PROPAGATION PHENOMENA FOR CNNS WITH ASYMMETRIC TEMPLATES AND DISTRIBUTED DELAYS

. The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward ( c ∗− ) and rightward ( c ∗ + ) spreading speeds for CNNs by appealing to the theory developed in [26, 27], and c ∗− + c ∗ + > 0. Especially, if cells possess the symmetric templates and the same delayed interactions, then c ∗− = c ∗ + > 0. Moreover, if the eﬀect of the self-feedback interaction αf (cid:48) (0) is not less than 1, then both c ∗− > 0 and c ∗ + > 0. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.


1.
Introduction. Classical digital computation methods have run into a serious speed bottleneck due to the kinds of their nature. In order to deal with this problem, based on some aspects of neurobiology and adapted to integrated circuits (see, e.g., [15,16]), neural networks were well proposed. In the brain, the active medium is provided by a sheet-like array of massively interconnected excitable neurons whose energy comes from the burning of glucose with oxygen. In cellular neural networks, the active medium is provided by the local interconnections of active cells, whose building blocks include active nonlinear devices powered by the batteries.
One class of locally coupled neural networks, called Cellular Neural Networks (CNNs for short), were first introduced in 1988 by L.O.Chua and L.Yang [7,8] as a novel class of information processing systems, which possesses some of the key features of neural networks (NNs) and which has important potential applications in such areas as image processing and pattern recognition (see, e.g., [6,7,8,35]). Cellular neural networks share the best features of both worlds; its continuous time feature allows real-time signal processing found wanting in the digital domain and its local interconnection feature makes it tailor made for VLSI implementation. CNN is simply an analogue dynamic processor array, made of cells, which contain linear capacitors, linear resistors, linear and nonlinear controlled sources. This circuit has been used sometimes to test the circuit robustness as well as for implementing the simplest propagating template. The circuit model of a one-dimensional standard CNN without input terms is Here the node voltage w n at n is is called the state of the cell at n. The quantity z is called a threshold or bias term and is related to independent voltage sources in electric circuits. The output function f (a nonlinearity) is given by For a positive integer r, the r-neighborhood N r (n) of a cell at n is defined as N r (n) = {k ∈ Z : |k − n| ≤ r}.
If r = 1, then N r (n) becomes the nearest and the next-nearest neighbors of n. For each n and k ∈ N r (n), A(n, k) constitutes the so-called cloning template, which measures the coupling weights for the cell at n from the cell at k and specifies the interaction between each cell and all its neighbor cells in terms of their state and output variables. When the template is the space invariant, each cell is described by simple identical cloning template, i.e., A(n, n + k) ≡ A(0, k) := a k (k ∈ N r (0)) or A(n, k) = A(n − k) (k ∈ N r (n)). Furthermore, the cloning template is called symmetric if a k = a −k (k ∈ N r (0)). This symmetry notion represents symmetric coupling weights between cells. Otherwise, the cloning template is called asymmetric. If r = 1, letting a k := A(0, k) (k ∈ N 1 (0)), then these numbers can be arranged in a 1 × 3 matrix form A := [a −1 , a 0 , a 1 ] and (1) can be written by dw n (t) dt = −w n (t) + z + a −1 f (w n−1 ) + a 0 f (w n ) + a 1 f (w n+1 ), n ∈ Z.
If a −1 = a 1 , (3) can be described by the space-invariant symmetric A-template. In modelling living neural networks, a typical construction is so-called interneuron. This means a time-delayed action, sometimes a delayed excitation, sometimes a delayed inhibition. The introduction of ideal delaying template elements in cellular neural networks (1) was motivated by this fact. Later, the synapse delay in general neural networks became widely used (see [9,20,35,36]). Some theoretical results concerning the dynamic range, the steady-state behavior and some impressive and promising application of cellular neural networks with or without delays have been presented in [3,6,7,8,23,32,35,36] and so on.
In CNNs, some experimental studies have revealed the propagation of traveling bursts of activity in slices of excitable neural tissue (see, e.g., [13,14,34] ). The underlying mechanism for propagation of these waves (i.e., travelling waves) is thought to be synaptic in origin rather than diffusive as in the propagation of action potentials. CNNs of synaptically generated waves and with the space-invariant asymmetric A-template have been widely investigated, (see, e.g., [17,18,20,21,29,41,43,44,45,46,47]). The CNNs are a special kinds of Lattice differential equations, whose propagation of these waves has also been well studied in [4,5,19,24,33,48].
