1 PHNG TRNH HM CAUCHY TNG QUT Trong bi ging ny chng ti s cp n mt lp bi ton phng trnh hm dng ( ), ) , ( ) ( ) ( ) ( y x H g y x f y f x f = + +(1) trongf vg lcchmphitmcnH lhmcho.Khi0 g th(1)tr thnh phng trnh hm Cauchy. Cc hm s y c xt l hm s thc, tc l tp xc nh v tp gi tr ca n l R hoc tp con ca R. 1.Phng trnh hm Cauchy v Pexider Xtphng trnh Cauchy ). ( ) ( ) ( y x f y f x f + = + (2) Bi ton 1.1: Chng minh rng tt c cc hmflin tc trn R tha mn phng trnh Cauchy (2) u c dng, ) ( cx x f = ycl hng s bt k. Bi ton 1.2: Chngminh rng tt c cc hmflin tc timt im thamn phng trnh Cauchy (2) u c dng, ) ( cx x f = ycl hng s bt k. Bi ton 1.3: Chng minh rng tt c cc hm fkhng m (khng dng) vixdng nh tha mn (2) u c dng, ) ( cx x f = ycl hng s dng (m) bt k. Bi ton 1.4: Chng minh rng tt c cc hmfb chn trn mt khong nh tha mn (2) u c dng, ) ( cx x f = ycl hng s. Biton1.5:Chngminhrngttccchmf bchnmtpha(bchntrn hocbchndi)trnmton[a,b]chotrcvthamn(2)ucdng , ) ( cx x f = ycl hng s. Biton1.6:Chngminhrngttccchmf niuthamn(2)uc dng, ) ( cx x f = ycl hng s. Bi ton 1.7: Chng minh rng tt c cc hmfkh tch trn mi on hu hn v tha mn (2) u c dng, ) ( cx x f = ycl hng s bt k. nh l 1: Gi sR R : fl nghim ca (2) vi0 ) 1 ( > = f c . Khi ta c cc khng nh sau l tng ng. (i)flin tc ti mt im.0x(ii)flin tc. 2 (iii)fl hm n iu tng. (iv)fkhng m vi mi. 0 > x (v)fb chn trn trn mt on hu hn. (vi)fb chn di trn mt on hu hn. (vii)fb chn trn (di) trn mt tp b chn c do Lebesgue dng. (viii)fb chn trn mt on hu hn. (ix). ) ( cx x f =(x)fkh tch Lebesgue a phng. (xi)fkh vi (xii)fo c Lebesgue. Ch : Mt cu hi t ra l c ti ti hay khng nhng nghim khng tuyn tnh ( cx x f = ) ( )caphngtrnhCauchy(2)?Lthuytcaphngtrnhhmc cu tr li thng qua khi nim c s Hamel. nhngha: Cho Sl mt tp con ca R. Khi mt con B caS c gil mtcsHamelcaSnuvimisthucSubiuthtuyntnhquacc phn t ca B mt cch duy nht vi cc h s l cc s hu t. nh l 2: Nghim tng qut ca phng trnh Cauchy (2) c dng ), ( ) ( ) ( ) (2 2 1 1 n nb f r b g r b g r x f + + + = trong B l c s Hamel ca R v,2 2 1 1 n nb r b r b r x + + + = eir Q veib B, vgl mt hm ty xc nh trn c s Hamel B. Phng trnh hm ) ( ) ( ) ( y g x h y x f + = +(3) c gi l phng trnh Pexider. Bi ton 1.8: Chng minh rng nghim tng qut ca (3) l, ) ( ) ( , ) ( ) ( , ) ( ) ( b t A t h a t A t g b a t A t f + = + = + + = trong A l hm bt k tha mn phng trnh Cauchy (2) vb a,l cc hng s bt k. 3 Biton1.9:ChoR R : , , h g f thamn(3),khinumttrongcchm h g f , ,l o c hoc b chn di, hoc b chn trn, hoc lin tc ti mt im th cc hmh g f , ,l lin tc. Hn na chng c dng , ) ( , ) ( , ) ( b ct t h a ct t g b a ct t f + = + = + + = yc b a , ,l cc hng s bt k. 2.Phng trnh hm Cauchy tng qut Trc ht chng ta xt mt s trng hp c bit ca (1)+: ) , ( xy y x H =(1) tr thnh ). ( ) ( ) ( ) ( xy g y x f y f x f = + + (4) + y xy x H1 1) , ( + =v: f g =). ( ) ( ) ( ) (1 1 + = + + y x f y x f y f x f (5) + xy y xy x xyy x H+ ++=2 2) () , (v: f g = .) () ( ) ( ) (2 2 ||.|

