Cac Dang Toan Co Ban Ve BPT Ham

1

CC DNG TON V BT PHNG TRNH HM C BN

Trnh o Chin

Trng Cao ng S phm Gia Lai

Cc bi ton v gii bt phng trnh hm thng l nhng bi ton kh. Trong nhng nm gn y, cc dng ton loi ny i khi xut hin trong cc thi chn hc sinh gii cc cp v Olympic Ton quc t. Chng hn Bi ton 3, trong IMO 2011 mi y: Gi s :f R R l mt hm gi tr thc xc nh trn tp cc s thc v tha mn

( ) ( ) ( )( )f x y yf x f f x+ + vi mi s thc x v y. Chng minh rng ( ) 0f x = vi mi 0x Bi vit ny cp n phng php gii mt lp cc bt phng trnh hm dng c bn. y l mt trong nhng phng php c th tham kho tm ti li gii cho mt bi ton v bt phng trnh hm. 1. Bt phng trnh hm vi cp bin t do. Xt hm bin s thc f tha mn cc tnh cht sau

( ) ( ) ( )f x y f x f y+ . Ta c th tm c hm f tha mn tnh cht trn nu f tha mn thm mt s iu kin ban u no , chng hn (xem [1])

( ) xf x a , 0a > . gii bi ton trn, trc ht ta cn gii cc bi ton sau Bi ton 1. Xc nh cc hm s ( )f x tha mn ng thi cc iu kin sau: (i) ( ) ( ) ( )f x y f x f y+ + , ,x y R" ; (ii) ( ) 0f x , x R" . Gii. T cc iu kin ca bi ton, thay 0x = ta thu c ( ) ( )0 2 0f f v

( )0 0f . Do ( )0 0f = . Vy nn

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( ) ( )( ) ( ) ( )0 0 0f f x x f x f x= = + - + - . Suy ra ( ) 0f x . Th li, ta thy hm s ( ) 0f x tha mn iu kin bi ra. Bi ton 2. Cho trc a R . Xc nh cc hm s ( )f x tha mn ng thi cc iu kin sau: (i) ( ) ( ) ( )f x y f x f y+ + , ,x y R" ; (ii) ( )f x ax , x R" . Gii. Xt hm s ( )g x ax= . rng

( ) ( ) ( )g x y g x g y+ = + . t ( ) ( ) ( )f x g x h x= + . Khi , ta thu c cc iu kin (i) ( ) ( ) ( )h x y h x h y+ + , ,x y R" ; (ii) ( ) 0h x , x R" . Theo Bi ton 1, ta c ( ) 0h x hay ( )f x ax= . Th li, ta thy hm s

( )f x ax= tha mn iu kin bi ra. By gi, ta tr li bi ton nu ban u. Bi ton 3. Cho trc 0a > . Xc nh cc hm s ( )f x tha mn ng thi cc iu kin sau: (i) ( ) ( ) ( )f x y f x f y+ , ,x y R" ; (ii) ( ) xf x a , x R" . Gii. Nhn xt rng ( ) 0f x > vi mi x R . Vy ta c th logarit ha hai v cc bt ng thc ca iu kin cho (i) ( ) ( ) ( )ln ln lnf x y f x f y+ + , ,x y R" ; (ii) ( ) ( )ln lnf x a x , x R" . t ( ) ( )ln f x xj= , ta thu c (i) ( ) ( ) ( )x y x yj j j+ + , ,x y R" ; (ii) ( ) ( )lnx a xj , x R" . Ta nhn c dng ca Bi ton 2. Vy ( ) ( )lnx a xj = . Suy ra ( ) xf x a= . Th li, ta thy hm s ( ) xf x a= tha mn iu kin bi ra.


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Nhn xt rng, cc bi ton trn vn gii c nu tp xc nh R ca cc hm s trn c thay bi mt khong m U cha 0 sao cho vi mi

,x y U th x y U+ . Mt cu hi t nhin c t ra: Trong Bi ton 3, c th thay hm s

( ) xg x a= bi hm s no bi ton cng c nghim khng tm thng ? Nhn xt rng - Vi 0 1a< < th 1xa x> + , 0x" < v 1xa x + , 0x" ; - Vi 1a th 1xa x> + , 0x" < ; 1xa x + , [ )0;1x" ; 1xa x + , 1x" . T , mt cch t nhin, tip theo ta xt hm s ( ) 1g x x= + . Ta c bi ton sau Bi ton 4. Gi s U l khong m cha 0 sao cho vi mi ,x y U th x y U+ . Xc nh cc hm s :f U R tha mn ng thi cc iu kin sau: (i) ( ) ( ) ( )f x y f x f y+ , ,x y U" ; (ii) ( ) 1f x x + , x U" . Gii. Bi (i), ta c

( ) 2 02 2 2x x xf x f f = +

, x U .