Recently, Yu et al. [46] extended the existence results of wave propagations in [17,18,20,21,29,41] to the time-delayed CNNs (DCNNs for short) with the space-invariant asymmetric A-templates and the general nonlinear output function w n (t) = −w n (t) + m i=1 a i τ 0 J i (y)f (w n−i (t − y))dy + α τ 0 J m+1 (y)f (w n (t − y))dy for n ∈ Z, m, l ∈ N. Furthermore, Yu et al. [47] studied propagation of these waves of (4) with the non-monotonic output function. More Recently, Wu and Hsu [43,44] further studied the existence of entire solutions for (4). Letting J i = δ(y − τ i ), i = 1, · · · , m + l + 1, (4) can reduce to the following multiple discrete delays Yu and Mei [45] investigated uniqueness and stability of travelling waves for (5) with the monotone output function. To the best of our knowledge, the existence, uniqueness and stability of travelling waves and the existence of entire solutions has been well studied, but the existence of the spreading speed and its coincidence with the minimal wave speed for DCNNs are still open. The asymptotic speed of spread ( spreading speed for short), as an important notion in biological invasions, was first introduced by Aronson and Weinberger [1,2] for reaction-diffusion equations. Since then, lots of works have shown the coincidence of the spreading speed with the minimal speed for travelling waves under appropriate assumptions for various evolution systems. Weinberger [40] and Lui [31] established the theory of spreading speeds and monostable traveling waves for monotone (order-preserving) operators. This theory has been greatly developed recently in [10,25,26,27,28,42] to monotone semiflows so that it can be applied to various discrete-and continuous-time evolution equations admitting the comparison principle. It is well known that many models with spatial structure are not monotone. The spreading speeds were also well obtained for some nonmonotone continuous-time integral equations, time-delayed reaction diffusion and lattice equations (see [11,12,22,38,39]).
Our main goals are to study the spreading behavior for monotone and nonmonotone DCNNs (4) with the asymmetric templates and the general output functions. More precisely, for the monotone case, we establish the existence of the leftward (c * − ) and rightward (c * + ) spreading speeds for CNNs by appealing to the theory developed in [26,27], and c * − + c * + > 0. Especially, if cells possess the symmetric space-invariant templates and the same delayed interactions, then c * − = c * + > 0 (see Remark 2.7). Moreover, if the effect of the self-feedback interaction αf (0) is not less than 1, c * + > 0 and c * − > 0 (see Remark 2.5). For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. The leftward and rightward spreading speeds are linearly determinate no matter whether CNNs are monotone or non-monotone. We further obtain the nonexistence of travelling wave solutions with the weaker condition than that in [46,47] (see Remark 2.2 (3)) and the spreading speed's coincidence with the minimal wave speed. With the help of the theory of the spreading speed, we can also obtain the existence of travelling waves with the weaker condition (see Remark 2.2 (1)).
The remaining part of the paper is organized as follows. In section 2, we study the existence of spreading speed and its estimate for the monotone DCNNs via the theory in [26,27]. In section 3, we investigate the spreading behavior and the existence of critical waves for the non-monotone DCNNs. Section 4 is devoted to some numerical simulations to illustrate our analytic results. For reader's convenience, we present the abstract results of [26,27] in the Appendix.
2. Spreading speeds: The monotone case. In this section, we investigate the existence of the left and right spreading speeds of DCNN model (4) with a monotone output function and further estimate it. We start with the definition of the left and right spreading speeds. Definition 1. A number c * + is called the rightward spreading speed for a function w n (t) w(t, n) : R + × Z → R + if lim sup t→+∞, n≥ct w n (t) = 0 for any c > c * + and if there exists some > 0 such that lim inf t→+∞, n≤ct w n (t) ≥ for any c < c * + . A number c * − is called the leftward spreading speed for a function w n (t) w(t, n) : R + × Z → R + if lim sup t→+∞, n≤−ct w n (t) = 0 for any c > c * − and if there exists some > 0 such that lim inf t→+∞, n≥−ct w n (t) ≥ for any c < c * − .
