Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 204

n

n n

∑

( QN u ) =− c x + ( a + αβ + ) f x ( ) +∧ T u +∨ H u + I

i i i ij ij ij j ij j ij j i

j = 1 j = 1

j = 1

We claim that

( QN u ) >= 0, i 1, 2, (7) , n .

i

On the contrary, suppose that there are some i such that (

QN u ) = 0

, namely

i

n

n n

∑

cx = ( a + αβ + ) f x ( ) + ∧ Tu + ∨ Hu I +

i i ij ij ij j ij j ij j i

j = 1 j = 1

j = 1

Then we have

1

n

n n

∑

hx a f x Tu HuI
c

== | | | ( ++ αβ )() +∧ +∨ + |

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 = 1 = 1

j j

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 +

(8)

ij j j

j = 1

Which is a contradiction. Therefore (7) holds and Lu Nu

=

n

QNu ≠∈ Ω 0, u ∩ KerL = Ω ∩ (9) R

ΦΩ :( ∩ KerL ) [0,1] × ∩ Ω KerL

Consider the homotopy

defined by

Φ= (, ) u μμ diag ( − c , − c , , − c u ) + (1 − μ ) QNu

1 2 n

Φ⋅ (,0) = JQN , Φ= (, ) 0 u

Note that

if , then we have