1988年L.O.Chua and L.Yang 提出 Cellular Neural Networks , 我們用functional differential equations 來描述一維的Delayed Cellular Neural Networks ，cell上的任一點i表示 internal state，它的 dynamic除了和自己有關，也會受到feedback template 的影響 。其中feedback template 之 dynamic 是透過a 、α、β乘上它所對應的 output function 影響其 dynamic。
我們討論general 的 output function ，以及滿足 boundary conditions 的 traveling wave， 利用 properties of characteristic equation 建構 upper and lower solution，並在Banach space上定義一個 operator，利用operator的四個特性和 monotone iteration method，證明小於臨界速度時traveling wave存在滿足boundary conditions 的 non-decreasing solution 。

This paper is concerned with the existence of monotonic traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice Z.
The dynamics of each given cell depends on itself and its neighbor cells with instantaneous feedback.The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in mixed type.
By using the monotone iteration method, we show the existence of non-decreasing traveling solutions when the speed is negative enough.

[1] S.-N. Chow, J. Mallet-Paret, and W. Shen,
Traveling waves in lattice dynamical systems,
J. Diff. Eqns., 149 (1998), pp. 248-291.
[2] L. O. Chua,
CNN: A Paradigm for Complexity,
World Scientific Series on Nonlinear Science, Series A,
Vol. 31, World Scientific, Singapore, 1998.
[3] L. O. Chua and T. Roska,
The CNN paradigm,
IEEE Trans. Circuits Syst., 40 (1993), pp. 147-156.
[4] L. O. Chua and L. Yang,
Cellular neural networks: Theory,
IEEE Trans. Circuits Syst., 35 (1988), pp. 1257-1272.
[5] L. O. Chua and L. Yang,
Cellular neural networks: Applications,
IEEE Trans. Circuits Syst., 35 (1988), pp. 1273-1290.
[6] T. Erneux and G. Nicolis,
Propagation waves in discrete bistable reaction-diffusion
systems, Physica D, 67 (1993), pp. 237-244.
[7] I. Gyori and G. Ladas,
Oscillating Theory of Delay Differential Equations with
Applications, Oxford University Press, Oxford, 1991.
[8] C.-H. Hsu and S.-S. Lin,
Existence and multiplicity of traveling waves in a lattice
dynamical system,
J. Diff. Eqns., 164 (2000), pp. 431-450.
[9] C.-H. Hsu, S.-S. Lin and W. Shen,
Traveling waves in cellular neural networks,
Internat. J. Bifur. and Chaos, 9 (1999), pp. 1307-1319.
[10] C.-H. Hsu and S.-Y. Yang,
On camel-like traveling wave solutions in
cellular neural networks,
preprint, 2002.
[11] H. Hudson and B. Zinner,
Existence of traveling waves for a generalized discrete
Fisher''s equations,
Comm. Appl. Nonlinear Anal., 1 (1994), pp. 23-46.
[12] J. Juang and S.-S. Lin,
Cellular neural networks: mosaic pattern and spatial chaos,
SIAM J. Appl. Math., 60 (2000), pp. 891-915.
[13] J. P. Keener,
Propagation and its failure in coupled systems of
discrete excitable cells,
SIAM J. Appl. Math., 47 (1987), pp. 556-572.
[14] J. Mallet-Paret,
The global structure of traveling waves in spatial
discrete dynamical systems,
J. Dyn. Diff. Eqns., 11 (1999), pp. 49-127.
[15] L. Orzo, Z. Vidnyanszky, J. Hamori and T. Roska,
CNN model of the feature linked synchronized activities
in the visual thalamo-cortical system,
Proc. 1996 Fourth IEEE Int. Workshop on CNN and
Their Applications, pp. 291-296, Seville, Spain,
June 24-26, 1996.
[16] P. Thiran, K. R. Crounse, L.O. Chua, and M. Hasler,
Pattern formation properties of autonomous cellular neural networks,
IEEE Trans. Circuit Syst., 42 (1995), pp. 757-774.
[17] P. Thiran,
Dynamics and Self-Organization of Locally Coupled
Neural Networks,
Presses Polytechniques et Universitaires Romandes, Lausanne,
Switzerland, 1997.
[18] F. Werblin, T. Roska and L.O. Chua,
The analogic cellular neural network as a bionic eye,
Internat. J. Circuit Theory Appl., 23 (1994), pp. 541-569.
[19] P. Weng and J. Wu,
Deformation of traveling waves in delayed cellular neural
networks, preprint, 2001.
[20] J. Wu and X. Zou,
Asymptotical and periodic boundary value problems of mixed FDEs
and wave solutions of lattice differential equations,
J. Diff. Eqns., 135 (1997), pp. 315-357.
[21] B. Zinner,
Existence of traveling wavefront solutions for discrete
Nagumo equation, it J. Diff. Eqns., 96 (1992), pp. 1-27.