Now we recall the existence of solution for (4) with the initial condition, its boundedness and comparison principle for the monotone output (see [43]). Lemma 2.1. Assume that (A) and (F 1)-(F 2) hold. For any w 0 = {w 0 n } n∈Z ∈ C K , where C K is defined in the Appendix, (4) has a unique global solution w(t) = {w n (t, w 0 )} n∈Z through w 0 with 0 ≤ w(t) ≤ K for any t ≥ 0.
2.1. Existence of spreading speeds. In order to obtain the existence of spreading speeds for CNNs with the monotone output f , we appeal to the theory of spreading speeds developed in [26,27]. Now we are concerned with the stability of an equilibrium of the spatially homogeneous system associated with (4) The linearized equation of (6) corresponding to the equilibrium 0 is The stability of the trivial solution of (7) is determined by the characteristic equation obtained by seeking solution of (7) of the form w(t) = e χt . Then χ must be a root of where The stability modulus of L is defined as Therefore, 0 is asymptotically stable if s(L) < 0 and unstable if s(L) > 0.
In order to obtain that the zero solution of (6) is unstable, we need to check that s(L) > 0. Indeed, consider an ordinary differential equation, which can be associated with (6) simply by ignoring delays, i.e.
We easily check that Q :C K →C K admits exactly two fixed points 0 and K. Note that if w is a solution of (6), then w n = w, n ∈ Z is a solution of (4). Thus, Q :C K →C K is monotone. Since 0 is unstable, and Q t admits exactly two fixed points 0 and K, it follows from the monotonicity of Q t that (A5) holds. Now we prove the condition (A2), that is, we need to verify that Q(t, w 0 ) = Q t (w 0 ) is continuous in (t, w 0 ) ∈ R + × C K with respect to the compact open topology. Proof of Claim 1. We need to show that, for given w 0 = {w 0 n } n∈Z ∈ C K and t ≥ 0, and any > 0, there is a δ := δ( , t 0 ) > 0 such that d(Q t (w 0 ), Q t0 (w 0 )) < whenever |t − t 0 | < δ.
with t 0 > 0 with respect to the compact open topology.
According to Steps 1-2, we obtain lim By Claims 1 and 2 and the inequality It is easy to see that Q 0 = I, and Q t1+t2 = Q t1 • Q t2 for all t ≥ 0. Thus Q t is a semiflow on C K . This completes the proof.
Next, we give the strong positivity of the solution with the initial condition.
Similarly, it follows from β 1 > 0 that Repeating these procedures, we obtain w n0±k (t) > 0 for any t > 0 and k ∈ N. Thus, w n (t, w 0 ) > 0 for all t > 0 and n ∈ Z.

ZHIXIAN YU AND XIAO-QIANG ZHAO
Similar to the process in Case 1, we easily verify that w n (t, w 0 ) > 0 for all t > 0 and n ∈ Z. This completes the proof. Note that in the statement of the general theorem on spreading speeds, it is often assumed that the initial data is larger than σ on a ball of radius r σ . Theorems A and C in the Appendix tell us that r σ can be chosen to be independent of the positive real number σ in the case where the monotone map Q either is subhomogeneous or can be approximated from below by a sequence of linear operators. It is easy to see that the solution map Q does not possess subhomogeneous property. Thus, we need to the following proposition, whose proof will be given in the later.
According to Theorems A-C, we have the following results.
Theorem 2.1. Assume that (A) and (F 1)-(F 2) hold. Let w(t) be a solution of (4) with the initial condition w 0 ∈ C K . Then there exist c * + and c * − being the rightward and leftward spreading speeds of Q 1 , respectively, such that the following statements are valid: Proof. The statement (i) follows from Theorem A (i). Note that Theorem A (ii) needs the condition c * + + c * − > 0, which will be stated in the Remark 2.6. According to Proposition 1, r σ in Theorem A (ii) can be chosen to be independent of σ > 0. Let r σ = r. For any c < c * + and c < c * − , if w 0 ∈ C K with w 0 n (θ) > 0 for all θ ∈ [−τ, 0] and n on an interval I with the length 2r, then there exists a positive number σ > 0 such that w 0 n (θ) > σ, for all θ ∈ [−τ, 0] and n ∈ I, and hence, can tell us that w n (t, w 0 ) > 0 for all t > τ and n ∈ Z. Fixing t 0 > 0, it follows that w n (t 0 , w 0 ) > 0 for all n ∈ Z. By taking w nt0 (· + θ, w 0 ) for all θ ∈ [−τ, 0] and n ∈ Z as a new initial data, we can obtain that statement (ii) holds. This completes the proof.
