Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

and Applied Analysis 3 Remark 1. If f satisfies the following hypotheses, (H1) f is a continuous odd function, f(0) = 0, f(x) = 1 for x ≥ 1,∑m i=1 a i + ∑ l j=1 β j > 0, αf󸀠(0) ≥ 1, and there exists L > 0 such that |f(u) − f(V)| ≤ L|u − V| for u, V ∈ [0, 1], (H2) f󸀠(0)u ≥ f(u) > 0 for u ∈ (0, 1]; in addition, f󸀠󸀠(0) exists and (α + a + β)f(u) > u for u ∈ (0, 1), then it is obvious that (F1)–(F3) hold and 0,K = ∑m i=1 a i +α+ ∑ l j=1 β j , and −K are three equilibria of (5). Remark 2. If a + α + β > 1, then the nonmonotonic output functions (6) and (7) satisfy assumptions (F1)–(F4). Our purpose is to investigate the existence of traveling waves for (5) with nonmonotonic output functions. A traveling wave solution of (5) is a special translation invariant solution of (12) with the form x n (t) = φ(n − ct), i ∈ Z, and t ∈ R for a wave profile φ(ξ), ξ = n − ct ∈ R, with a given wave speed c ∈ R. Letting ξ = n− ct, it follows that φmust be a solution of the following wave profile equation: −cφ 󸀠 (ξ) = −φ (ξ) + m


Introduction
Recently, there have been an increasing activity and interest in the study of nervous systems, namely, in the study of equations modeling neural networks, which are applied to a broad scope of fields such as image and video signal processing, robotic and biological versions, and higher brain functions (see [1][2][3][4][5][6] for more details).The methodology of cellular neural networks (for short, CNN) was first proposed by Chua and Yang [7][8][9] as an achievable alternative to fully connected neural networks in electric circuit systems.The dynamic evolution of the network is governed by the assumed dynamics of the individual processing units and their reciprocal interactions.The infinite system of ordinary differential equations for CNN distributed in a one-dimensional integer lattice can be described by where  is the piecewise-linear output function given by The quantity  is called a threshold or bias term and is related to independent voltage sources in electric circuits.The real constant coefficients   , , and   of the output function  constitute the so-called space-invariant template that measures the synaptic weights of self-feedback and neighborhood interaction.In recent years, when the output function  is defined as (2), some incisive mathematical analyses have subsequently been done; see [1][2][3][7][8][9] and the references cited therein.
However, to be more realistic, the neural models should be incorporated into time delays, since the transmission of information from one neuron to another is not instantaneous.In models of electronic neural networks, the dynamics of each given cell depends on itself and its nearest left or right neighbors where delays exist in left or right neighborhood interactions due to, for example, finite switching speed and finite velocity of signal transmission (see [7]).Such for  ∈ Z, ,  ∈ N.Here   ( = 1, . . ., ), , and   ( = 1, . . ., ) are nonnegative constants;  may be a positive constant or infinity; and the output function () is nondecreasing.Here the dynamics of each given cell depends on itself and its nearest  left or  right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission.Moreover, we point out that the CNN models with discrete time delays or no delays case can be included in this model by choosing suitable kernel functions (see [26]).
Note that some known results on the existence of traveling waves for (5) can be obtained only when the output functions are nondecreasing.But the output functions may not be nondecreasing.For example, let us consider the following two nonmonotonic output functions: It is obvious that the previous results in [2,3,10,12,27] cannot be applied for CNN or DCNN models with nonmonotonic output functions like ( 6) or (7).Therefore, it gives us the motivation to consider the existence of traveling wave solutions for DCNN model (5) with nonmonotonic output functions.Throughout this paper, we assume that there exists a  > 0 such that  and  satisfy the following assumptions.
Our purpose is to investigate the existence of traveling waves for (5) with nonmonotonic output functions.A traveling wave solution of ( 5) is a special translation invariant solution of (12) with the form   () = ( − ),  ∈ Z, and  ∈ R for a wave profile (),  =  −  ∈ R, with a given wave speed  ∈ R. Letting  =  − , it follows that  must be a solution of the following wave profile equation: Under assumptions (F1)-(F2), it is easily seen that 0 and ± are three equilibria of (13).We are mainly interested in the existence of traveling waves satisfying one of the following asymptotic boundary conditions: Since the output functions are nonmonotonic, our main idea is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions.Then we can apply the results in [26] and Schauder's fixed point theorem to derive the existence of traveling wave solutions.Now we state our main results.Theorem 3. Assume (F1)-(F3) and (J) hold.If  > 0 and ( + )  (0) > 1, then there exists  * 1 < 0 such that the following statements hold.
Remark 7. In the above theorems, we obtain that the traveling waves either converge to the nontrivial equilibrium ( or −) or oscillate on the nontrivial equilibrium at infinity.Furthermore, we give a sufficient condition for the convergence of traveling waves to the nontrivial equilibrium.
But under what conditions the wave will oscillate on the nontrivial equilibrium at infinity is an interesting problem.
On the other hand, similarly to the proofs of nonexistence of traveling waves in [26], we can also obtain the same conclusions for nonmonotonic DCNN models.
Remark 8.The different results can be obtained through the choice of the signs for  and  and ( 5) reduces to kinds of CNN model with mixed delays by choosing suitable kernel functions.Please refer to [26].
The remainder of this paper is organized as follows.In Section 2, we recall some properties of the characteristic function of (13).Then, in Section 3, we construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions.Based on the construction of two nondecreasing functions, we devote Section 4 to proofs of the existence of traveling waves for (5).Our approach is to squeeze the nonmonotonic output functions and Schauder's fixed point theorem in a suitable Banach space.Furthermore, according to the construction of a wave profiles set, we can obtain the exponential asymptotic behavior of traveling waves in the infinity.