Nu ( )0 0f x = , th

( ) 20 0 000 2 2 2x x xf x f f = = +

.

Do 0 02xf =

. Quy np, ta c 0 0

2

xf n

=

vi mi s nguyn dng n .

Tuy nhin, t (ii) suy ra rng rng ( ) 0f x > vi mi x U v x gn 0. Do iu trn l mu thun. Vy

( ) 0f x > , x U . Tip theo, t (i) v (ii), ta s thy rng f kh vi ti mi im x U v

( ) ( )'f x f x= . Tht vy, t (i) v (ii), vi 0h > nh, ta c

( ) ( ) ( ) ( ) ( )f x h f x f x f h f x+ - - ( )( ) ( ) ( )1f h f x hf x= - . Do

( ) ( ) ( )f x h f x f xh

+ - .


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Mt khc, cng t (i) v (ii), vi 0h > nh, ta c ( ) ( ) ( ) ( ) ( ) ( )1f x f x h h f x h f h h f x h= + - + - - + .

Suy ra ( ) ( ) ( ) ( ) ( )1 1h f x hf x h f x h- + - + .

Do ( ) ( ) ( ) ( )( )1hf x h f x h f x - + - ,

hay ( ) ( ) ( )

1f x h f x f x

h h+ -

-

.

Vy, vi 0h > nh, ta c

( ) ( ) ( ) ( )1

f x h f x f xf x

h h+ -

-

.

Tng t, bt ng thc trn cng ng i vi chiu ngc li, vi 0h < nh. Do , ta c

( ) ( ) ( )' lim0

f x h f xf x

hh

+ -=

tn ti v bng ( )f x , vi x U . T , vi x U , ta c

( ) ( ) ( )0

' 'f x f x f xx xe e

-= =

.

Do ( ) . xf x C e= (C l hng s).

Hn na, t (i) ta c ( ) ( )20 0f f hay ( )0 1f v t (ii) ta c ( )0 1f . Do ( )0 1C f= = . Th li, hm ( ) xf x e= tha mn cc iu kin (i) v (ii). Nh vy, vi ( ) xg x a= hoc ( ) 1g x x= + , Bi ton 3 v Bi ton 4 u gii c. Mt cu hi tip theo c t ra: Vi nhng lp hm ( )g x no th bi ton tng qut l gii c ? Ta c kt qu sau nh l 1. Gi s U l khong m cha 0 sao cho vi mi ,x y U th x y U+ . Nu hm s :f U R tha mn ng thi cc iu kin sau (i) ( ) ( ) ( )f x y f x f y+ , ,x y U" ; (ii) ( ) ( )f x g x , x U" ;


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trong ( )g x l hm s cho trc kh vi ti 0, ( )0 1g = , ( )' 0g k= , th

( ) kxf x e= . Chng minh. Tng t li gii Bi ton 4, t cc iu kin cho, ta suy ra

( ) 0f x > vi mi x U . Gi s rng ( )f x l hm s tha mn cc iu kin ca nh l. Th th, vi 0h > nh, ta c

( ) ( ) ( ) ( ) ( )f x h f x f x f h f x+ - - ( )( ) ( ) ( )( ) ( )1 1f h f x g h f x= - - . Do

( ) ( ) ( ) ( ) ( )0f x h f x g h g f xh h

+ - - .

Mt khc, cng t (i) v (ii), vi 0h > nh, ta c ( ) ( ) ( ) ( )f x f x h h f x h f h= + - + - ( ) ( )f x h g h + - .

V hm ( )g x kh vi ti 0 nn n lin tc ti im . Do , vi 0h > nh, ta c ( ) 0g h- > . Khi , vi 0h > nh, ta c

( ) ( ) ( )( ) ( )1g h

f x h f x f xg h- -

+ - --

( ) ( )( ) ( )

0.

g h gf x

h g h- -

=- -

.

Vy vi 0h > nh, t cc kt qu trn, ta c ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

0 0g h g f x h f x g h gf x f x

h h g h- + - - -

- -


( ) ( ) ( )' lim0

f x h f xf x

hh

+ -=

tn ti v bng ( ) ( ) ( )' 0g f x kf x= , vi x U . T , vi x U , ta c

( ) ( ) ( ) ( ) ( )0

' 'f x f x kf x kf x kf xkx kx kxe e e

- -= = =

.

