[PDF] [PDF] Cours Équations fonctionnelles Pierre Bornsztein Table des matières

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f(x+y) =f(x) +f(y) ????R? ??X???Y?

1 =f(-1)?

?????f-1? ????f-1:Y→X?? ?????? ??? ? R -?? ? ??????? ????R+? ?????f-1:x?→ -⎷ x? a,b?I? ??a < b?????f(a)6f(b)??????f(a)< f(b)?? ????a,b?I? ??a < b?????f(a)>f(b)??????f(a)> f(b)?? ????? ???I? ??a < b?????f(a)< f(b)? ?? ??b < a?????f(b)< f(a)? ????? ???? ???? ??? ???? ?? ? Y? ??????x,y?X???? ???f(x) =a??f(y) =b? ??????f-1(a) =x??f-1(b) =y? ??? ??????? ???Y? ????x?R? ?? ???f(f(x)) =x? ? ??f(a)> a?????? ?? ?? ???? ??????f(f(a))> f(a)? ?? ????a > f(a)?? ??? ??? ?? ? ????f(a) =a?J ?? ???? ????x?X? ?? ?(-x)?X? ?? ???? ????x?X? ?? ?f(-x) =f(x)??????f(-x) =-f(x)?? n??? ?????? ????x,y?R? f(2x) =f³ sin³π 2 x+π 2 y´´ +f³ sin³π 2 x-π 2 y´´ f(x2-y2) = (x+y)f(x-y) + (x-y)f(x+y) f(2x) = 2f³ sin³π 2 x´´ f(-2x) =f³ sin³ 2 x+π´´ +f³ sin³ 2 x-π´´ = 2f³ sin³π 2 x´´ =f(2x) ??? ??????f??? ?????? f(-y2) =yf(-y)-yf(y) =y(f(-y)-f(y)) = 0 ?? ???? ????x?X? ?? ?x+T?X ?? ???? ????x?X? ?? ?f(x) =f(x+T) ? ????a?? ????? ????f:R→R????? ???? ???? ???? ????x? f(x+a) =1 2 +p f(x)-(f(x))2 1 2

6f(x)61???? ????x?R?

????g:x?→f(x)-1 2 ? ?????? ???? ???? ????x?06g(x)61 2 g(x+a) =r 1 4 -(g(x))2 ??? ?????(g(x+a))2=1 4 -(g(x))2? ?? ???? (g(x+ 2a))2=1 4 -(g(x+a))2= (g(x))2 ???? ???? ??? ??????? ???????f:x?→1 2

¯¯sin¡π

2 x¢¯¯+1 2 ?J x,y? f(x-f(y)) = 1-x-y 2 2 ?J x-y=a? f(x) =ax+f(0)? ?? ??? ?????? ???f??? ????? f(x+y) =x+f(y) ??x??y? ???? ?? ?????? ?? ?????? ???? ?????? ?????? ??? ?????? ???? ????x,y?R? x+f(y) =y+f(x)? ?? ??? ??????? ?f(y)-f(x) =y-x? ?? ?????? ????? ???f??? ? f(x+y) =f(x) +f(y) ?????? ????f(1) =a? ???? ?????f(2) =f(1+1) =f(1)+f(1) = 2a? ?? ????f(3) =f(2+1) =f(2)+f(1) = n? ???? ???? ???????p,q????q >0,?? ? ? ap=f(p) =fµ q×p q =qfµp q ????f³p q =ap q ???f(0)??f(1)? ?????? ???? ???? ????x,y?N? f(f(n) +f(m)) =n+m ????n=m? f(mn) =f(m)f(n) f(1)< f(2) = 2? ????f(0) = 0??f(1) = 1? ?? ????f(3) = 3+k? ??k?N? ?????f(6) =f(2)f(3) = 6+2k? ????f(5)65+2k? ??? ?????f(10) =f(2)f(5)610+4k? ????f(9)69+4k??f(18)618+8k? ?? ??????f(15)6

15+8k? ????? ??????? ????? ?? ?f(5)>5+k????f(15) =f(3)f(5)>(5+k)(3+k)? ??