Properties of the Characteristic Function
In this section, we recall some properties of the characteristic function for (13).The characteristic function arises from the linearized equation of ( 13) at the equilibrium solution 0 and is given by The main properties of the characteristic function are stated in the following lemmas, and all the proofs can be found in our recent work [26].Lemma 9. Assume that  > 0 and ( + )  (0) > 1.Then there exists a unique  * 1 < 0 such that (1) if  ≤  * 1 , then there exist two positive numbers  1 () and λ1 () with (3) if  >  * 1 , then Δ(, ) < 0 for all  ≥ 0.
According to the results of the above lemmas, in Section 4, we can see that the roots of characteristic equation play crucial roles in studying the behavior solutions of (13) near the equilibrium 0.

Construction of Nondecreasing Functions
In this section, we will construct two nondecreasing functions such that  lies between the two functions.Based on the construction of the nondecreasing functions, we can apply the results in [26] to derive the existence results.

Existence of Traveling Wave Solutions
Now, applying the techniques developed in [26] to the functions  − () and  + (), we prove the main theorems in the sequel.

Traveling Waves with (BC1) and (BC3)
. First, we consider the existence of traveling wave solutions satisfying condition (BC1).Once there exist traveling waves with (BC1), then we can easily obtain that (5) admits traveling wave solutions satisfying condition (BC3).Let us define the operator  : (R, [0, ]) → (R, [0, ]) by where  > −1/ > 0 is a constant and According to (F1) and  < 0,  is well defined.It is easy to see that a fixed point  of  or a solution of the equation is a traveling wave solution of (5).Let  ± be defined as in (30) with  replaced by  ± ; that is, where According to Lemma 13, it is easily seen that  ± and  ± are nondecreasing on (R, [0, ]) and Now we define the functions  + () and  − () by Since  * − ≤ , it is obvious that  − () ≤  + ().Similar to the proof of Lemma 3.1 in [26], it is easily seen that the following lemma holds.
By Lemmas 13-15 and (33), we easily verify that For a given number  > 0, let It is easy to see that (  , ‖ ⋅ ‖   ) is a Banach space.Then we define the following set: where  ∈ (0,  1 ).By (39) and the definitions of  − ,  + , we easily check that Γ is a nonempty closed convex subset of   .Thus, we have the following assertion.
Proof of Theorem 3. (1) By Lemma 3.4, Schauder's fixed point theorem implies that there exists  ∈ Γ such that  = ().Since we can easily obtain that, for  <  * 1 , lim Moreover, we have Next, we prove that In fact, if  1 = According to the above argument and taking the limit as  → +∞, we obtain Since  is arbitrary, it follows that Similarly, we can obtain If  1 <  2 ≤ , then (61) and (F2) imply that which leads to a contradiction.On the other hand, if  ≤  1 <  2 , then (60) and (F2) also imply that which also gives a contradiction.Hence we conclude that  1 <  <  2 .If, in addition, (F4) holds, then we claim that lim  → +∞ () = .Suppose the claim is false; that is, Then we consider the following three cases.Similar to the discussion of Case 1, the case cannot happen.
( (66) It is easy to check that (66) has the same form as ( 13) and hence we can obtain the result of part (ii).The proof is complete.
Now we start the proofs of Theorems 4 and 6.