\|+ ++= + +xy y xy x xyf y x f y f x f (6) Tanhnthyrngccphngtrnh(4)-(6)lnhngdngbitonquenthuc trong l thuyt phng trnh hm.Nhn xt:+ Nugl hm hng, tc lc x g = ) (th vi bt c hmH cho no (1) u tng ng vi. ) ( ) ( ) ( c y x f y f x f = + +R rng nghim tng qut ca phng trnh trn l, ) ( ) ( c x A x f + = y) (x Al mt hm cng tnh bt k, ngha l) (x Al nghim ca phng trnh Cauchy (2). Vy nghim ca (1) trong trng hp ny l c x A x f + = ) ( ) (v. ) ( c x g =+ Tng t trong trng hpc y x H = ) , ( , cng thc nghim ca (1) l ) ( ) ( ) ( c g x A x f + =v gl hm bt k 4 vi) (x Acng l mt hm cng tnh ty . Chng ta gi nghim) , ( g fca phng trnh (1) l tm thng nufl afin, tc l, ) ( ) ( c x A x f + =trong ) (x Al cng tnh vcl hng s.+rngccphngtrnh(4)-(6)ldngphngtrnh(1)vi) , ( y x H c dng sau ( ), ) ( ) ( ) ( ) , ( y x y x y x H + + = (7) vdthy(4)-(6)tongngvivicchn 1 2 , ln , ) (= x x x x v . , , 2 / ) (1 = u e u uuBy gi chng ta xt mt trng hp c bit ca (1) c dng nh sau ( ). ) ( ) ( ) ( ) ( ) ( ) ( y x y x g y x f y f x f + + = + + (8) R rng nul afin th vi migphng trnh (8) u c nghimfl afin, tc l (8) c nghim tm thng, v vy chng ta xtkhng l afin v k hiuIl ) , ( + (hoc| ), ,+ ), , ( + ), , ( ( | , ) trong . 0 > Bi ton 2.1: Cho I : R l gii tchvkhng afin, hy tm tt c cc nghim ca phng trnh (8). Gii: Gi sfvgl nghim ca (8). t), ( ) ( ) ( ) , ( y x f y f x f y x F + + =thFl nghim ca phng trnh ). , ( ) , ( ) , ( ) , ( z y F z y x F z y x F y x F + + = + + (9) Dofl nghim ca (8) nn ta c( ), ) ( ) ( ) ( ) , ( y x y x g y x F + + = v vy ng thc (9) tng ng vi

( ) ( )( ) ( ). ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) (z y z y g z y z y x x gz y x z y x g y x y x g+ + + + + + + =+ + + + + + + (10) t + + =+ + + + =+ + =), ( ) ( ) (), ( ) ( ) (), ( ) ( ) (z y z y wz y x z y x vy x y x u (11) chng ta c th vit li phng trnh (10) di dng sau ). ( ) ( ) ( ) ( w g w v u g v g u g + + = +(12) 5 Taschrarng(11)lmtphpibin,tclnhthcJacobicaphpi bin (11) l khc. 0Tht vy,( )( )( )( )( )( ). ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ') ( ' ) ( ' ) ( ' ) ( ' 0) ( ' ) ( ' 0 ) ( ' ) ( '0 ) ( ' ) ( ' ) ( ' ) ( 'det( ' ) ( ' ) ( ' ) ( ' 0) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( '0 ) ( ' ) ( ' ) ( ' ) ( 'det) , , () , , (z y z x y z y x y xz y y z y x z y x xz y z z y yz y x z z y x y xx y y x xz y z z y yz y x z z y x y x z y x y xy x y y x xz y x Dw v u D+ + + + + + + + =((((

+ + + + + + + + =((((

+ + + + + + + + + ++ + = Gistntimttpconmkhcrngca 3I mnhthcthcJacobing nht bng 0,khi dogii tch nn suy ranh thc Jacobi ng nht bng 0 trn,3Itc l, u , v w l cc hm ph thuc. Mt khc, ta c ( )( ). ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ) ( ' ( ' ) ( 'y) (x ' - (y) ' y) (x ' - (x) 'det) , () , (z y x y x y xz y x y x z y x y x y x Dv u D+ + + =((