Do ( ) . kxf x C e= (C l hng s).


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Hn na, t (i) ta c ( ) ( )20 0f f hay ( )0 1f v t (ii) ta c ( )0 1f . Do ( )0 1C f= = . Vy ( ) kxf x e= . Ta c iu phi chng minh. R rng ( ) kxf x e= tha mn iu kin (i). Nu gi thit bi ton c thm iu kin ( ) kxg x e , vi x U , th hm s ( ) kxf x e= tha mn tt c cc iu kin ca bi ton. T kt qu trn, ta c H qu 1. Gi s U l khong m cha 0 v :f U R tha mn iu kin (i) vi mi , x y U sao cho x y U+ . Nu f kh vi ti 0, ( )0 1f = v

( )' 0f k= th ( ) kxf x e= , x U . Chng minh. p dng nh l 1, vi ( ) ( )g x f x= , x U" , ta c iu phi chng minh. H qu 2. Gi s F l hm xc nh trn khong m U cha 0 v tha mn

( ) ( ) ( )F x y F x F y+ + vi mi , x y U sao cho x y U+ . Nu F b chn trn bi mt hm G kh vi ti 0 v tha mn ( )0 1G = , th ( )F x kx= , x U , trong k l mt hng s.

Chng minh. p dng nh l 1, vi ( ) ( )F xf x e-= v ( ) ( )G xg x e-= , ta c c iu phi chng minh. Tng t phng php chng minh nh l 1, ta c kt qu sau y nh l 2. Gi s U l khong m cha 0 sao cho vi mi ,x y U th x y U+ . Nu hm s :f U R tha mn iu kin sau

( ) ( ) ( )f x y f x g y+ , ,x y U" , trong ( )g x l hm s cho trc kh vi ti 0, ( )0 1g = , ( )' 0g k= , th mi nghim ca bt phng trnh hm trn u c dng ( ) kxf x Ce= , C l hng s. H qu 3. Ta c ( ) kxf x e= v ( ) kxg x e= l nghim duy nht ca h bt phng trnh hm


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( ) ( ) ( )( ) ( ) ( )

;

,

f x y f x g y

g x y g x f y

+

+

vi iu kin ( )0 1f = , ( )g x l kh vi ti 0, ( )0 1g = v ( )' 0g k= . Chng minh. T bt phng trnh hm th nht, p dng nh l 2 ta c

( ) kxf x Ce= (C l hng s). V ( )0 1f = , nn 1C = . Do ( ) kxf x e= . Tng t, t bt phng trnh hm th hai, p dng nh l 2 ta cng c

( ) kxg x e= . R rng ( ) kxf x e= v ( ) kxg x e= tha mn h bt phng trnh hm cho, vi nhng iu kin nu. H qu c chng minh. nh ngha 1. Hm ( )g x xc nh trn mt khong m U cha 0 c gi l hm ta bi l ti 0 nu tn ti mt hm ( )k x xc nh trn U sao cho

( ) ( )0 0k g= , ( )' 0k l= tn ti v ( ) ( )k x g x vi mi x U . H qu 4. Bt phng trnh hm

( ) ( ) ( )f x y f x g y+ , trong g l mt hm cho trc xc nh trn I vi ( )0 1g = v l hm ta bi l ti 0, c nghim khng m f khi v ch khi ( )lxe g x trn I v trong trng hp ny mi nghim khng m u c dng ( ) lxf x Ce= , trong

0C l hng s. Chng minh. Gi s ( )f x l mt nghim khng m ca bt phng trnh hm cho. V ( ) ( )g x k x trn U, nn ta c

( ) ( ) ( )f x y f x k y+ , trong ( )k x tha mn ( )' 0k l= v ( ) ( )0 0 1k g= = . p dng nh l 2 vo bt phng trnh hm ny, ta c ( ) lxf x Ce= , trong 0C l hng s. R rng, ( ) lxf x Ce= l mt nghim khng m ca bt phng trnh hm cho nu ( )lxe g x trn U. T H qu 4 , ta c th sng tc ra cc bi ton, chng hn sau y


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Bi ton 5. Tm tt c cc hm s ( )f x , xc nh trn khong m ( ) ; e- , tha mn h bt phng trnh hm sau

( ) ( ) ( )( )

.log ;

.

f x y f x f y

f x x e

+

+

Gii. Trn khong m ( ) ; e- , mi nghim dng f c suy ra bi bt phng trnh hm th hai. Ngoi ra, t h bt phng trnh hm cho, ta c ( )0f e= . p dng H qu 4 i vi trng hp ( ) ( )logg x f x= l hm ta

bi 1e

ti 0, qua hm ( ) ( )logk x x e= + . Do , theo chng minh ca H qu 4

, ta c ( ) ( )1

0 .

