eigenvalues of the operator A n by ξ n,1 , ..., ξ n,kn . Then, by direct computation, A is non-singular if and only if ∀ j=0,1,..., pm 2 ∀ n=1,2,...,p ∀ r=1,...,kn µ j,n,r := −2 + 2 cos 2πj pm + ξ n,r = 0.
Assume that condition (51) is satisfied and take the G-isotypical decomposition i (here Z 2 acts by multiplication), while V − 0 is modelled on the one-dimensional G-representation, where D m acts trivially. Denote by σ − (A ) the set of all real negative eigenvalues µ of the operator A . Then, formula (74) from Appendix B reads: Example 7.5. Suppose that condition (α) is satisfied and consider the operator

ZALMAN BALANOV, CARLOS GARCÍA-AZPEITIA AND WIESLAW KRAWCEWICZ
where the basic degrees are given by (f D 5 ) there exists M > 0 such that for all n = 1, ..., p and z ∈ R k , one has: We have the following then x > ε.
Proof. The above statement is clearly sure for λ = 0. By condition (f D 4 ), there exists a constant α > 4 such that Assume for contradiction that there exists a solution 0 = x = (x n ) to (57) such that x ≤ ε, so for some n o ∈ Z, we have 0 < |x no | = max{|x n | : n ∈ Z} ≤ ε.
Then, it follows from (58) that and consequently we get a contradiction.
The following Nagumo-type result is essentially used in what follows.
Proof. Put R := pmM 2 + 1 and assume, for contradiction, that x = (x n ) ∈ E k pm is a solution to (59) for some λ ∈ [0, 1] and x ≥ R. Suppose that |x no | = max{|x n | : n = 1, ..., pm]}, then clearly, |x no | > M , x no , 2 x no−1 ≤ 0 and by assumption (f D 5 ), one has: We are now in a position to formulate the main result of this section.
(ii) Finally, if for an element x ∈ E k pm , one has G x = D mj , then where |X| denotes the number of elements in the set X. Since all (D z mj ), j = 1, ..., l, are maximal, the multiplicity result follows from the existence part.
with periodic boundary conditions u n = u n+m . As usual, we distinguish between two cases: focusing DNLS equation satisfying σ = −1 and defocusing DNLS satisfying σ = 1. As is well-known (and easy to see), equation (63) admits a standing wave of the form u n (t) = e iωt x n , with x n ∈ R, if and only if Equation (64) is equivalent to the following one: To study solubility of (65), one can apply Theorem 7.8. For the defocusing case σ = 1, the assumptions (f D 0 )-(f D 5 ) are satisfied if ω > 4. In the focusing case σ = −1, the linear map F (0) has positive eigenvalues for ω < 0, meaning that (f D 4 ) and (f D 5 ) are satisfied for ω < 0. Therefore, in both cases (σ = 1 and ω > 4 or σ = −1 and ω < 0), Theorem 7.8 provides the existence of a G-orbit of standing waves for (63) with symmetry D mj for each j = 1, ..., l. In particular, in both cases, there are 2(p 1 + ... + p l ) orbits of pm-periodic standing waves for (63). Remark 7.9. In Theorem 7.8 the solutions with m and m are necessarily different if m and m are relative prime, i.e.,, this theorem gives the existence of at least one orbit of 2p j periodic solutions for each prime number m = p j . Therefore, if p is the minimal period of the nonlinear function f (n, x), then there is an infinite number of subharmonic solutions with minimal period pp j for each prime p j . This result applies to the non-homogeneous equation 7.5. Concluding remarks. Multiple periodic/subharmonic solutions to second order difference equations is a subject of great importance. Keeping in mind that the corresponding theory is far away from being complete, we conclude this section by outlining several directions which, as we believe, may lead to a progress in this field.
(a) Impact of symmetries. Theorem 7.8 is dealing with the minimal (spacial) symmetry requirements on the nonlinear map f . However, if one has a network of identical "discrete oscillators" coupled symmetrically, then symmetries of coupling may have a deep impact on symmetries of actual dynamics of the system. Studying this impact using all the arsenal of the equivariant degree theory could be a very interesting problem.