+ + + + + ++ += T lgiitchvkhnglafinnn y x Dv u D, () , (khngthngnhtbng0trn con m khc rng ca.2IVy ta nhn c , u , v w l cc hm ph thuc v , uvl cc hm c lp, suy ra tn ti hm kh visao cho ( ) . , ,, ) , , ( ), , , ( ) , , ( I z y x z y x v z y x u z y x w e = Vy chng ta nhn c ( ) . z) y, (x, , ) ( ) ( ) ( ), ( ) ( ) () ( ) ( ) (3I z y x z y x y x y xz y z ye + + + + + + =+ + Thay i vai tr caxvycho nhau trong h thc trn suy ra 3z) y, (x, ), ( ) ( ) ( ) ( ) ( ) ( I z x z x z y z y e + + = + + hay. z) y, (x, ), ( ) ( ) ( ) (3I z x x z y y e + = + rngvtricangthctrnkhngphthucvox nnvbnphil mt hm s ch theo binz , iu ny khng th xy ra v gi thitkhng l afin. Vykhngtntimttpconmkhcrngca 3I mnhthcJacobing nht 0 trn , ngha l, u , v w l cc hm c lp vi mi. ) , , (3I z y x eBi vw 6 l c lp iviu vvnn phng trnh (12)c th xem nh l phng trnh Pexider ) ( ) ( ) ( v u h v g u g + = +vi mi gi tr ca) , , ( w v unhn c t 3) , , ( I z y x equa php i bin (11). Tuy nhin do, u , v w l cc hm c lp nn phng trnh Pexider c tp xc nh l tpmkhcrngtrongR2.TheoktqucaphngtrnhPexidervimi ) ( ) ( ) ( y x y x u + + = vi 2, ) , ( I y x e hmgc dng , ) ( ) (1c u A u g + = y 1Al hm cng tnh vcl hng s bt k. C2: Ta c (12) tng ng vi ( ) ( ) ( ) ), ( ) ( ) ( ) ( ) ( ) ( ) ( w g w w v w u g w g w w v g w g w w u g + + = + + + hay ). ( ) ( ) ( ) ( ) ( ) ( w g w t g w g w s g w g w t s g + + + = + +t) ( ) ( ) ( w g w z g z + = ta nhn c ), ( ) ( ) ( y x y x + = +hay ), ( ) (1x A x = trong ) (1x Al mt hm cng tnh bt k. Suyra), ( ) ( ) (1x A x g y x g = + t) ( ) ( ) (1x g x A x g + = khitanhnc ) ( ) ( x g y x g = + , tc lc x g ) (vicl hng s. Vy , ) ( ) (1c u A u g + =trong 1Al hm cng tnh vcl hng s bt k. Thay biu thc ca hmgny vo phng trnh (8), ta nhn c ( )c y x A y A x Ac y x y x A y x f y f x f+ + + =+ + + = + +) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1 1 11 hay tng ng vi ( ) ( ) ( ). ) ( ) ( ) ( ) ( ) ( ) (1 1 1c y x A y x f c y A y f c x A x f + + = + iu ny chng tc A f 1 l hm cng tnh, tc l 7 , , ) ( ) ( ) (2 1I x c x A x A x f e + + = trong 2Acng l mt hm cng tnh bt k.Ngc li, r rng cc hmfvgxc nh nh trn tha mn phng trnh (8). Vy cng thc nghim ca phng trnh (8) l

{ }e + + e + =e + + =, , | ) ( ) ( ) ( , ) ( ) (, , ) ( ) ( ) (12 1I y x y x y x s c s A s gI x c x A x A x f y,1A2Al cc hm cng tnh bt k vcl hng s ty . Biton2.2:Xt= I Rv, 2 / ) (2x x = khiphngtrnh(8)trvphng trnh (4). Theo Bi ton 2.1 th nghim tng qut ca phng trnh (4) le + +||.|

\| =e + =, , ) (2) (, ) ( ) (2211RRx c x AxA x f, s c s A s g trong ,1A2Al cc hm cng tnh vcl hng s bt k. rng 1Al hm cng tnh nn. ), (2122121R e =||.|

\| x x AxAT nu t 1 121A B =th nghim tng qut ca phng trnh (4) c vit di dng gn hn nh sau e + + =e + =RRx c x A x B f(x)s c s B s g, ) ( ) (, , ) ( 2 ) (2211 vi,1B2Al cc hm cng tnh vcl hng s ty . V d 2.1: TmR R : flin tc trn R tha mn=e + + = +. 1 ) 1 (, , , 2 ) ( ) ( ) (fy x xy y f x f y x f R Bnh lun: T hmftha mniu kin , 2 ) ( ) ( ) ( xy y f x f y x f + + = + e y x, R ta suy rafl nghim ca Bi ton 2.2 vi. 2 ) ( s s g =Theo kt qu ca bi ton ny ta ce = s s s B , ) (1R v), ( ) (2x A x x f + =trong ) (x Al mt hm cng tnh 8 bt k. V vy ta s dng php th 2) ( ) ( x x g x f + =th s nhn c phng trnh Cauchy i vi n mi. gGii: t, ) ( ) (2x x g x f + =khi doflin tc trn R nngcng lin tc trn R v iu kin ca bi ton tng ng vi =+ = +. 0 ) 1 (), ( ) ( ) (gy g x g y x g y l phng trnh Cauchy, nn nghim ca bi ton l. , 0 ) ( R e = x x gvy bi ton cho c duy nht nghim l. ) (2x x f = V d 2.2: TmR R : ftha mn = |.|

\|=e + + = +. 0 ,1) (, , , 2 ) ( ) ( ) (4xxf x x fy x xy y f x f y x f

R Gii: t 2) ( ) ( x x g x f + =ta c bi ton tr thnh (2.2.1) T iu kin th nht ca (2.2.1) suy ra e = t t g t g ), ( 2 ) 2 ( R(2.2.2) v . 0 ,2121= |.|

\|= |.|

\|ttgtg(2.2.3) Mt khc theo iu kin th hai ca (2.2.1) th 0 ,) ( 14= = |.|

\|ttt gtgv. 0 ,16) 2 (214= = |.|

\|ttt gtgTh hai biu thc trn vo (2.2.3) ta nhn c . 0 ), ( 16 ) 2 ( = = t t g t gHn na, ch rng t iu kin th nht ca (2.2.1) ta suy ra, 0 ) 0 ( = gdo ta c = |.|

