Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 202

1

ω

∫

QZ z : ∋ ztdt Z ( ) ∈ , NXx x Z : ∋ Θ⋅ ∈ ( , )

ω

0

Then system (1) can be reduced to operator equation Lx Nx

=

. It is easy to see

ω

∫

KerL R =

n

,Im L = { z Z ∈ : z t dt ( ) = 0}

1

, which is closed in Z . that

ω

0

dim KerL co = dim Im L n = < ∞ PQ are continuous projectors such that

, and

It follows that L is a Fredholm
mapping of index zero. Furthermore, the generalized inverse (to L )
:Im

Im P KerL KerQ = , = Im L = Im( I Q − ).

K L ∩ is given by KerP DomL

p

t ω s

∫ ∫ ∫

(( )() Kz t zsds = () − zvdvds ()

1

p i i ω i

0 0 0

Therefore, ap pl ying the Arzela-Ascoli theorem, one can easily show that N is a L -
compact on

Ω Ω⊂ X

. Since Im QK rL

=

.we
take the isomorphism J of Im Q onto KerL to be the identity mapping. Now we
need only to show that, for an appropriate open bounded subset, applying the
continuation theorem corresponding to the operator equation

with any bounded open subset

Lx = λλ Nx , ∈ ( , ) . Let

'

xt () =Θ ( , ), xti = 1,2, (2) , . n

i i

xxtX =∈ () λ ∈ (0,1)

Assume that is a solution of system (2) for some .

Integrating (2) over the interval [0,

ω ]

, we obtain that

ω ω

∫

'

∫

0 = x t dt ( ) = λ Θ ( x t dt , )

(3)

i i

0 0

Hence

n

n n

ω ω

∑

∫ ∫

ctxtdt () () = { atf xt () ( ()) + ∧ α () ( ( tf xt t Ttut − τ ())) + ∧ () ()

i i ij j j ij j j ij ij j

0 0

j = 1 j = 1

j = 1

n n

+∨ β () tfxt t Htut Itdt ( ( − τ ())) +∨ () () + ()}

ij j j ij ij j i

j = 1 j = 1

Noting assumption (A1), we get

n n

∑ ∑

+

| xc a | ≤ (| | + | αβ | + | |) px a | | + (| | + | αβ | + | |) q

i − i ij ij ij j j ij ij ij j

j = 1 j = 1

n n

+ +

+∧ | Tu Hu I || | +∨ | || | + | |

(4)

ij j ij j i

j = 1 j = 1

It follows that