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This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer lattice

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 [

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 [

Note that some known results on the existence of traveling waves for (

Throughout this paper, we assume that there exists a

There exists

If

If

Our purpose is to investigate the existence of traveling waves for (

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 [

Assume (F1)–(F3) and (J) hold. If

For each

For each

Assume (F1)–(F3) and (J) hold. If

For each

For each

Assume (F1)–(F3) and (J) hold. If

For any

For any

Assume (F1)–(F3) and (J) hold. If

For any

For any

In the above theorems, we obtain that the traveling waves either converge to the nontrivial equilibrium (

The different results can be obtained through the choice of the signs for

The remainder of this paper is organized as follows. In Section

In this section, we recall some properties of the characteristic function for (

Assume that

if

if

if

Assume that

if

if

if

Assume that

for any

for

Assume that

for any

for

According to the results of the above lemmas, in Section

In this section, we will construct two nondecreasing functions such that

Let us define functions

Assume that (F1)–(F3) hold.

There exists

It is obvious that the assertions of

(2) By the definition of

(4) Obviously,

Now, applying the techniques developed in [

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 (

Let us define the operator

Let

Now we define the functions

Assume that (F1)–(F3) hold. Then, for any

Since

Assume that (F1)–(F3) hold. Then, for any

The proof is the same as that of Theorem 1.1 of [

By Lemmas

Assume that (F1)-(F2) hold. Then we have

(1) For

(2) For any

Next, we claim that

Now we start the proofs of Theorems

(1) By Lemma 3.4, Schauder’s fixed point theorem implies that there exists

Now let us consider

According to the above argument and taking the limit as

If, in addition, (F4) holds, then we claim that

If

Similar to the discussion of Case 1, the case cannot happen.

By (

Thus,

(2) Since

Let us assume that

By arguments similar to those of the proof of Theorem

For

First, we consider

Assume that (F1)–(F3) hold. Then, for any

Assume that (F1)–(F3) hold. Then, for any

Now, fixing a number

Assume

Now we start the proofs of Theorems

According to Lemma

Next, we consider the case

In this paper, the motivation of our work is to consider the existence of traveling wave solutions for DCNN model with nonmonotonic output functions. We have developed a technique to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. By adopting Schauder’s fixed point theorem, the existence of traveling wave solutions has been derived.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Zhi-Xian Yu is partially supported by the National Natural Science Foundation of China, Shanghai Leading Academic Discipline Project (no. XTKX2012), Innovation Program of Shanghai Municipal Education Commission (no. 14YZ096), and Hujiang Foundation of China (B14005). Rong Yuan and Ming-Shu Peng are partially supported by the National Natural Science Foundation of China and RFDP. Cheng-Hsiung Hsu is partially supported by the National Science Council and NCTS of Taiwan.