\|=e + = +. 0 ,1) (, ), ( ) ( ) (4xxg x x g, y x y g x g y x g

R9 e = t t g t g ), ( 16 ) 2 ( R. Kt hp ng thc ny vi (2.2.2) ta nhn ce = t t g , 0 ) ( R.Th li ta c 2) ( x x f =tha mn, vy bi ton c duy nht nghim. ) (2x x f = Nhnxt:Rngbucthhailmttrong4rngbucisquenthucca phng trnh hm Cauchy cng tnh. Chng ta c kt qu quen thuc sau. "ChoR R : ftha mn (2), khi cc khng nh sau l tng ng (i). , ), ( ) ( ) ( Re + = y x x yf y xf xy f(ii). 0 ), (1 12= = |.|

\|x x fx xf (iii). ), ( 2 ) (2Re = x xf x f(iv)( ) ( ) . , ) ( 4 ) (2 222Re = x x f x x f " V d 2.3: Xc nh tt c cc hmR R : ftha mn . , , 2012 ) ( ) ( ) ( R e + + = + y x xy y f x f y x fGii: t, 1006 ) ( ) (2x x g x f + =khi iu kin cho tng ng vi . , ), ( ) ( ) ( R e + = + y x y g x g y x gVy nghim ca bi ton l 21006 ) ( ) ( x x A x f + =viA l hm cng tnh bt k. Biton2.3:Bygitaly), , 0 (+ = I , ln ) ( x x = ) ( ) (xe h x g = thphng trnh (8) tr thnh . 0 , ), ( ) ( ) ( ) (1 1> + = + + y x y x h y x f y f x fGii: p dng kt qu ca Bi ton 2.1, ta nhn c nghim ca bi ton cho l > + + = + =0 , ) ( ) (ln ) (, , ) ( ) (2 11x c x A x A x fs c s A e hs hay > + + => + =, 0 , ) ( ) (ln ) (, 0 , ) (ln ) (2 11x c x A x A x fs c s A s h y 2 1, A Al cc hm cng tnh vcl hng s bt k. Bnh lun: Khi lyf h =th ta nhn c cng thc nghim ca (5) l 10 ), (ln ) ( x A x f =trongAlhmcngtnhty.Cnkhilyf h = thtacktlunphng trnh 0 , ), ( ) ( ) ( ) (1 1= + = + + y x y x f y x f y f x f ch c nghim tm thngx c x f = , ) (vicl hng s bt k. Ch : Ta bit rng nghim ca phng trnh Cauchy logarit 0 , ), ( ) ( ) ( = + = y x y f x f xy f (13) c dng , 0 |), | (ln ) ( = = x x A x ftrong A l hm cng tnh bt k. Tht vy, t (13) cho0 = = = t y xv0 = = = t y xta c , 0 ), ( 2 ) (v ) ( 2 ) (2 2= = = t t f t f t f t fdo . 0 ), ( ) ( = = t t f t f(2.3.1) Gi s (13) tha mn vi, 0 , > y xkhi t s te y e x = = ,v) ( ) (se f s g =th (13) tng ng vi phng trnh Cauchy ). ( ) ( ) ( t g s g t s g + = +Suy ra. 0 ), (ln ) ( > = x x A x fMt khc theo (2.3.1) tanhn c nghim tng qut ca (13) l , 0 |), | (ln ) ( = = x x A x ftrong A l mt hm cng tnh bt k. VychngtanhnckhngnhCcphngtrnh(5)v(13)ltng ng vi mi. 0 , > y x Bi ton 2.4: Chngminh rng cc phng trnh (5) v(13) l tng ngvi nhau trn). , 0 (+Gii: Gi sfl nghim ca (5), khi t (5) vi 1 = x ysuy ra 0 ), ( ) ( ) ( ) (1 1 1> + = + + x x x f x x f x f x fhay11 0 ), ( ) (1> =x x f x f(2.4.1) v. 0 ) 1 ( = f (2.4.2) t), ( ) ( ) ( ) , ( y f x f y x f y x F + =dof l nghim ca (5) nn0 , ), ( ) , (1 1> + = y x y x f y x Fv d dng kim tra c ) , ( ) , ( ) , ( ) , ( z y F z y x F y x F z y x F + + = + +hay . 0 , , ,1 1 1 1 1 1 1 1> ||.|

\|+ +||.|

\|++ =||.|

\|+ +||.|

\|++z y xz yfz y xfy xfz y xf(2.4.3) Xt11 1= +z y vi0 , > z y , khi ta c1 , > z yv , hn na ) (1 12y x yx xy yz y x + += ++v .1 122xyx xy yx z y += ++ p dng (2.4.2) vo (2.4.3) ta nhn c . 1 , 0 ,) (22 2> > ||.|

