New result on exponential stability for neutral stochastic l(4)

H.Chen,L.Wang/AppliedMathematicsandComputation239(2014)320–325323

#"#ZtnnnnXXXXci

xiðtÞ¼xið0ÞÀdijxjðÀhð0ÞÞeÀcitþdijxjðtÀhðtÞÞþeÀciðtÀsÞaijÀdijxjðsÀhðsÞÞds

nj¼10j¼1j¼1j¼1

ZtnX

ÀciðtÀsÞ

þehijxjðsÞdBðsÞ

j¼1

"

¼I1iþI2iþI3iþI4i:

Byusingthefundamentalinequalities:ðaþbÞ6m1Àpa2þð1ÀmÞ1Àpbðm2ð0;1ÞÞandðaþbþcÞ63ða2þbþc2Þfor

anyrealnumbersa;bandc.Andtakingmathematicalexpectation,weobtainforalltP0,

2

2

2

2

EfjI2ij2g3½EfjI1ij2gþEfjI3ij2gþEfjI4ij2g

EfjxiðtÞjg6;þ

jdiij1Àjdiij

2

ð9Þ

where

8 29 !()

=nn< XX 2

EfjI1ij2g6E xið0ÞÀdijxjðÀhð0ÞÞ eÀ2cit621þndijEsupjwiðhÞj2eÀcit;

;: h2½Àh;0 j¼1j¼1EfjI2ij2g6n

nX2

dijsupEfjxjðtþhÞj2gj¼1

Àh6h60

ð10Þ

ð11Þ

and

EfjI3ij2g6E

8 <Z:Z

t0

t

!29 nn=XXcicici

eÀðtÀsÞeÀtÀsÞ aijÀdij jxjðsÀhðsÞÞjds

;nj¼1 j¼1

1

6ci

"#2nnXXci

eÀciðtÀsÞnaijÀdijEfjxjðsÀhðsÞÞj2gds

nj¼10j¼1

"#2

Znn1tÀciðtÀsÞXciX

enaijÀdijsupEfjxjðsþhÞj2gds:6

ci0nj¼1h2½Àh;0 j¼1

ð12Þ

FromthepropertyofBrownianmotion,itfollows

dhw;wit¼t;& Z

0t

tP0Z

andforanycontinuousfunctionf:R!R,thefollowingformula

t

EfiðsÞdBðsÞfjðsÞdBðsÞ

'

¼E

&Z

t

'

fiðsÞfjðsÞdhB;Bis;

tP0

holds.Thus,weobtain

8 29Z

=nn< ZttXX 22ÀciðtÀsÞÀciðtÀsÞ

EfjI4ijg¼E ehijxjðsÞdBðsÞ 6ehijEfjxjðsÞj2gds:

;: 00j¼1j¼1

Pn2 ()Pn2

61þnj¼1dijnj¼1dij

EfjxiðtÞj2g6supEfjxjðtþhÞj2gEsupjwiðhÞj2eÀcitþ

jdj1Àjdiijiih2½Àh;0 Àh6h60

"#2ZtnnX3ciXÀciðtÀsÞ

þenaijÀdijsupEfjxjðsþhÞj2gdsð1ÀjdiijÞci0nj¼1h2½Àh;0 j¼1

ZtnX32

eÀciðtÀsÞhijsupEfjxjðsþhÞj2gdsþ

ð1ÀjdiijÞ0h2½Àh;0 j¼1

ð13Þ

Substituting(10)–(13)into(9),itfollows

forallt2½0;þ1Þandi¼1;2;...;n.AnditiseasilyseenthatthereexistsomepositivenumbersMi>0ði¼1;2;...;nÞsuch

Pn2

6ð1þndÞ2j¼1ijÀcit

thatEfjxiðtÞjg6Mie,whereMi¼Efsuph2½Àh;0 jwiðhÞj2g,forallt2½Àh;0 andi¼1;2;...;n.1Àjdiij

Therefore,byvirtueofLemma1and(7),wehave

EfjxiðtÞj2g6M0ieÀat;

tP0;

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