(b) Bifurcation. Parameterized families of discrete analogs of classical continuous systems have been studied by many authors using different techniques. For an elementary and, at the same time, rigorous introduction into this topic and related local and global bifurcations, we refer to [42] (see also references therein). It seems to be interesting to apply the equivariant degree methods to study bifurcation phenomena in discrete systems with/without symmetries.
(c) Stability. Stability of periodic solutions to discrete systems is attracting a great attention for a long time (see, for example, [29], where this problem is discussed for discrete Lotka-Volterra competition system; see also references therein). Stabilization of unstable periodic orbits in continuous systems is a classical control problem. An elegant method of non-invasive control due to Pyragas [78] is based on using a delayed phase variable (see also [36,50] for the equivariant setting). A possible extension of Pyragas control techniques to discrete systems could be very useful.
(d) Applications. Last but not least. Extension of the application field of discrete systems is a problem of formidable complexity and importance.

8.
A new direction. Continuous-time difference equations appear often as natural descriptions of observed evolution phenomena in some fields of the natural sciences, see [81,82]. Our further question is how we study the existence and multiplicity of periodic solutions to difference equations with continuous variable by variational and topological methods. For example, how to study the existence and multiplicity of periodic solutions and homoclinic solutions to the second order difference equations with continuous variable where f ∈ C(R × R, R), and there exists some positive constant τ such that for any (t, z) ∈ R × R, f (t + τ, z) = f (t, z), and the discrete Hamiltonian systems with continuous variable ∆x where Unfortunately, so far there has been no results in this direction.
Appendix A. Some results from critical point theory. For readers' convenience, we firstly introduce some preliminaries in the critical point theory. The following theorems will be used frequently later, which can be found in the monograph [24,79] or [75]. Let H be a real Hilbert space, J ∈ C 1 (H, R), i.e.,, J is a continuously Fréchetdifferentiable functional defined over H. J is said to be satisfying Palais-Smale condition (PS condition for short), if any sequence {x n } ⊂ H for which {J(x n )} is bounded and ∇J(x n ) → 0(n → ∞) possesses a convergent subsequence in H.
Let B r denote the open ball in H of radius r about 0 and let ∂B r denote its boundary.  Let X be a real Banach space. Σ is the set of subsets of X, which is closed and symmetric with respect to 0, i.e., Σ = {A ⊂ X : A is closed and x ∈ A if and only if − x ∈ A}.
For any A ∈ Σ, the genus γ of A is defined by γ(A) = min{k ∈ N : there is an odd map ϕ ∈ C(X, R k \{0})}.
When there does not exist such a finite k, set γ(A) = ∞. Finally set γ(∅) = 0. The genus is also called the Z 2 geometrical index.
Lemma A.4. Let X k be a subspace of X with dimension k, S 1 be the unit sphere of X, then γ(X k ∩ S 1 ) = k.
Then there exist y 0 ∈ J −1 ((−∞, r)) and λ > 0 such that the equation y = λ∇J(y) + y 0 (70) has at least three solutions in H. Then there exist y 0 ∈ J −1 (r) and λ ∈ R such that the equation (70) has at least three solutions in H.
for the trivial irreducible representation(it is well-known that if G is finite, then there are only finitely many non-equivalent irreducible G-reprersentations). Take an orthogonal G-representation V represented it as a direct sum where the G-invariant subspaces V j satisfy the property: every irreducible Gsubrepresentation V of V j is equivalent to V j . Decomposition (72) is unique and is called the isotypical decomposition of V ; each subspace V j is called the isotypical component of V modeled on V j . It is well-known that if A : V → V is a G-equivariant linear operator, then A(V j ) ⊂ V j for all j = 0, 1, . . . , r. Moreover, for a real eigenvalue µ of A, the generalized eigenspace E(µ) is G-invariant. The integer is called V j -isotypical multiplicity of the eigenvalue µ of A. If A : V → V is a Gequivariant non-singular linear operator, denote by σ − (A) the set of all its negative eigenvalues. Using the multiplicativity property (P 5 ), one can easily obtain the following computational formula for G-deg(A, B 1 (0): where deg Vj := G-deg(−Id , B(V j )), B(V j ) stands for the unit ball in V j . The degrees deg Vj are called basic degrees of G.