\| +=||.|

\| ++||.|

\|+ +y xxyx xy yfxyy xfy x yx xy yft xyy xu+= , ) (2y x yx xy yv+ +=th h thc trn tr thnh ). ( ) ( ) ( v f u f uv f + =T xyy xu+=v 1) ( 1+ = y xyxvsuy ra

1 =uxxy(2.4.4) v 12= +uxu xy xhay,1) (21u xuxy x= + do u xuxv22) 1 (1 = , vy .) 1 (22xuxuv u= Do0 > unn ta nhn c, 1 0 < < vhn na kt hp vi (2.4.4) ta c 12 v uy=11. V 1 1 = y u xnn ( ) v u ux =11 hay tng ng vi ,) 1 (1 + +=v u uv uxvy ta thu c. 0 1 > + v uiu ny c thamn nu. 1 > uVy vi0 , > y xth1 > uv, 1 0 < < vtc l ta c . 1 0 , 1 ), ( ) ( ) ( < < > + = v u v f u f uv fT (2.4.2) ta d thy ng thc trn cng ng vi1 = uhc, 1 = vni cch khc ta chng minh c . 1 0 , 1 ), ( ) ( ) ( s < > + = v u v f u f uv f (2.4.5) XtP s 1th1 01s = P P f P f(2.4.6) XtQ P s s 1 ,khi1 01s + = Q P Q f P f PQ fMt khc vi mi1 , > Q Pth theo h thc trn v (2.4.1) ta c ). ( ) ( ) ( ) ( ) ( ) (1 1 1 1 + = = = Q f P f Q f P f PQ f Q P fVy ta thit lp c , 0 , ), ( ) ( ) ( > + = v u v f u f uv fni cc khc mi nghim ca (5) u l nghim ca (13). 13 C2: Vi0 , > v uta ly0 > ln sao cho1 > u v1 + + + = + = + y x y f y x f x f y y x x f y x fhayfl nghim ca (5). Vy ta chng minh c (5) v (13) tng ng trn ). , 0 (+ V d 2.4: Tm tt c cc hmR R : +flin tc tha mn . 0 , ), ( ) ( ) ( ) (1 1 1> + = + + + y x y x f y x f y f x fGii: R rng phng trnh cho tng ng vi . 0 ,), ( ) ( ) ( ) (1 1> + = + + y x y x f y x f y f x fTheo kt qu ca Bi ton 2.4 th phng trnh trn tng ng vi Vy bi ton tr thnh tm tt c cc hmR R+: fsao cho . 0 , ), ( ) ( ) ( > + = y x y f x f xy fDo0 , > y xvfln tc trn +Rnn t s te y e x = = ,v) ( ) (se f s g =th bi ton tng ng vi tm tt c cc hmR R : glin tc tha mn , , ), ( ) ( ) ( t s t g s g t s g + = +tc lcs s g = ) (vicl hng s bt k. Vy nghim ca bi ton cho l , 0 , ln ) ( > = x x c x f . 0 , ), ( ) ( ) ( > + = y x y f x f xy f14 ycl hng s ty . V d 2.5: Tm tt c cc hmR R : ftha mn = == = += + + = = = + = + . 0 ) 0 ( v 1 ) 1 (0, x |, | ln 2 ) ( ) (, 0|, | ln ) 1 ( ) ( ) (, 1 , 0 ,|, | ln ) ( ) ( ) (2 1 21 1 1f fx x f x fx x x x f x x fxy y x y x y f x f y x f Bnh lun: Ch rng phng trnh u ca h tng ng vi , , 0 |, | ln ) ( ) ( ) (1 1y x x,y y x y x f y f x f = = + = + +

yldng(5)vi| | ln ) ( x x f = ,hnnadthy| | ln ) ( x x f = lmtnghim ring ca phng trnh trn. Gii:t, 0 ), ( | | ln ) ( = + = x x g x x f v) 0 ( ) 0 ( g f = tacbitonchotng ng vi = = = = = = + = +

, 1 ) 1 (, ), ( ) (, 0 ), ( ) (, 1 , 0 ), ( ) ( ) (1 21 1gx x g x gx x g x x gxy y y g x g y x g hay == = + = +. 1 ) 1 (, 0), ( ) (, ,), ( ) ( ) (1 2gx x g x x gy x y g x g y x g (2.5.1) T phng trnh u ca (2.5.1) ta c. ), ( ) ( v 0 ) 0 ( x x g x g g = = rng , 1 , 0 ,111) 1 (1= =xx x x x do . 1 , 0 ,111) 1 (1= |.|