? ????n?N????? ?? ??????? ???f(2n+ 1) = 2n+ 1? ????? ? f

¡2n+1+ 2¢=f(2)f(2n+ 1) = 2n+1+ 2

n >0? ???? ????n?N? ?? ???f(n)>g(n)? ??????? ???f=g? ????? ???? ???? ??????a? ?? ?f(a) =g(a)?J ?????? ?? ?0< b??f(0)< f(b)? ?????? ?? ?f(c) +f(0) = 2f(b)? ?? ????f(c) =f(b) + (f(b)-f(0))> f(b)> f(0)? ?? ??? ?????? ?? ??????a= 0? ?? ?????a < b < c??f(a) +f(c) = 2f(b)?J ????a? ?? ?limn→∞f(xn) =f(a)? ???? ???? ???? ????x? f(x) =fµ x 2+1 4 ????g:x?→x2+1 4 ? ?? ??? ?????? ?? ??????? ???? ???? ????a?£-1 2 ,1 2 a6g(a)61 2 ?? ???? ???? ????a?£1 2 1 2

6g(a)6a???

? ????x?£-1 2 ,1 2 2 2 2 ? ??? ????? ?? ?????(f(xn)) 2 2 ,1 2 2 2 2 I?? ????[a]?? ?????? ??????? ?? ????a? ?????[a]6a <[a] + 1? a = lima→-∞[a] a 1? 10 n??x?n=[10nx]+1 10 n? ???? ???? ????x,y?R? f(x+y) =f(x) +f(y) q? ?? ?f(q) =qf(1)? ????x? ?? ? ????f(xn) =xnf(1)???? ????n? ?? ????limn→+∞f(xn) =xf(1)? ??? x nf(1) =f(xn)6f(x)6f¡x?n¢=x?nf(1) ?? ??????? ??????n????+∞? ?? ?????xf(1)6f(x)6xf(1)? ????f(x) =xf(1)? ??? ? |f(x)-f(r)|67(x-r)2 i? {0,1,...,n}? ?? ????ai=a+ib-a n a i+1-ai=b-a n |f(a)-f(b)|6n-1X i=0|f(ai)-f(ai+1)|67n-1X i=0(ai+1-ai)2=7(b-a)2 n |x-rn|61 10 n? ????? ? |f(x)-c|=|f(x)-f(rn)|67(x-rn)267 10 2n R? ???I? ?? ???? ???? ?????x,y?f(x+y) +f(x-y) = 2f(x)f(y) ??limx→+∞f(x) = 0 ???? ????a,y????? ? f(a+ 2y) +f(a) = 2f(a+y)f(y)

0?????? ???? ???? ???? ????x?

f(x)-2f(tx) +f¡t2x¢=x2 ?????? ???? ???? ??????n>1? g(tx)-g¡t2x)¢=t2x2 g¡tn-1x¢-g(tnx) =t2(n-1)x2 g(x)-g(tnx) =x2³

1 +t2+...+t2(n-1)´

=x21-t2n 1-t2 ?????? ??? ???? ???? ????x?g(x) =x2

1-t2? ??????f(x)-f(tx) =x2

1-t2? f(x)-f(tnx) =x2

1-t2³

1 +t2+...+t2(n-1)´

=x21-t2n (1-t2)2 ??? ?? ??????? ??????n????+∞? ?? ?????f(x)-f(0) =x2 (1-t2)2? ???? ????x?R? ??????f??? ?? ?? ?????x?→x2 f n+1(x) =f(fn(x))? n?N,?? ???Un+2=aUn+1+bUn? U n+k=a1Un+k-1+a2Un+k-2+...+akUn ???? ???? ???? ??????n>0? U n=kX i=1λ iαni f:R+→R+????? ???? ???? ???? ????x>0? f(f(x)) +af(x) =b(a+b)x ??????n>0?xn+1=f(xn)? ?????xn+2+axn+1-b(a+b)xn= 0? X x?[0,1]? f(2x-f(x)) =x ?????? ?? ? ??????? ???[0,1]? ?? ????? ???? ????x?[0,1],?? ?f(g(x)) =x? ???? ???? ??????n>0? ?? ????gn??n????? ?????? ??g? ????x?[0,1]? ?? ?g(g(x)) = ????x?[0,1]??n?N?gn(x) =ng(x)-(n-1)x=n(g(x)-x) +x? g(x) =x? ??????f(x) =x? J ???? ??????n >0? f(f(n)) =n+ 1987 ?? ????g=f◦f? f(b)?????a+ 1987 =f(f(a)) =f(f(b)) =b+ 1987? ????a=b? m= 1??f(k) =f(f(n)) =n+1987? ???? ?? ?????? ???? ?? ? ?????k=f(n)? ???? ?????? ???????J ??f???????f(a) =a? ?? ???? ???? ?????x,y?f(xf(y)) =yf(x)? ??limx→+∞f(x) = 0 x f(a) =f(b)? ?????af(1) =f(f(a)) =f(f(b)) =bf(1)? ?? ????a=b? ?? ??? ?????? ???f ??????? ????? ???? ????y >0? ?? ?f³ f³y f(1)´´ =y? ?? ??? ?????? ???y????? ?? xf(x).?????? ???? ????x >0? ?? ?xf(x)?F? ?????f(1)>0? ?? ? ?????c= 1? ?? ????f(1) =f(c) = 1? ??????x,y?F? ?? ? ????f(x) =x??f(y) =y? ???? ?????f(xy) =f(xf(y)) =yf(x) = yx? ?? ??? ?????? ???xy?F? ??????? ????? ???????1?F? ?? ? ?????? ????x?F?

1 =fµ1

x =fµ1 x =xfµ1 x ?? ??? ?????? ???f¡1 x

¢=1

x x ?F? ???? ???? ??????n>0? ?? ?f(an) =an? ???? ?????limn→+∞f(an) = +∞? ?? ??? ??????? ?? ????b=1 a

P(0) = 0??? ???? ????x?R?

P

¡x2+ 1¢= (P(x))2+ 1

I???????P(0) = 0? ?? ?????? ???P(1) = 1? ????P(2) = 2??P(5) = 5? ???? ?????? ??? ??? ???? ???? ??????n>0?P(an) =an? ???f(y)?? ?? ???? ??? ??????? ???y? ?????? ???? ???? ???? ?????x,y>0? f(x)f(y) =yαf³x 2 +xβf³y 2 ? ????x=y= 0? ?? ?????f(0) = 0? y

αf³x

2 +xβf³y 2 =xαf³y 2 +yβf³x 2 xα-xβ´ f³y 2 yα-yβ´ f³x 2 ? ??α?=β?????? ????x,y?R+?\{1}? ?? ? ? f ¡x 2 x

α-xβ=f¡y

2 y

α-yβ

2 x c??? ???? ???? ???? ????x?R+?\{1 2 f(x) =c³

2αxα-2βxβ´

2 ,1}? ??????? ? ? c 2³

2αxα-2βxβ´³

2αyα-2βyβ´

=cyα³ xα-xβ´ +cxβ³ yα-yβ´ =c³ yαxα-xβyβ´ y?R+?\{1 2 ??? ?????f(x) = 0???? ????x?R+\{1 2 2 ??(f(2))2=¡2α+ 2β¢f(1) = 0? ????f(1) =f¡1 2 2

¢=1

2 x-α(f(x))2? ?????? ???? ???? ?????x,y >0? f(x)f(y) =1 2 yαx-α(f(x))2+1 2 xαy-α(f(y))2 f(x) x

α=f(y)

y c

2xαyα= 2c³xy

2 c= 21-α? 2 f

¡x3+x¢6x6(f(x))3+f(x)

f(n+k) +f(k-n) = 2f(k)f(n) f

µx+y

2 =f(x) +f(y) 2 ??f(0) = 0??f(1) = 1? ??? ???? ????x,y?[0;1]????x6y? ?? ?f¡x+y 2

¢= (1-a)f(x) +af(y)?

f(xf(y)) =f(x)quotesdbs_dbs44.pdfusesText_44

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