x xe ef x f e e

+= = . Th li, ta thy hm s ( )

1 xef x e

+= tha mn

h bt phng trnh hm cho trn khong m ( ) ; e- . Bi ton 6. Trn khong m cha 0 c mt nghim ca h bt phng trnh hm

( ) ( ) ( )

( )

. ;2.

f yf x y f x e

f x x

+

Gii. Gi s ( )f x l mt nghim xc nh trn mt khong m cha 0 no . Th th, bi bt phng trnh hm th hai, ( )f x l khng m. T h bt phng trnh cho suy ra ( )0 0f = . p dng H qu 4 i vi trng hp

( ) ( )f xg x e= l hm ta bi 0 ti 0, qua hm ( )2xk x e= . Hn na, v

( )0 0f = , nn ta c ( ) 0f x tha mn bt phng trnh hm th hai trn khong khng m cha 0. nh l sau y cho ta kt qu v vic gii mt dng bt phng trnh hm c bn khc nh l 3. Gi s U l khong m cha 0 sao cho vi mi ,x y U th x y U+ . Xt bt phng trnh hm

( ) ( ) ( ) ( ) ( )f x y f x g y f y g x+ + , ,x y U" , trong ( )g x l mt hm gii ni, kh vi ti 0, ( )0 1g = v ( )' 0g k= . Th th

( ) 0f x l hm s duy nht tha mn bt phng trnh cho, vi iu kin


9

( )lim 00

f xxx

=

.

Chng minh. Gi s rng ( )f x l nghim ca bt phng trnh cho, vi iu kin

( )lim 0

0

f xxx

=

.

Th th, vi 0h > nh, ta c ( ) ( ) ( ) ( ) ( )f x h f x g h f h g x+ +

hay ( ) ( ) ( )( ) ( ) ( ) ( )1f x h f x g h f x f h g x+ - - + .

Do ( ) ( ) ( ) ( ) ( ) ( ) ( )0f x h f x g h g f hf x g x

h h h+ - -

+ .

Mt khc, ta c ( ) ( ) ( ) ( ) ( ) ( )f x f x h h f x h g h f h g x h= + - + - + - +

hay ( ) ( ) ( )( )g h f x f x h- - + ( ) ( ) ( ) ( ) ( )g h f x f x f h g x h- - + - +

V hm ( )g x kh vi ti 0 nn n lin tc ti im . Do , vi 0h > nh, ta c ( ) 0g h- > . Vy, vi 0h > nh, ta c

( ) ( ) ( )( ) ( ) ( ) ( )( )

1g h f x f h g x hf x h f xh g h

- - + - ++ -

- -.

( ) ( )( ) ( )

( )( ) ( )

0. .

g h g f hf x g x h

h g h h g h- - -

= + +- - - -

.

Vy vi 0h > nh, t cc kt qu trn, ta c ( ) ( ) ( ) ( ) ( ) ( ) ( )0g h g f h f x h f xf x g x

h h h- + -

+

( ) ( )( ) ( )

( )( ) ( )

0. .

g h g f hf x g x h

h g h h g h- - -

+ +- - - -

.


( ) ( ) ( )' lim0

f x h f xf x

hh

+ -=

tn ti v bng ( ) ( ) ( )' 0g f x kf x= , vi x U , v


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( )lim 00

f xxx

=

v ( )g x l mt hm gii ni. T , vi x U , ta c

( ) ( ) ( ) ( ) ( )0

' 'f x f x kf x kf x kf xkx kx kxe e e

- -= = =

.

Do ( ) kxf x Ce= (C l hng s).

Hn na, t iu kin ( )

lim 00

f xxx

=

,

suy ra rng 0C = . Vy ( ) 0f x l hm s duy nht tha mn bt phng trnh cho, vi iu kin

( )lim 0

0

f xxx

=

.