\| |.|

\|=||.|

\|xxgxgx xgp dng phng trnh th hai v th ba ca (2.5.1) cho h thc trn ta thu c . 1 , 0), ( 2 ) (2 2= + = x x xg x x g15 Hn na, r rng h thc trn cng ng cho, 1 , 0 = xvy . ), ( 2 ) (2 2x x xg x x g + =(2.5.2) T 2 2) ( ) ( 4 y x y x xy + =ta suy ra . ,), ) (( ) ) (( ) 4 (2 2y x y x g y x g xy g + =Kt hp h thc trn vi (2.5.2) ta c , , )), ( ) ( )( ( 2 ) ( )) ( ) ( )( ( 2 ) ( ) ( 42 2y x y g x g y x y x y g x g y x y x xy g + + + + + =hay tng ng vi . , , ) ( ) ( ) ( y x xy x yg y xg xy g + =By gi ta s dng h thc trn vi0 = xv 1 = x yth nhn c . 0, 1 ) ( ) ( ) 1 ( 11 1= + = = x x g x x xg gp dng phng trnh th hai ca (2.5.1) mt ln na, ta c ng thc trn tng ng vi , 0, ) ( = = x x x gvy

== +=. 0, 00, , | | ln) (xx x xx f (2.5.3) Th li, ta thy) (x fcho bi (2.5.3) tha mn bi ton cho, vy bi ton cho c duy nht nghim cho bi cng thc (2.5.3). V d 2.6: Tm tt c cc hmR R+: ftha mn ( ) ( ) . 0 , , ) 1 )( ( ) ( ) ( ) 1 )( (1 1 1 1> + + = + + + + y x xy y x f y x f y x f xy y x f Gii: Vi0 , > y xt1 + = y x uv,1y x v + = khi0 , > v u v) 1 )( (1 1+ + = + xy y x v u ,). 1 )( (1 1+ + = + xy y x v u Vyf tha iu kin ca bi ton khi v ch khi . 0 ,), ( ) ( ) ( ) (1 1> + = + + v u v u f v u f v f u fSuy ra nghim ca bi ton l , 0), (ln ) ( > = x x A x f16 trong A l hm cng tnh bt k. Bi ton 2.5: Tm tt c cc hmR R+: , h fbit . 0 , ,) () ( ) ( ) (2 2> ||.|

\|+ ++= + + y xxy y xy x xyh y x f y f x f (14) Gii: rng phng trnh (14) l trng hp ring ca (8) vi0 , ) (1> =x x x v. 0 ), ( ) (1> =s s h s g Do theo kt qu ca Bi ton 2.1 th nghim ca bi ton cho l > + + => + =, 0 , ) ( ) ( ) (, 0 , ) ( ) (21111x c x A x A x fs c s A s h y 2 1, A Al cc hm cng tnh vcl hng s bt k. Bnhlun:Nutalyf h = thphngtrnh(14)trvphngtrnh(6).Ni cch khc, tp hp tt c cc hmftha mn (6) vi mi0 , > y xl , ) ( ) (1c x A x f + = trong A l hm cng tnh vcl hng s bt k. V d sau s a ra mt cch gii s cp phng trnh (6). V d 2.7: Tm tt c cc hmR R : flin tc tha mn ( ) . , 0 , ,) () ( ) ( ) (2 21 1 1y x y xy x xyxy y xf y x f y f x f = = ||.|

\|++ += + + (15) Gii: Gi sfl nghim ca (15), t( ) . , 0 , , ) ( ) ( ) ( ) , (1 1 1y x y x y x f y f x f y x F = = + + =

D dng kim tra c . 0 , , , 0 , , ), , ( ) , ( ) , ( ) , ( = + + = = = + + = + + z y x z y y x z y x z y F z y x F z y x F y x F (2.7.1) Vfl nghim ca (15) nn ( ),1 1 1, ,1 1 1) , (||.|

\|+ + ++= +||.|

\|+ + =z y x z y xf z y x Fy x y xf y x Fv 17 .1 1 1) , ( ,1 1 1) , (||.|

\|+ + =||.|

\|+ +++ = +z y z yf z y Fz y x z y xf z y x F Th cc biu thc trn vo ng thc (2.7.1) ta nhn c . 0 , , ,1 1 11 1 1 1 1 1 1 1 1= ||.|

\|+ + +||.|

\|+ +++ =||.|

\|+ + +++||.|

\|+ +z y xz y z yfz y x z y xfz y x z y xfy x y xf

(2.7.2) t,1 1 1,1 1 1,1 1 1z y x z y xwz y z yvy x y xu+ + ++=+ + =+ + = th (2.7.2) tng ng vi , , , ), ( ) ( ) ( ) ( w v u v w u f v f w f u f + + = +hay ( ) ( ) ( ) , , , ), ( ) ( ) ( ) ( ) ( ) ( ) ( w v u v f v v w v u f v f v v w f v f v v u f + + = + + + hay . , , ), ( ) ( ) ( ) ( ) ( ) ( v t s v f v t f v f v s f v f v t s f + + + = + + t, ), ( ) ( ) ( z v f v z f z g + = th ng thc trn tr thnh . , ), ( ) ( ) ( t s t g s g t s g + = +Mt khc,vflin tc nnglin tc, suy raas s g = ) (vial hng s bt k. Vy ta c . , , ) ( ) ( y x ax y f y x f = +tx x h ax x f + = ), ( ) ( , v th vo ng thc trn ta nhn c , , ), ( ) ( y x y h y x h = + suy rab x h = ) ( , trong bl hng s bt k.Th lib ax x f + = ) (ta thy tha mn (15), vy nghim ca bi ton l , ) ( b ax x f + = yavbl cc hng s ty . T kt qu ca Bi ton 2.1 ta dn n nh l sau. 18 nhl3:ChoR I : lgiitchvkhngafin,H cdng(7)vimi I y x e , trongR D : vi{ } I y x y x y x D e + + = , : ) ( ) ( ) ( .Khimi nghim ca (1) tha mn e + =e + + =, , ) ( ) (, , ) ( ) ( ) (12 1D s c s A s gI x c x A x A x f