Theorem 2.2. Assume that (A) and (F 1)-(F 2) hold. Then we have the following results.

Remark 2.2. (1) For the existence of travelling waves, Theorem 2.2 (i) and (ii)
(in this case, we need to make a variable change, that is, c is replaced with −c) is consistent with Theorem 1.2 (i) in [46] and Theorem 1.1 (i) in [46], respectively. Moreover, the condition (a + α + β)f (0) > 1 in this paper is weaker than the conditions (α + β)f (0) > 1 in Theorem 1.1 [46] and (a + α)f (0) > 1 in Theorem 1.2 [46]. (2) According to Theorems 2.1 and 2.2, the minimal wave speed is consistent with the spreading speed. Moreover, the spreading speed is linearly determinate. (3) Authors in [46] only obtained the non-existence of travelling wave with the condition of its exponential behavior for any c < c * ± . In this paper, we not only obtain the same results by the theory of spreading speed, but also the nonexistence of travelling wave without its exponential behavior.
is an odd function, we easily obtain the following results. Theorem 2.3. Assume that (A) and (F 1)-(F 2) hold. Let w(t) be a solution of (4) with the initial condition w 0 ∈ C −K . c * + and c * − are given in Theorem 2.1. Then the following statements are valid: Theorem 2.4. Assume that (A) and (F 1)-(F 2) hold. Then we have the following results.
2.2. The estimate of spreading speeds. In this subsection, we estimate the rightward spreading speed c * + and the leftward spreading speed c * − in Theorem 4. In order to estimate the asymptotic speeds of spread, we consider the linearized equation of (4) at the zero solution, Let {M t } t≥0 be the solution semiflow associated with (10). Thus, for each t > 0, the map M t satisfies the assumptions (C1)-(C5). Note that f (0)u ≥ f (u) for u ∈ (0, K]. Thus Q t (w 0 ) is a lower solution of linear system (10) for t ∈ [0, +∞) and it follows that For each φ ∈ C([−τ, 0], R), Let η(t, φ) be the unique solution of the linear delay equation with the initial condition η(θ, φ) = φ(θ), ∀θ ∈ [−τ, 0]. It is easily checked that χ is the solution map at time t of equation (11). It can be checked that B n0 χ is a strongly positive linear operator and compact inC for n 0 > τ . Therefore, (C6) holds for B χ .
Since (11) is a cooperative and irreducible delay equation, we can obtain that its characteristic equation . It is obvious that η(t, ψ) = e λ(χ)t , ∀ t ≥ 0. Then we have Thus, e λ(χ)t is the principle eigenvalue of B t χ with the positive eigenfunction ψ. Letting t = 1, e λ(χ) is the principle eigenvalue of B 1 χ := B χ . Since (C6) holds, it follows from Lemma 3.7 in [26] that λ(χ) is convex on R. Thus, according to the convexity of λ(χ) and λ(0) > 0, and as a direct result of Lemma 3.8 in [26], we obtain the following result.
3. The spreading speeds: Non-monotone case. Throughout this section, we assume that there exists a b > 0 such that f satisfies the following assumptions. (H2) (a + α + β)f (u) > u for u ∈ (0, K) and (a + α + β)f (u) < u for u ∈ (K, b]. Define functions f − (u) and f + (u) by Then f − (u) and f + (u) satisfy the following properties (see [47]). (i) f − (u) and f + (u) are nondecreasing and |f Proof . We omit the proof since it is essentially the same as that of the monotone case (see, Theorem 3.1 [43]). This completes the proof.
We also have the following proposition.