2. Bt phng trnh hm dng cng - nhn tnh. Phn ny cp n vic gii cc h bt phng trnh hm, vi cc dng sau y - Dng cng:

( ) ( )f a x f xa+ + , ( ) ( )f b x f xb+ + , x R ; - Dng cng - nhn:

( ) ( )f a x f xa+ , ( ) ( )f b x f xb+ , x R ; - Dng nhn - cng:

( ) ( )f ax f xa + , ( ) ( )f bx f xb + , x I , I R ; - Dng nhn:

( ) ( )f ax f xa , ( ) ( )f bx f xb , x I , I R ; trong , , , a b a b l cc s thc cho trc. Ch rng, nu ( )f aa = , ( )f bb = , th h bt phng trnh hm dng cng trn l s thu hp ca bt phng trnh hm Cauchy c in

( ) ( ) ( )f x y f x f y+ + , , x y R . Trc ht, ta nhc li rng, mt tp hp M tr mt trong tp s thc R nu nh trong mi ln cn ca mt im ty ca tp R u c t nht mt im ca tp M . Chng hn, tp Q cc s hu t l tp tr mt trong tp R .


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Tnh cht sau y l mt kt qu quen thuc (nh l Kronecker), c th tm thy chng minh cc ti liu l thuyt c bn Nu a v b l cc s thc khng thng c vi nhau, th tp

{ } ; , A ma nb m n Z= + tr mt trong R . Hn na, ta c th chng minh c cc kt qu sau y B 1. Gi s , a b R v 0a b< < l cc s cho trc. K hiu

{ } ; , A ma nb m n N= + .

1) Nu b Qa

, th tp A tr mt trong R .

2) Nu b Qa

, th tn ti 0d > sao cho

{ } ; A kd k Z= . Dng nhn ca b ny nh sau B 2. Gi s , a b R v 0 1a b< < < l cc s cho trc. K hiu

{ } ; , m nM a b m n N= . 1) Nu log

logb Qa

, th tp M tr mt trong ( )0, .

2) Nu loglog

b Qa

, th tn ti 0d > sao cho

{ } ; kM d k Z= . By gi, ta chng minh cc nh l sau y

nh l 4. (Dng cng) Gi s , , , a b a b l cc s thc cho trc tha mn

0a b< < , a ba b

= ,

v gi s rng hm :f R R lin tc ti t nht mt im.

1) Nu b Qa

, th f tha mn h bt ng thc hm

( ) ( )f a x f xa+ + , ( ) ( )f b x f xb+ + , x R (1)

khi v ch khi ( ) ( )0f x px f= + , x R , trong :paa

= .


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2) Nu b Qa

, th tn ti duy nht mt nghim hm lin tc :f R R ca

h phng trnh hm tng ng ( ) ( )f a x f xa+ = + , ( ) ( )f b x f xb+ = + , x R (2)

sao cho [ ] 00,f fd = , trong

{ }: min 0 ; ,d ma nb m n N= + > tn ti, l s dng v [ ]0 : 0,f d R l hm lin tc cho trc tha mn iu kin

( ) ( )0 0 0f d d faa

= + .

Hn na, nu 0f l n iu nghim ngt, th n trng vi hm f trn on [ ]0, d . Chng minh. 1) T (1), d dng suy ra

( ) ( )f ma x m f xa+ + , ( ) ( )f nb x n f xb+ + , , m n N x R . Trong bt ng thc u tin trn, thay x bi nb x+ , ta c

( ) ( ) ( )f ma nb x m f nb x m n f xa a b+ + + + + + . Do

( ) ( )f ma nb x m n f xa b+ + + + , , m n N x R . t

:pa ba b

= = ,

ta c th vit bt ng thc ny di dng ( ) ( )f t x pt f x+ + , t A , x R , (3) trong , theo B 1, tp

{ } ; , A ma nb m n N= + tr mt trong R . Gi s rng 0x l im m ti hm f lin tc v x l mt gi tr thc ty . Bi tnh cht tr mt ca A trong tp R , tn ti mt dy ( )tn sao cho

nt A ( )n N , 0lim t x xnn= -

+.

T bt ng thc (3), ta c ( ) ( )n nf t x pt f x+ + , n N .

Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c


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( ) ( ) ( )0 0f x p x x f x - + , x R . By gi, chng minh phn o, thay x bi x t- trong (3), ta dc

( ) ( )f x pt f x t + - , t A , x R . Chn mt im x R c nh ty v, bi tnh tr mt ca A trong R , mt dy ( )tn sao cho

t An ( )n N , 0lim t x xnn= -

+.

Th th, ta c ( ) ( )n nf x pt f x t + - , n N .

Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c bt ng thc ( ) ( ) ( )0 0f x p x x f x - + , x R .

Do , ta c ( ) ( ) ( )( )0 0f x p x f x px= + - , x R .

Ta c iu phi chng minh. 2) T (2), d dng suy ra ( ) ( ) ( )f ma nb x ma nb f x

aa

+ + = + + , , m n N , x R . (4)

Theo B 1, phn 2, s { }: min 0 ; , d ma nb m n N= + >

l xc nh v l s dng. Hn na { } { }0 ; , ; ma nb m n N kd k Z+ > = .