(16) y 2 1, A A lcchmcngtnhvc lhngsbtk.Hnnanu kh ngc thgc dng ). ( , ) ( ) (11D s c s A s g e + =Ngclitrongtrnghp khngkhngcthphngtrnh(1)vivtri ca n l o c ch c nghim tm thng. Chngminh:Cngthc(16)csuyratktqucaBiton2.1vtrng hp khngctanhnckhngnhngaytcngthcnghim(16).By gixttrnghp khngkhngc,khitnti 2 1 2 1: , s s D s s = e nhng ) ( ) (2 1s s = ,kthpvi(16)tasuyra) ( ) (2 1 1 1s A s A = .Mtkhctheogithit hm s ( ) c y x y x A y x f y f x f y x F + + + = + + = ) ( ) ( ) ( ) ( ) ( ) ( ) , (1 o c v dolin tc v khng afin nn hm cng tnh 1Ao c, vy ta nhn c , ) (1bs s A =trongb lhngs.Tuynhin,tngthc) ( ) (2 1 1 1s A s A = vi 2 1s s = suyra . 0 = bVy (1) ch c nghim tm thng. V d 2.8: Tm tt c cc hmR R : , g flin tc sao cho . , ), ( ) ( ) ( ) (2 2y x y x g y x f y f x f = + + (17) Gii: Ta c (17) l trng hp ring ca (1) vi) , ( y x Hc dng (7) trong . sss x x x RR e = e = ,4) ( , , ) (22 Ta c khng kh ngc trn R vglin tc nn) ( ) ( ) ( ) ( ) , (2 2y x g y x f y f x f y x F = + + =lin tc hayFo c. p dng nh l 3 ta nhn c nghim ca (17) l19 e =e + =. , ) (, , ) ( ) (RRs b s gx b x A x f Mt khc theo gi thitflin tc nnx b ax x f + = , ) ( . Vy nghim ca bi ton l = + =, , ) (, , ) (s b s gx b ax x f trong b a,l cc hng s ty . V d 2.9: Tm tt c cc hmR R+: , g flin tc tha mn. 0 , ), ( ) ( ) ( ) (2 2> = + + y x y x g y x f y f x fGii: Do0 , > y xnn + R R : v. 0 , 2 ) (1> =s s s Vy bi ton cho c nghim > + => + + =0 , 2 ) (, 0 , ) (2s c s a s gx c bx ax x f vic b a , ,l cc hng s bt k. Nhn xt: Gi s chng ta tm c mt nghim ring khng tm thng) , (0 0g fca phng trnh (1), tc l ta c ng thc( ) ) , ( ) ( ) ( ) (0 0 0 0y x H g y x f y f x f = + + . Do 0gkh ngc nn ng thc trn tng ng vi ( ), ( ) ( ) ( ) , (0 0 010y x f y f x f g y x H + + = suyratrongtrnghpnyhmH cdng(7),vytacthcoiH dng(7)l iu kin cn (1) c nghim khng tm thng. Theo nh l 3 th mi nghim ca (1) c dng + =+ + =, ) ( ) (, ) ( ) ( ) (0 12 0 1c s g A s gc x A x f A x f trong 2 1, A Al cc hm cng tnh vcl hng s bt k. Mt cu hi t ra l nuHkhng c biu din dang (7) th (1) c nghim khng tm thng khng? Bi ton 2.6: Cho hmR + + ) , 0 ( ) , 0 ( : Hxc nh nh sau 20

> > e > e > + e s + >+=. 1 , ), 1 )( 1 (, 1 ), 1 , 0 ( , 1, 1 ), 1 , 0 ( , 1, 1 : ) 1 , 0 ( , , 2, 1 : 0 , ,1) , (y x y xx y yy x xy x y x y xy x y xy xy x H

(18) Chngminhrng) , ( g f lnghimca(1)nuvchnutnticchmcngtnh R R : ,2 1A Av hng sbsao cho> + + e +=, 1 , ) ( ) 1 (), 1 , 0 ( , ) () (2 12x b x A x Ax b x Ax f

>e + s =. 1 ,), 1 , 0 ( , ) 1 (, 0 ), 1 () (11s bs b s As A bs g

Hn na,Hkhng c biu din dng (7) vi kh ngc. Gii:(i) Xt trng hp0 , > y xv. 1 s + y xKhi (1) tr thnh ,1) ( ) ( ) (||.|

