Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 203

1

n

1

n

∑ ∑

+

| x | ≤ | a | + | αβ | + | |) | p x | + | a | + | αβ | + | |) q

i − ij ij ij j j ij ij ij j

c c

j = 1 j = 1

i i

1 {

n n

+ +

+∧ | Tu Hu I || | +∨ | || | + | |} (5)

ij j ij j i

c

j = 1 j = 1

i

i xt is continuously differentiable for

() i = 1, 2, ; it is certain that , n

Note that each

t

∈ [0, ω ]

such that |()||()|
ii ii

xt xt − =

T

FFFF = (, , ,

there exists . Set

12 n

where

(

n

n n

∑

+ +

F =+ ω ){ | a | + + | α | | β |) q +∧ | T u || | +∨ | H u || | + | I |}

i ij ij ij j ij j ij j i

c

j = 1 j = 1

j = 1

i

− 1 T

ρ ()1 K < ( EKFh hh h − ) = = ( , , , ) ≥ 0

In view of and Lemma 1, we have

12 n

n

∑

h is given by

h = k h Fi + , = , , Set , . n where

i ij j

j = 1

T n

Ω= {( xx x R x hi , , , ) ∈ :| | < , = 1, 2, (6) , } n

12 n i i

t ∈+ [, t t ω ]

Then, for , we have

i i

t t +

∫

+

∫

+

| xt xt D xt dt xt ( )| | ≤ ( )| + | ( )| ≤ | ( )| + D xt dt | ( )|

i i i i i − i

t t

i i

1

n

1

n

∑ ∑

+

≤ (| a | + | αβ | + | |) p | x | + {(| a | + | αβ | + | |) q

ij ij ij j j ij ij ij j

c c

j = 1 j = 1

i i

n n

+ ω

t

+ +

∫

+

+∧ | Tu Hu I Dxt t || | +∨ | || | + | |} + | ( |

ij j ij j i i

j = 1 j = 1 t

i

n n

1 1

∑ ∑ + |

+

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

α |

ij ij ij j j ij ij

c c

j = 1 j = 1

i i

n n

+ +

+ | β |) q + ∧ | T || u Hu | + ∨ | || | + | I |}

ij j ij j ij j i

j = 1 j = 1

n

∑

≤ kh F h + =

ij j i i

j = 1

i hi

, = , , , are independent of , n λ ∀∈ (0,1), x ∈∂Ω Clearly, . Then for

T n

such that Lx

= λ Nx uxx x L R = (, , , ∈ ∂Ω ∩ = ∂Ω ∩ . u

. When

12 n

is a constant vector with |

xhi | == , , , : Note that QNu , n = JQNu

, when

i i

uK rL ∈

. it must be