D , (4) c dng

( ) ( )f kd x kd f xaa

+ = + , k Z , x R .

D dng thy rng h phng trnh hm ny tng ng vi phng trnh

( ) ( )f d x d f xaa

+ = + , x R .

By gi, ta xc nh ( ]1 : , 2f d d R bi cng thc

( ) ( )1 0

:f x d f x daa

= + - , ( ], 2x d d . Gi s rng

( )(: , 1nf nd n d R+ ( )n N c xc nh. Th th, ta xc nh

( ) ( )(1 : 1 , 2nf n d n d R+ + + bi h thc truy hi


14

( ) ( )1 :n nf x d f x daa

+ = + - ,

( ) ( )( 1 , 2x n d n d + + , n N . Tng t, gi s

( ) ( )1 0:f x d f x daa

- = - + + , [ ),0x d - . Gi s rng ta c nh ngha

( ) ( ) ): , 1nf x nd n d R- = - - + ( )n N . Th th, ta nh ngha

( ) ( ) ( )1 : nnf x d f x daa

-- + = - + + ,

( ) )1 ,x n d nd - + - , n N . D dng kim tra c rng :f R R , xc nh bi

( )( ) ( ) )

( ) [ ]( ) ( )(

0

, khi , 1 ,

, khi 0, ,

, khi , 1 ,

n

n

f x x nd n d

f x f x x d

f x x nd n d

- - - +=

+

, n N ,

tha mn h (2), l hm lin tc v [ ] 00,f fd = .

nh l c chng minh hon ton. nh l 5. (Dng cng-nhn) Gi s , a b R v , 0a b > l cc s cho trc tha mn

0a b< < , log loga b

a b=

v gi s rng hm :f R R lin tc ti t nht mt im.

1) Nu b Qa


( ) ( )f a x f xa+ , ( ) ( )f b x f xb+ , x R (5)

khi v ch khi ( ) ( )0 pxf x f e= , x R , trong log:pa

a= .

2) Nu b Qa

, th tn ti duy nht mt nghim hm lin tc :f R R ca

h phng trnh hm tng ng ( ) ( )f a x f xa+ = , ( ) ( )f b x f xb+ = , x R (6)


{ }: min 0 ; ,d ma nb m n N= + >


15

tn ti, l s dng v [ ]0 : 0,f d R l hm lin tc cho trc tha mn iu kin

( ) ( )0 0log

0d

af d f e

a

= . Hn na, nu 0f l n iu nghim ngt, th n trng vi hm f trn on [ ]0, d . Chng minh. 1) T (5), d dng suy ra

( ) ( )mf ma x f xa+ , ( ) ( )nf nb x f xb+ , , m n N , x R .

Trong bt ng thc u tin trn, thay x bi nb x+ , ta c ( ) ( ) ( )m m nf ma nb x f nb x f xa a b+ + + .

Do

( ) ( )m nf ma nb x f xa b+ + , , m n N , x R . t

log log:pa b

a b= = ,

ta c th vit bt ng thc ny di dng ( ) ( )ptf t x e f x+ , t A , x R , (7) trong , theo B 1, tp { } ; , A ma nb m n N= + tr mt trong R . Gi s rng 0x l im m ti hm f lin tc v x l mt gi tr thc ty . Bi tnh cht tr mt ca A trong tp R , tn ti mt dy ( )nt sao cho

nt A ( )n N , 0lim nn t x x+ = - . T bt ng thc (7), ta c

( ) ( )nnptf t x e f x+ , n N .

Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c

( ) ( ) ( )00p x xf x e f x- , x R .

By gi, chng minh phn o, thay x bi x t- trong (7), ta dc ( ) ( )ptf x e f x t - , t A , x R .


16

Chn mt im x R c nh ty v, bi tnh tr mt ca A trong R , mt dy ( )tn sao cho

t An ( )n N , 0lim t x xnn= -

+.

Th th, ta c

( ) ( )nptnf x e f x t - , n N .

Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c bt ng thc

( ) ( ) ( )0 0p x xf x e f x- , x R .

Do , ta c

( ) ( ) ( )( )0 0p xf x e f x px= - , x R . Ta c iu phi chng minh. 2) Chng minh tng t chng minh phn 2 ca nh l 4. nh l c chng minh hon ton. nh l 6. (Dng nhn-cng) Gi s , , , a b a b l cc s thc cho trc tha mn

0 1a b< < < , log loga b

a b= ,

v gi s rng hm :f I R lin tc ti t nht mt im.