\|++ + = +y xg y x f y f x ftacthcoivphicaphngtrnhtrnnhlmthmcay x + tcltanhn c phng trnh Pexider, v vy ta nhn c nghim ca (1) l ) 1 , 0 ( , ) ( ) (2e + = x b x A x f v , 1 , 2 ) (1) (2s + + + =||.|

\|++ + y x b y x Ay xg y x f hay ng ng vi ) 1 , 0 ( , ) ( ) (2e + = x b x A x fv, 1 , ) ( > = s b s g (2.6.1) trong 2Al hm cng tnh vbl hng s bt k. (ii)) 1 , 0 ( , e y xv1 > + y x : Dob x A x f + = ) ( ) (2 vb y A y f + = ) ( ) (2 nn (1) tr thnh , 2 ) ( ) 2 ( ) (2b y x A y x g y x f + + = + +hay. 1 , 2 ) ( ) 2 ( ) (2> + = + t b t A t g t f (2.6.2) 21 (iii): 1 ), 1 , 0 ( > e y x p dng (2.6.1) suy ra (1) tr thnh ( ) ). ( ) ( ) 1 ( ) (2y x f y f x g b x A + = + +Tacoishngucavphicaphngtrnhtrnlmthmcax thphng trnh trn cng c dng Pexider, do nghim ca n l 1 , ) ( ) (3> + = y c y A y f(2.6.3) v ), 1 , 0 ( ), ( ) 1 ( ) (3 2e = + x x A x g b x Ahay tng ng vi ), 1 , 0 ( ), 1 ( ) 1 ( ) (3 2e + = s s A b s A s g (2.6.4) y 3Al hm cng tnh vcl hng s.Mt khc t (2.6.2) v (2.6.3) ta suy ra ). 1 , 0 ( , ) 2 ( 2 ) 2 ( ) (3 2e + = s c s A b s A s g Kt hp vi (2.6.4) ta nhn c , ) 2 ( 2 ) 2 ( ) 1 ( ) 1 (3 2 3 2c s A b s A s A b s A + = + hay ( ). ) 1 ( ) 1 (2 3A A b c = (2.6.5) (iv): 1 , > y xp dng (2.6.3) cho (1) ta nhn c ( ) , 1 , , ) 1 )( 1 ( > = y x c y x g ni cch khc . 0 , ) ( s = s c s g (2.6.6)t 2 3 1: A A A = vkthpvicccngthc(2.6.1),(2.6.3)-(2.6.6)tanhnc khng nh ca bi ton. GisH cdng(7)vi , no,trong khngc,khivi 0 , > y xv1 s + y xth ta c .1) ( ) ( ) (1||.|

\|+= + +y xy x y x t ) ( ) ( ) ( ) , ( y x y x y x F + + = , khi 22 . 1 : 0 , ,1) , (1s + > ||.|

\|+=y x y xy xy x FCh rng ta c ), , ( ) , ( ) , ( ) , ( z y F z y x F z y x F y x F + + = + +do . 1 : 0 , , ,1 1 1 11 1 1 1s + + > ||.|

\|++||.|

\|+ +=||.|

\|+ ++||.|

\|+ z y x z y xz y z y x z y x y x Vy , 1 : 0 , , ,1 11 1s + + > ||.|

\|+=||.|

\|+ z y x z y xz y y x iunylvlviz x = ,suyraH khngthcbiudindng(7)vi kh ngc. Nhn xt: Cho hmHc dng (18), khi tt c cc hmR +) , 0 ( : , g flin tc tha mn (1) l> + + e +=1 , ) 1 (), 1 , 0 ( ,) (x b cx x ax b cxx f

v >e + s =. 1 ,), 1 , 0 ( , ) 1 (, 0 ,) (s bs b s as a bs g

Ktlun:Bigingnychochngtacchtipcnikhikhhinivmtlp phng trnh hm. Tuy nhin, qua chng ta hon ton c th xy dng c mt s bi ton v phng trnh hm m c th cho hc sinh ph thng gii theo cch s cp (cc v d trong bi ging). Trc khi kt thc, chng ta c th ch thm nhng v d sau v hy vng bi ging phn no gip cc thy, c trong vic chun b cc bi ging bi dng hc sinh gii Ton. (1)Ly 3) ( x x = v uu3) ( = thtanhncbiton:Tmttccchm R g f +R : ,tha mn . 0 , ,) (1) ( ) ( ) ( > ||.|

\|+= + + y xy x xyg y x f y f x f 23 (2)Chn1 ) ( =xe x vu u = ) ( tacbiton:Tmttccchm R R : , g ftha mn ( ) . , , ) 1 )( 1 ( ) ( ) ( ) ( y x e e g y x f y f x fy x = + +Ti liu tham kho 1.J. Aczel, Lectures on FunctionalEquations and Their Applications, Academic Press New York and London, 1966. 2.B.Ebanks,GeneralizedCauchydifferencefunctionalequations,Aequationes Math, 70 (2005) 154-176. 3.C. G. Small, Functional Equations and How to Solve Them, Springer, 2007.