Existence of Periodic Solution for Fuzzy Cellular Neural Networks with Time-Varying Delays - Advanced Research on Computer Education, Simulation and Modeling - page 201

2 Periodic Oscillatory Solutions

In this section, we will consider the periodic oscillatory solutions of system (1),
we give the following assumption.

τ

ij CR

∈ (,[0, ) ∞ ω

for ,

ij = , , , . n

(A2) are periodic functions with periodic

cCR a THuICRR ∈ ( , (0, ∞ )), , αβ , , , , , ∈ ( , )

(A3) are periodic functions

i ij ij ij ij ij j i

We will use the coincidence degree theory to obtain the existence of an

ω

j f CRRi

∈ (, ,, j = 1,2, ,. n with common periodic and

ω

-

periodic solution to system (1).

Lemma 3. [12] Let L be a Fredholm mapping of index zero and let N be L -compact
on

Ω

. Suppose that

λ ∈ (0,1)

, every solution x of Lx

= λ Nx

is such that x

∉∂Ω

(a) for each .

QNx ≠ 0

for each x

∈∂Ω ∩ KerL

and deg{

JQN , Ω ∩ Ker L , 0} ≠ 0

(b) .

Then the equation Lx Nx

∩ .
To be convenience, in the rest of paper, for a continuous function

=

has at least one solution lying in DomL

Ω

g :[0, ω ] , R

we denote

1

ω

∫

g

+

= max g t ( ) g

−

= min g t ( ) g = g t dt ()

, ,

ω

0

t ∈ [0, ω ] t ∈ [0, ω ]

k =+ ( ωαβ )(| a | + | | + | |) p

1

Theorem 1. Under assumptions (A1); (A2) and (A3), ,

ij c ij ij ij j

Kk × = ()

ij n n

ρ ()1 K < ω

.Suppose that , then system (1) has at least an -periodic

solution.

T n

XZxt xtxt xt CRRxt xt == {()(( ,( ,,( ) = ∈ (, ):( + = ω ) ( }

Proof. Take .

1 2 n

x = max max | x t ( ) | ⋅

and denote . Equipped with the norm , both

1 ≤≤ in t ∈ [0, ω ] i

X and Z are Banach space. For any xt X () ∈

, it is easy to check that

n

n

∑

Θ= − ( , ): xt ctxt atf xt () () + () ( ()) + ∧ α () ( ( tf xt t It − τ ())) + ()

i i i ij j j ij j j ij i

j = 1

j = 1

n n n

+∨ β ()( tft t Ttut Htut Z − τ ( ) +∧ ()() +∨ ()() ∈

ij j ij ij j ij j

j = 1 j = 1 j = 1

Let

1

ω

∫

LDomL x Xx CRR x x Z : =∈ { : ∈ ( ,

n

)} ∋

'

( ) ⋅∈ . PX x : ∋ xtdt X ( ) ∈

ω

0