1) Nu loglog

b Qa


( ) ( )f ax f xa + , ( ) ( )f bx f xb + , x I , (8) th i) Trng hp ( )0,I = :

( ) ( )log 1f x p x f= + , 0x > , ii) Trng hp ( ),0I = - :

( ) ( ) ( )log 1f x p x f= - + - , 0x < ,

trong :log

pa

a= .

2) Nu loglog

b Qa

, th tn ti duy nht mt nghim hm lin tc :f I R

ca h phng trnh hm tng ng ( ) ( )f ax f xa= + , ( ) ( )f bx f xb= + , x I (9)



17

{ }: min 1 ; ,m nd a b m n N= > tn ti, ln hn 1 v [ ]0 : 1,f d R l hm lin tc cho trc tha mn iu kin

( ) ( )0 0.log 1logf d d faa

= + .

Hn na, nu 0f l n iu nghim ngt, th n trng vi hm f trn on [ ]1,d . Chng minh. 1) i) Gi s rng ( )0,I = . T (8), chng minh tng t nh cc phn trn, ta c

( ) ( )m nf a b x m n f xa b + + , , m n N , 0x > . t

:log log

pa b

a b= = ,

ta c th vit bt ng thc ny di dng

( ) ( ) ( )logm n m nf a b x p a b f x + , , m n N , 0x > , hay ( ) ( )logf tx p t f x + , t M , 0x > , (10) trong , theo B 2, tp

{ } ; , m nM a b m n N= tr mt trong I . Gi s rng 0 0x > l im m ti hm f lin tc v 0x > l mt gi tr ty . Bi tnh cht tr mt ca M trong tp I , tn ti mt dy ( )tn sao cho

t Mn ( )n N , 0lim xtn xn

=+

.

T bt ng thc (10), ta c

( ) ( )logf t x p t f xn n + , n N . Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c

( ) ( )00 logxf x p f xx

+ , 0x > .


18

By gi, chng minh phn o, thay x bi xt

trong (10), v chn mt

dy ( )tn sao cho t Mn ( )n N ,

0

lim xtn xn=

+.

Th th, ta c

( ) ( )00

log xf x p f xx

+ , 0x > .

Do , ta c ( ) ( )0 0log logf x f x p x p x= - + , 0x > .

Phn i) c chng minh. ii) Gi s rng ( ),0I = - . Ta xt hm ( ): 0,g R xc nh bi cng thc ( ) ( )g x f x= - , 0x < , tha mn h (8) v chng minh tng t nh chng minh phn i). 2) Phn ny chng minh tng t nh chng minh nh l 4, phn 2. H qu 5. Gi s , , , a b Ra b tha mn cc gi thit ca nh l 4, phn 1. Nu hm ( ) ( ): ,0 0,f R - U tha mn h bt ng thc (8) v trong mi khong ( ),0 , ( )0, tn ti t nht mt im m ti hm f lin tc, th

( )( ) ( )

( ) ( ) ( )log 1 , khi 0, ,

log 1 , khi ,0 ,

p x f xf x

p x f x

+ = - + - -

trong :log

pa

a= .

Ch 1. Gi s , , , a b a b l cc s thc cho trc tha mn 0 1a b< < < v

log loga ba b

= . Nu 0 I , th khng tn ti hm no tha mn h (8).

Tht vy, trong bt ng thc (8) nu t 0x = , th 0 a , 0 b , mu thun vi gi thit 0ab < . nh l 7. (Dng nhn) Gi s , , , a b a b l cc s thc cho trc tha mn

1a b< < , log loglog loga b

a b= ,


19

v gi s rng hm :f I R lin tc ti t nht mt im.

1) Nu loglog

b Qa


( ) ( )f ax f xa , ( ) ( )f bx f xb , x I , (11) th i) Trng hp ( )0,I = :

( ) ( )1 pf x f x= , 0x > , ii) Trng hp ( ),0I = - :

( ) ( )( )1 pf x f x= - - , 0x < ,

trong log:log

paa

= .