Proposition 3.1. Assume that (A) and (H1)-(H2) hold. For any w 0 , w 0 + ∈ C b and w 0 − ∈ C u * − , with w 0 − ≤ w 0 ≤ w 0 + , let w(t, w 0 ) be the solution of (4) through w 0 and w + (t, w 0 Then v n (t) and Z(t) := sup n∈Z v n (t) are continuous and bounded on [0, +∞). Suppose the assertion does not hold, then there exists t 0 > 0 such that Z(t 0 ) > 0 and where δ is chosen such that It is easy to see that there exists a sequence For any ∈ (0, t 0 ), take L := max By (25), we obtain Thus, there is k such that for all k > k , it holds Thus, we obtain t k ∈ [t 0 − , t 0 ] for all k ≥ k . Thus, it holds lim

From (25), we can obtain
According to Lemma 3.2, we have Thus, it follows from (25) that Taking k → ∞ in (28), we obtain According to (24) and (29), it follows that Z(t 0 ) ≤ 0, which is a contradiction. Therefore, we have w n (t) ≥ w − n (t) for all t ≥ 0 and n ∈ Z. Similar to the above argument, we can also prove w n (t) ≤ w + n (t) for all t ≥ 0 and n ∈ Z. This completes the proof.
3.1. Existence of spreading speeds. In this subsection, we mainly study the existence of spreading speeds by using the results of spreading speeds for the monotone case and squeezing the given output function between two appropriate nondecreasing functions.
In addition, if f (u) u is strictly decreasing for u ∈ [u * − , b] and the property (P) holds, then lim t→∞,−c t≤n≤ct w n (t) = K, where the property (P) means that for Proof . Since f (0) > 0, there exists a positive number δ ∈ (0, u * − ) such that f ± (u) = f (u) for all u ∈ (0, δ) and hence (f ± ) (0) = f (0). By Theorem 2.1, it follows that the spreading speeds are determined by the linearized systems at 0 for the systems and w n (t) Therefore, c * + and c * − are the right and left spreading speed for (30) on C b , respectively, and are also for (31) on C u * − . (i). For given w 0 ∈ C b with compact support, according to Lemma 2.2, it follows that (ii). For any w 0 ∈ C b , we havew 0 ∈ C u * − , wherẽ w 0 (θ) = min{w 0 (θ), u * − } for any θ ∈ [−τ, 0]. According to Lemma 2.2 andw 0 (θ) ≤ w 0 (θ) for θ ∈ [−τ, 0], it follows that For any c < c * + and c < c * − , Theorem 2.1 (ii) implies if w 0 ∈ C b \{0}, then lim t→∞,−c t≤n≤ct w + (t, w 0 ) = b, and ifw 0 ∈ C u * − \{0}, then lim t→∞,−c t≤n≤ct w − (t,w 0 ) = u * − . Therefore, for any c < c * + and c < c * − , if w 0 ∈ C b \{0}, then Next, we prove the upward convergence. Setting we obtain that F ∈ C(R 2 + , R), which is nondecreasing in the first variable and nonincreasing in the second one. Furthermore, F satisfies Now we verify that lim t→∞,n≤ct w n (t) = K for any c < c * + . Letting β ≤ c * + , we define where w n (t) is the solution of (4). We choose sequences (t k ) n∈N , (n k ) k∈N such that t k ≥ 0, n k ≤ βt k for all k ∈ N and t k → +∞ for k → +∞, and We note that (4) has an equivalent form as follows: J m+1 (y)F (w n k (t k + s − y), w n k (t k + s − y))dy ds β j τ 0 J m+1+j (y)F (w n k +j (t k + s − y), w n k +j (t k + s − y))dy ds .
According to (A) and the boundedness of w n , we can apply Fatou's lemma to (34) and obtain that Let β < β < c * . For any given s ∈ R and y ∈ [0, τ ], there exists a sufficiently large number k ∈ N such that n k + l ≤ β (t k + s − y), then it follows that Similarly, we can obtain and Therefore, (36)- (38) and the monotone properties of F imply that Since 0 −∞ e s ds = 1 and τ 0 J i (y)dy = 1, i = 1, 2, · · · , m + l, it follows from (39) that In a similar way, we can obtain Let c < c < c * + . Setting and by the properties of F , we have and According to the definition of F , there exist Hence, by (43)- (45), it follows that which implies that Thus, we have By (48) and the monotonicity of f (u) u on [u * − , b], it is easily seen that u 0 ≤ K ≤ v 0 . On the other hand, by the property (P) and (48), we can obtain u 0 = v 0 . Thus u 0 = K = v 0 = V * (c, c ) = V * (c, c ). It follows from the definitions of V * (c, c ) and V * (c, c ) that W * (c) = K = W * (c).
