Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 205

1 −

c μ

n

n n

∑

| x | = | ( a + αβ + )() f x + ∧ T u + ∨ H u + I |

i ij ij ij j ij j ij j i

j = 1 j = 1

j = 1

i

1 {(|

n n

∑ ∑

≤ a | + | αβ | + | |) p h + (| a | + | αβ | + | |) q

ij ij ij j j ij ij ij j

c

j = 1 j = 1

i

n n

+∧ | Tu Hu I || |

+

+∨ | || |

+

+ | |}

ij j ij j i

j = 1 j = 1

1

n

1

n

∑ ∑

+

≤+ ( ω { | a | ++ | α | | β |)| p h | }( ++ ω { | a | ++ | α | | β |) q

ij ij ij j j ij ij ij j

c c

j = 1 j = 1

i i

n n

+ +

+∧ | Tu Hu I || | +∨ | || | + | |}

ij j ij j i

j = 1 j = 1

n

∑

= kh F +

(10)

ij j j

j = 1

∩ .
It follows from the property of invariance under homotopy that

Φ≠ (, ) 0,(, ) ( u μ u μ ∈Ω KerL ) [0,1] ×

Therefore

deg{ JQN , Ω ∩ KerL , 0} = deg{ Φ⋅ ( , 0), Ω ∩ KerL , 0) = deg{ Φ⋅ ( ,1), Ω ∩ KerL , 0)

= deg{ diag ( − c , − c , , − c )} ≠ 0

1 2 n

Ω

Thus, we have shown that satisfies all the assumptions o f Lemma 3. Hence,

Lu Nu = ω

-periodic solution on DomL

∩ . This completes the Ω

has at least one

proof.

3 Conclusion

In this paper, we have studied the existence of the periodic solution for fuzzy cellular
neural networks with time-varying delays. Some sufficient conditions set up here are
easily verified. The obtained criteria can be applied to design periodic oscillatory
fuzzy cellular neural networks.

Acknowledgement

This work is partially supported by the Doctoral Foundation of Guizhou College of
Finance and Economics (2010), the Scientific Research Foundation of Guizhou
Provincial Scientific and Technological Department, and the Scientific Research
Foundation of Hunan Provincial Education Department (10B023).