2) Nu loglog

b Qa

, th tn ti duy nht mt nghim hm lin tc :f I R

( ( )0,I = hoc ( ),0I = - ) ca h phng trnh hm tng ng ( ) ( )f ax f xa= , ( ) ( )f bx f xb= , x I , (12)


{ }: min 1 ; ,m nd a b m n N= > tn ti, ln hn 1 v [ ]0 : 1,f d R l hm lin tc cho trc tha mn iu kin

( ) ( )0 0

loglog1 af d f d

a

= . Hn na, nu 0f l n iu nghim ngt, th n trng vi hm f trn on [ ]1,d . Chng minh. 1) i) Gi s rng ( )0,I = . T (11), chng minh tng t nh cc phn trn, ta c

( ) ( )m n m nf a b x f xa b , , m n N , 0x > . t

log log:log log

pa ba b

= = ,

ta c th vit bt ng thc ny di dng


20

( ) ( ) ( )pm n m nf a b x a b f x , , m n N , 0x > , hay ( ) ( ).pf tx t f x , t M , 0x > , (13) trong , theo B 2, tp

{ } ; , m nM a b m n N= tr mt trong I . Gi s rng 0 0x > l im m ti hm f lin tc v 0x > l mt gi tr ty . Bi tnh cht tr mt ca M trong tp I , tn ti mt dy ( )tn sao cho

t Mn ( )n N , 0lim xtn xn

=+

.

T bt ng thc (13), ta c

( ) ( )pf t x t f xn n , n N . Cho n , bi tnh lin tc ca hm f ti 0x , ta thu c

( ) ( )00pxf x f x

x

, 0x > .

By gi, chng minh phn o, thay x bi xt

trong (13), v chn mt

dy ( )tn sao cho t Mn ( )n N ,

0

lim xtn xn=

+.

Th th, ta c

( ) ( )00

.pxf x f x

x

, 0x > .

Phn i) c chng minh. ii) Gi s rng ( ),0I = - . Ta xt hm ( ): 0,g R xc nh bi cng thc ( ) ( )g x f x= - , 0x < , tha mn h (11) v chng minh tng t nh chng minh phn i). 2) Phn ny chng minh tng t nh chng minh nh l 4, phn 2. Ch 2. Gi s , , , a b a b l cc s thc cho trc tha mn 0 1a b< < < v log loglog loga b

a b= . Nu I R= hoc [ )0,I = hoc ( ],0I = - v :f I R tha mn

h (11), th ( )0 0f = .


21

Tht vy, bi mt trong hai gi thit 1a b< < hoc 1b a< < v, hn na,

( )( )0 1 0f a- v ( )( )0 1 0f b- , ta suy ra ( )0 0f = . T Ch ny, ta c Ch 3. i) Gi s [ ): 0,f R tha mn h (11). Nu ( )0,f v , , , a b a b tha

mn tt c cc gi thit ca nh l 7, phn 1, th

( ) ( ) ( )1 khi 0, ,0 khi 0,

pf x xf xx

= =

trong log:log

paa

= .

ii) Gi s ( ]: ,0f R- tha mn h (11). Nu ( ),0f - v , , , a b a b

tha mn tt c cc gi thit ca nh l 7, phn 1, th

( ) ( )( ) ( )1 khi ,0 ,0 khi 0,

pf x xf xx

- - -= =

trong log:log

paa

= .

H qu 6. Gi s , , , a b Ra b tha mn cc gi thit ca nh l 7, phn 1. i) Nu hm ( ) ( ): ,0 0,f R - U tha mn h bt ng thc (11) v trong mi khong ( ),0- , ( )0, tn ti t nht mt im m ti hm f lin tc, th

( )( ) ( )

( )( ) ( )

1 khi 0, ,

1 khi ,0 ,

pf x xf x

pf x x

= - - -

trong log:log

paa

= .

ii) Nu hm :f R R tha mn h bt ng thc (16) v trong mi khong ( ),0- v ( )0, tn ti t nht mt im m ti hm f lin tc, th


22

( )( ) ( )

( )( ) ( )

1 khi 0, ,0 khi 0,

1 khi ,0 ,

pf x xf x x

pf x x

= =

- - -

trong log:log

paa

= .

Ch 4. Ta lun c cc nh l tng t nh cc nh l 4- nh l 7, vi hm f tha mn cc bt ng thc c du ngc li. Pleiku, 7 / 2011 T..C

TI LIU THAM KHO [1] Nguyn vn Mu, Bt ng thc, nh l v p dng, Nh xut bn Gio dc, 2006. [2] Trnh o Chin, Mt s dng bt phng trnh hm dng c bn, K yu Hi ngh khoa hc v cc chuyn chuyn Ton bi dng hc sinh gii Trung hc ph thng, H Ni - Nam nh, 26-28/11/2010. [3] Th. M. Rassias, Functional equations, inequalities and applications, 73 - 89, Kluwer Academic Publishers, 2003. [4] PI. Kannappan, Functional equations and with applications, 617 - 636, Springer Monographs in Mathematics, 2009.


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