Similarly, we can prove lim t→∞,n≥−c t w n (t) = K. This completes the proof.
We would like to point out that the functions of logistic type f (u) = pu(1 − u K ) and Ricker type f (u) = pue −qu both satisfy (H1)-(H2) and (P) under the suitable parameters (see [22]).

3.2.
Existence of travelling waves. In this subsection, we mainly study the existence of critical travelling waves for non-monotone CNNs (4), and non-existence of travelling waves with the help of the spreading theory. Moreover, under the condition that f (u) u is strictly decreasing and the condition (P) holds, we can also obtain the travelling wave solution converges to the positive equilibrium K at negative infinity, which are different from the assumption (F4) in [47].
In order to obtain the existence of traveling waves, we need to impose the following assumption (see [46,47]).

ZHIXIAN YU AND XIAO-QIANG ZHAO
Hence, the limit of g 2 (η) exists as η → +∞. According to the similar argument, we can obtain the limits of both also exist as η → +∞. By the above argument, we can obtain lim η→+∞ φ (η) exists and lim η→+∞ φ (η) = 0 according to +∞ 0 φ (θ)dθ < ∞. For c < c * + , let w n (t) = φ(n − ct). Choosing ac < c * + and by Theorem 2.1 (ii), it follows that Hence, lettingc ∈ (c, c * ) and n =ct This implies that which is a contradiction to lim Similarly, we can prove that (ii) also holds. This completes the proof.  The evolution of the solution is shown in Figure 2. Figure 3 gives the evolution of solution for the leftward case. From Figures 2 and 3, we may verify that the rightward wave speed is less than the leftward wave speed. A natural question then arises: under what A-templates constructions, the observation is true.
To observe the special case mentioned in the last section, we let α = 0.35, β 1 = 0, β 2 = 0, a = 0.4, f (x) = 4 3 xe 1−0.5x . Figure 4 suggests the rightward spreading is positive, the leftward should be no more than zero, which we leave for the future investigation.
Appendix. In this Appendix, we first introduce some necessary notations and assumptions and then recall the abstract results developed in [26,27] on spreading speeds and travelling wave solutions for abstract monotone evolution systems. As  mentioned in [27], this theory is a generalization of that in [26] to continuous-time semiflows under a weaker compactness assumption.
Let C be the set of all bounded and continuous functions from [−τ, 0] × Z to R. Clearly, any number in R and any element in the spaceC := C([−τ, 0], R) can be regarded as a function in C. For any r > 0, we set [0, r] := {u ∈ R : 0 ≤ u ≤ r} and C r := {u ∈ C : 0 ≤ u ≤ r}. For any u(θ) = {u(θ, j)} j∈Z {u j (θ)} j∈Z , v(θ) = {v(θ, j)} j∈Z {v j (θ)} j∈Z , we write u(θ) ≥ v(θ) (u(θ) v(θ)) provided u j (θ) ≥ v j (θ) (u j (θ) v j (θ)), ∀ j ∈ Z, θ ∈ [−τ, 0], and u > v provided u ≥ v but u = v. We equip C with the compact open topology, that is, u m → u in C means that the sequence of u m j converges to u j , as m → ∞, uniformly for j in any compact set of Z. Define  Then (C, · ) is a normed space. It follows that the topology in the metric space (C r , · ) is the same as the compact open topology in C r . Moreover, C r is a complete metric space.
Recall that a family of operators {Q t } ∞ t=0 is said to be a semiflow on a metric space (C r , d) provided Q t has the following properties: It is easy to see that the property (iii) holds if Q(·, v) is continuous on [0, ∞) for each v ∈ C r , and Q(t, ·) is uniformly continuous for t in bounded interval in the sense for any v 0 ∈ C r , bounded interval I and > 0, there exists δ = δ(v 0 , I, ) > 0 such that if d(v, v 0 ) < δ, then d(Q t [v], Q t [v 0 ]) < for all t ∈ I. Define the translation operator T y by T y [u](j) = u(j − y) for any given y ∈ Z.
Let Q : C K → C K . In order to state the theory developed in [26,27], we need the following assumptions on Q: