f (φ (x)) = af (x)+
b.
f(φ(x)) = a(x)f(x) + b(x), φ(x)
a(x), b(x) f
X, Y X ∩ Y = ∅ f : X → Y
x
∗
∈ X f f(x
∗
) = x
∗
.
f
(T f)(x) = f(x),
f T
f(x + y) = Af(x) + Bf(y), A, B
X, Y X ∩ Y = ∅ f : X → Y
x
∗
∈ X f f (x
∗
) = x
∗
.
f F ix(f).
f : R → R f (x) = x
3
F ix(f) = {−1, 0, 1} .

g : R → [−1; 1] g(x) = sinx
x
∗
= 0, F ix (g) = {0} .
h : R → R h(x) = x + 1 F ix(h) = ∅.
[a; b] g(x) = f(x) − x. g(x)
f (a) , f (b) ∈ [a; b] g (a) .g (b) = (f (a) − a) (f (b) − b) ≤ 0.
g(x) = 0 x
∗
∈ [a; b] g(x
∗
) = 0
f (x
∗
) = x
∗
.
X
X.
f(x)
x
X.
X g(X)
x ∈ X. g(X)
g(X)
g(x) =
f(x)
x
, x ∈ X
g(X) X. g(X)

X, g(X) X.
F (u) φ (x, y, s, t)
X × X × R × R R
F (u) u
∗
F (φ (x, y, f (x) , f (y))) = φ (y, x, f (y) , f (x)) (x, y ∈ X )
X
φ (x, x, f (x) , f (x)) = u
∗
.
f(x) y = x ∈ X
F (φ (x, x, f (x) , f (x))) = φ (x, x, f (x) , f (x)) (∀x ∈ X).
φ (x, x, f (x) , f (x)) F x ∈ X.
F u
∗
φ (x, x, f (x) , f (x)) = u
∗
(∀x ∈ X).
F (u) = u u
∗
φ,
f R,
f(x + f(y)) = f (x) + y (∀x, y ∈ R).
y = x = 0 f(f(0)) = f(0). f(0)
f. f(0)
f R. φ (x, y, f (x) , f (y)) = x + f (y)
f x+ f (x) = f (0)
f (x) = f (0) − x. f(0) = c,
f (x) = c − x c f (x) = c − x
c − (x + c − y) = c − x + y → c = 0.

f(x) = −x. f(x) = −x R
(x + f(y)) = f(x) + y. f(x) = −x
f : (0, +∞) → (0; +∞)
f(xf(y)) = yf(x) x, y
lim
x→+∞
f(x) = 0.
y = x = 1 f(f(1)) = f(1). f(1)
f. x = 1 y = f(1), f (f (f (1))) = f(1)
2
f(1) f f (1) = f(1)
2
. f
f(1) = 1. u
∗
f.
x
∗
> 0 f y = x = x
∗
f

(x
∗
)
2

= (x
∗
)

2
. (x
∗
)
2
f.
(x
∗
)
n
f x = x
∗
, y = (x
∗
)
n
f

(x
∗
)
n+1

= (x
∗
)
n+1
. (x
∗
)

n
f
n x
∗
> 1 lim
n→∞
(x
∗
)
n
= +∞, lim
n→∞
f((x
∗
)
n
) =
lim
n→∞
(x
∗
)
n
= +∞, f x
∗
> 1.
x
∗
0 < x
∗

< 1
1 = f(1) = f(
1
x
∗
.x
∗
) = f(
1
x
∗
f(x
∗
)) = x
∗
f(
1
x
∗
) → f(
1
x
∗
) =
1
x
∗
1
x
∗

1
x
∗
> 1,
u
∗
= 1 f.
φ (x, y, f (x) , f (y)) = xf (y)
xf (x) = 1 ↔ f (x) =
1
x
(x > 0).
f
f(xf(y)) =
1
x
y
=
y
x
= yf(x).
lim
x→+∞
f(x) = lim
x→+∞
1
x
= 0.
f (x) =
1

x
f : (−1; +∞) → (−1; +∞)
f(x + f(y) + xf(y)) = y + f(x) + yf(x).
f(x)
x
(−1; 0) (0; +∞).
x
∗
= 0, x
∗
∈ (−1; 0) , x
∗
∈ (0; +∞) x = y = x
∗
f(2x
∗
+ x
∗2
) = 2x
∗
+ x
∗2
. x
∗
∈ (−1; +∞) 2x
∗
+ x
∗2
∈ (−1; +∞) .
x

∗
2x
∗
+ x
∗2
−1 < x
∗
< 0 −1 < 2x
∗
+ x
∗2
< 0. x
∗
(−1; 0) 2x
∗
+ x
∗2
= x
∗
→ x
∗
∈ {−1; 0} .
(−1; 0).
x
∗
∈ (0; +∞) 0 < 2x
∗
+ x
∗2
. (0; +∞) f

x
∗
2x
∗
+ x
∗2
= x
∗
→ x
∗
∈ {−1; 0} . f
(0; +∞) .
x
∗
x = y = 0 f(f(0)) = f(0). f(0)
f(0) = 0.
φ(x, y, f(x), f(y)) = x + f(y) + xf(y)
0 = x + f (x) + xf (x) ↔ f(x) = −
x
x + 1
f(x + f(y) + xf(y)) =
y−x
x+1
y + f(x) + yf(x) =
y−x
x+1
.
(
f(x)
x

)

=
1
(x+1)
2
> 0
(∀x > −1)
f(x)
x
(−1; 0) (0; +∞) .
f(x) = −
x
x+1
F (u) u, φ(x, y, z, t)
x, y, s, t X ×X ×R×R X
R.
F ix(F )
F (φ (x, y, f (x) , f (y))) = φ (y, x, f (y) , f (x)) (x, y ∈ X) .
φ ∈
F (φ (x, x, f (x) , f (x))) = φ (x, x, f (x) , f (x)) (∀x ∈ X) .
φ(x, x, f(x), f(x)) ∈ F ix(F) ∀x ∈ X. φ
g(x) = φ(x, x, f(x), f(x)) X, X g(X)
R. g(X) g(X)
g(X) g(X)
F ix(F ), g(X)
R
(f(x)(x + 2y) + f(y))
3
= f(x) + f(y)(y + 2x) (∀x, y ∈ R).

F (u) = u
3
F (u) R.
φ (x, x, f (x) , f (x)) = f (x) (x + 2y) + f (y) X = R. φ =
s (x + 2y) + t x, y, s, t. F (u) = u
3
F ix (F ) = {0; −1; 1} .
(3x + 1) f (x) = u
∗
∈ {0; −1; 1} .
u
∗
= 0 f(x) = 0 ∀x ∈ R. f(x) = 0
R u
∗
∈ {−1; 1}
f(x) =
u
∗
3x + 1
(x = −
1
3
), lim
x→−1/3
f(x) = ∞.
f(x) R
(3x + 1)f (x) = u
∗
u

∗
∈ {−1; 1} .
f (x) = 0
f R
π sin(y
2
f(x) + f(y)) = 2(x
2
f(y) + f(x)) ∀x, y ∈ R.
φ(x, y, f(x), f(y)) = y
2
f(x) + f(y),
X = R F (u) =
π
2
sin u. F (u) R
F ix (F ) =

0; −
π
2
;
π
2

, φ (x, y, s, t) = 0.x + y
2
s + t
x, y, s, t.
(x

2
+ 1)f(x) = u
∗
u
∗
∈

0; −
π
2
;
π
2

.
u
∗
= 0 f(x) = 0 x ∈ R. f(x) = 0
R
u
∗
∈ { −
π
2
,
π
2
} f(x) =
u
∗

x
2
+1
. R
x, y. f(x) =
u
∗
x
2
+1
π sin(u
∗
y
2
(y
2
+ 1) + x
2
+ 1
(y
2
+ 1)(x
2
+ 1)
) = 2u
∗
x
2
(x
2

+ 1) + y
2
+ 1
(y
2
+ 1)(x
2
+ 1)
.
y = 0, π sin(u
∗
) = 2u
∗
(x
2
+
1
x
2
+1
)
x.
x, x, y f(x) =
u
∗
x
2
+1
f(x) = 0
N

0
→ R R.
x
∗
f(x
∗
) = x
∗
x
∗
f(x
∗
+ n) = x
∗
+ f(n) n.
f(0) = 0, f (m) =

m
x
∗

x
∗
+ f(m −

m
x
∗

x

∗
) m,
x
x f
f(i)vii = 1, , x
∗
− 1.
n = 0 f (0) = 0.
f(kx
∗
+ r) = kx
∗
+ f(r) k
r ∈ {0, 1, , x
∗
− 1}. k = 1.
k ≥ 1, k + 1.
f ((k + 1) x
∗
+ r) = f (x
∗
+ kx
∗
+ r) = x
∗
+f (kx
∗
+ r) = x
∗
+kx

∗
+f (r) = (k + 1) x
∗
+
f (r) .
f(kx
∗
) = kx
∗
+ f(r) k m
m =

m
x
∗

x
∗
+ m −

m
x
∗

x
∗
.
k =

m

x
∗

.
r = m −

m
x
∗

x
∗
= m − kx
∗
∈ {0, 1, , x
∗
− 1} m = kx
∗
+ r.
f(kx
∗
+ r) = kx
∗
+ f(r) =

m
x
∗

x

∗
+ f(m −

m
x
∗

x
∗
).
f(0) = 0, f (m) =

m
x
∗

x
∗
+ f(m −

m
x
∗

x
∗
)
m
f(x
∗

) = x
∗
n
x
∗
+ n = x
∗
+

n
x
∗

x
∗
+ n −

n
x
∗

x
∗
= (1 +

n
x
∗

)x

∗
+ n −

n
x
∗

x
∗
> 0.
f(x
∗
+ n) = f((1 +

n
x
∗

x
∗
) + n −

n
x
∗

x
∗
) = (1 +


n
x
∗

)x
∗
+ f(n −

n
x
∗

x
∗
)
= x
∗
+

n
x
∗

x
∗
+ f(n −

n
x
∗


x
∗
) = x
∗
+ f(n).
N
0
→ R R.
x
∗
f(x
∗
) = a = 0.
f(x
∗
+ n) = f(x
∗
) + f(n) n.
f f(0) = 0, f (m) = a

m
x
∗

+ f(m −

m
x
∗


x
∗
)
m. f(i)
i = 1, , x
∗
− 1.
g(m) =
f(m)
a
.x
∗
. g
g(x
∗
) = x
∗
.
g(x
∗
+ n) = x
∗
+ g(n) n.
g(0) = 0, g(m) =

m
x
∗


x
∗
+ g(m −

m
x
∗

x
∗
) ↔
f(m) = a

m
x
∗

+ f(m −

m
x
∗

x
∗
) m.
k = x
∗
x
n+k

− x
n
= a, x
0
= 0, x
1
= a
1
, , x
k−1
= a
k−1
f : N
0
→ [1; +∞)
f(3) = 8.
f(n) = f(3).f(n − 3) n ≥ 3.
g (n) = log
2
f (n) . g N
0
→ [0; +∞)
g(3) = 3.
g(3 + n) = 3 + g(n) n.
g
g(n) = 3

n
3


+ g(r), r = n − 3

n
3

∈ {0, 1, 2}.
g g(1) = a, g(2) = b a, b ∈ [0; +∞)
g(n) = 3.

n
3

+ g(r) =





3.

n
3

, khi r = 0
3.

n
3

+ a, khi r = 1

3.

n
3

+ b, khi r = 2
r = n(mod3) = n − 3.

n
3

f f(n) = 2
g(n)
g
N
0
→ N
0
f(m + f(n)) = f(f(m)) + f(n) (∗)
m, n.
m = n = 0 f(f(0)) = f(f(0)) + f(0).
f (0) = 0. m = 0 f(f(n)) = f(n). f (n)
n. F ix (f) = {0}
f(n) = 0 n f(n) ≡ 0 N
0
F ix(f) F ix(f)
x
∗
. n = x
∗

f(x
∗
+ m) = x
∗
+ f(m)
m.
w r = w −[
w
x
∗
].x
∗
,
w = f(w) = x
∗

w
x
∗

+ f(r) → f(r) = w − x
∗

w
x
∗

= r.
r r < x
∗

x
∗
F ix(f)nnr = 0. w
w = kx
∗
k f(n) n ∈ N
0
f f(n) = k
n
x
∗
n ∈ N
0
.
f(n) =

n
x
∗

x
∗
+ f(r) =

n
x
∗

x
∗

+ k
r
.x
∗
= x
∗
(

n
x
∗

+ k
r
)
r = n − x
∗

n
x
∗

∈ {0, 1, , x
∗
− 1}. k
n
=

n
x

∗

+ k
r
. k
0
= 0,
x
∗
− 1 k
1
, , k
x
∗
−1
f(n) = x
∗
(

n
x
∗

+ k
r
), r ∈ {0, 1, , x
∗
− 1}(∗∗)
f(n) ∈ N
0

.
r(n) = n(modx
∗
) = n − x
∗

n
x
∗

,
f(m + f(n)) = f(m + x
∗
(

n
x
∗

+ k
r(n)
)) = x
∗
(

m
x
∗

+


n
x
∗

+ k
r(n)
+ k
r(m)
)
= x
∗
(

m
x
∗

+ k
r(m)
) + x
∗
(

n
x
∗

+ k
r(n)

) = f(m) + f(n) = f(f(m)) + f(n)

f(m)
x
∗

=

m
x
∗

+ k
r(m)
, r
f(m)
= 0
f(n) ≡ 0 N
0
f (φ (x)) = af (x) + b.
f (φ (x)) = af (x) + b, x ∈ X.
a, b ∈ R, X φ(x)
X φ(x) ⊂ X
f
0
(x)
f(x) = f
0
(x) + g(x)
g(x) gφ(x) = ag(x).

h(u) = au + b a, b a = 1, b = 0 x
∗
h(u) f(x) = g(x) + u
,
g(x) gφ(x) = ag(x), x ∈ X.
u
∗
k(u) = αu + β α, β 0 < α = 1)
f(k(x)) = γf(x), 0 < γ = 1, (x ∈ R).
f(x) =





γ
log
α
(x−u
∗
)
.s(log
α
(x − u
∗
)), x > u
∗
0, x = u
∗
γ

log
α
(u
∗
−x)
.s
1
(log
α
(u
∗
− x)), x < u
∗
.
s(z), s
1
(z) z
f (x) = g (x) + f
0
(x) .
g(φ(x) f
0
(φ(x)) = a (g (x) + f
0
(x)) + b ↔ g (φ (x)) = ag (x) x ∈ X
f
0
(x) = u
∗
x = t + u

∗
f(α(t + u
∗
) + β) = γf(t + u
∗
) ↔ f(αt + u
∗
) = γf(t + u
∗
)
r(t) = f(t + u
∗
)
r(αt) = γr(t) (−∞ < t < +∞).
0 < γ = 1 r(0) = 0.
t > 0,
r(t) = t
log
α
γ
φ(t) → φ(αt) = φ(t).
t = α
z
φ(t) = φ(α
z
) = s(z). s(z)
s(z + 1) = s(z). s
t > 0
r(t) = γ
log

α
t
.s(log
α
t).
t < 0,
r(t) = (−t)
log
α
γ
φ(−t) → φ(−αt) = φ(−t).
t < 0,
t

= −t
r(t) = γ
log
α
(−t)
.s
1
(log
α
(−t))
s
1
(z)
r(t) =






γ
log
α
t
.s(log
α
t), t > 0
0, t = 0
γ
log
α
(−t)
.s
1
(log
α
(−t)), t < 0
s (z) , s
1
(z) z
t = x˘u
∗
f(x) =






γ
log
α
(x−u
∗
)
.s(log
α
(x − u
∗
)), x > u
∗
0, x = u
∗
γ
log
α
(u
∗
−x)
.s
1
(log
α
(u
∗
− x)), x < u
∗
s (z) , s

1
(z) z
R
f (x + a) = f (x) + b
a, b
φ(x) = x + a.
f
0
(x) =
b
a
x
f (x) =
b
a
x + g (x)
g(x)
g(x)
|a| R.
f(x) R
f (x + c) = −f (x) + b
b, c
φ(x) = x + c h(u) = −u + b
u
∗
=
b
2
.
f (x) =

b
2
+ g (x)
g(x)
g (x + c) = −g (x) (x ∈ R)
g (x + 2c) = −g (x + c) (x ∈ R)
x ∈ R g(x)
2 |c| R.
g (x) = q (x + c) − q (x)
q(x) R 2 |c| .
2 |c| ,
q(x) = −
g(x)
2
q(x) 2 |c| R
q(x + c) − q(x) = −
g(x+c)
2
+
g(x)
2
= −
−g(x+2c)
2
+
g(x)
2
=
g(x)
2

+
g(x)
2
= g(x)
f(x) =
b
2
+ q(x + c) − q(x)
q(x) 2 |c| R.
f(x) R
f(αx + β) = af(x) + b, α, β, a, b ∈ R, a, α ∈ (0; +∞)\{1}
h (x) = au + b u
∗
=
b
1−a
.
f(x) =
b
1 − a
+ g(x)
g(x)
g(αx + β) = ag(x)(x ∈ R)
g(x) =





a

log
α
(x−u
∗
)
.s(log
α
(x − u
∗
)), x > u
∗
0, x = u
∗
a
log
α
(u
∗
−x)
.s
1
(log
α
(u
∗
− x)), x < u
∗
u∗ =
b
1−a

s(z) s
1
(z) z
g(x)
x
0
∈ X
x
1
= fx
0
, x
n
= fx
n−1
= f
n
x
0
(n = 1, 2, )
{x
n
}
∞
n=0
x
0
.
d(T x, Ty) ≤ kd(x, y) (∀x, y ∈ X)
k [0; 1). T x

∗
. x
0
X {x
n
}
∞
n=0
T x
0
,
d(x
n
, x
∗
) ≤
k
n
1 − k
d(x
0
, T x
0
)
d(x
n
, x
n+1
) ≤ kd(x
n−1

, x
n
) ≤ ≤ k
n
d(x
0
, x
1
)
d(x
n
, x
n+p
) ≤
p

i=1
d(x
n+i−1
, x
n+i
) ≤ (
p

i=1
k
n+i−1
)d(x
0
, x

1
) = k
n
k
p
−1
k−1
d(x
0
, x
1
)
≤
k
n
1−k
d(x
0
, x
1
) =
k
n
1−k
d(x
0
, T x
0
)
0 ≤ k < 1 ε > 0 n

0
= n
0
(ε)
n > n
0
d(x
n
, x
n+p
) ≤
k
n
1 − k
d(x
0
, T x
0
) < ε (∀p ∈ N
∗
)
{x
n
}
∞
n=0
X, d). (X, d)
{x
n
}

∞
n=0
x
∗
∈ X.
T x
∗
= lim
n→∞
x
n
= lim
n→∞
T x
n−1
= T ( lim
n→∞
x
n−1
) = T x
∗
.
x
∗
T. p
d(x
n
, x
∗
) ≤

k
n
1 − k
d(x
0
, T x
0
).
x
∗
y
∗
T.
d(T x
∗
, T y
∗
) ≤ kd(x
∗
, y
∗
) ↔ (1 − k)d(x
∗
, y
∗
) ≤ 0
0 ≤ k < 1 d(x
∗
, y
∗

) = 0 y
∗
= x
∗
,
x
∗
T
D φ D D. C
∞
(D)
D
f = sup {|f(x)| : x ∈ D} .
h, g ∈ C
∞
(D), h < 1
h(x)f(φ(x)) = f(x) + g(x)
C
∞
(D).
D R
C
∞
(D) C(D)
D
T : C
∞
(D) → C
∞
(D)

T f(x) = h(x)f(φ(x)) − g(x), f ∈ C
∞
(D).
φ, h, g T
T C
∞
(D). f
1
, f
2
∈ C
∞
(D),
T f
1
− T f
2
 = h(x).(f
1
(φ(x)) − f
2
(φ(x))
= sup {|h(x).(f
1
(φ(x)) − f
2
(φ(x)))| : x ∈ D} ≤ h f
1
− f
2


h < 1 T T
T f = f C
∞
(D).
R
f(x
2
+ 1) = 2f(x) + 3
1
2
f(x
2
+ 1) = f(x) +
3
2
h(x) =
1
2
, g(x) =
3
2
, φ(x) = x
2
+ 1 D = R h =
1
2
< 1.
C
∞

(R) .
f
0
(x) ≡ −3
C
∞
(R) f
0
(x) ≡ −3.
f(φ(x)) = a.f(x) + b, (a, b ∈ R, |a| > 1)
φ(x) R C
∞
(R)
f
0
(x) ≡
b
1−a
R
x
2
f(
x
3
x
2
+ 1
) = (2x
2
+ 1)f(x) + 2x

2
+ 2
x
2
2x
2
+ 1
f(
x
3
x
2
+ 1
) = f(x) +
2(x
2
+ 1)
2x
2
+ 1
h(x) =
x
2
2x
2
+1
, g(x) =
2(x
2
+1)

2x
2
+1
lim
x→±∞
h(x) =
1
2
, lim
x→±∞
g(x) = 1, h(x), g(x)
R h, g ∈ C
∞
(R), h =
1
2
< 1, φ(x) =
x
3
x
2
+1
R.
C
∞
(R) . f
0
(x) ≡ −2
f
0

(x) ≡ −2

−
π
2
;
π
2

sin x.f(cos x) = 2f(x)
s
inx
2
f(cos x) = f(x)
h(x) =
s
inx
2
, φ(x) = cos x, g(x) = 0, D =

−
π
2
;
π
2

h =
1
2

< 1 D R.
C(D). f(x) ≡ 0
f(x) ≡ 0.
f(cos x) = a.f(x) +
s
inx
a |a| > 1
1
a
f(cos x) = f(x) +
s
inx
a
h(x) =
1
a
, φ(x) = cos x, g(x) =
s
inx
a
h =
1
|a|
< 1
f(x
∗
) x
0
f
0

(x) = x
sup
0≤x≤1



f
∗
(x) −
cos(cos x)−sin(cos x)−a sin x
a
2



≤
1
|a|
2
(|a|−1)
sup
0≤x≤1
|ax − cos x +
s
inx|
R
(D
n
)
∞

n=1
R
∞

n=1
D
n
= R. φ R φ(D
n
) ⊂ D
n
n h(x)
R sup {|h(x)| : x ∈ R} < 1, g(x) R
R
h(x)f(φ(x)) = f(x) + g(x)
q(x) R q
n
(x)
q D
n
C(D
n
) f
n
D
n
n x ∈ D
n
. m < n (D
n

)
∞
n=1
f
n
x ∈ D
m
. D
m
f
n
D
m
f
m
∞

n=1
D
n
= R
f(x) R
f(x) = f
n
(x) khi x ∈ D
n
.
f(x) k(x)
k
n

(x) C(D
n
) k
n
(x)
f
n
(x) D
n
f
n
C(D
n
)
k(x) ≡ f(x).
R.
φ(x) R
(D
n
)
∞
n=1
D
1
= [−m, m], m |φ(x)| ≤ m (∀x ∈ R),
D
n
= [−m − n, m + n] n n ≥ 2.
R
f(

x
x
2
+ 1
) = 2f(x) −
2x
3
+ x
x
2
+ 1
1
2
f(
x
x
2
+ 1
) = f(x) −
2x
3
+ x
2(x
2
+ 1)
h(x) ≡
1
2
, g(x) = −
2x

3
+x
2(x
2
+1)
,
φ(x) =
x
x
2
+1
.
h(x), g(x) φ(x)
R |φ(x)| ≤
1
2