# МОНОТОННЫЕ ОПЕРАТОРЫ В ПАРАНОРМИРОВАННЫХ ПРОСТРАНСТВАХ С КОНУСОМ

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• 14, 2008

517

. . , . .

- . - .

1. X (, = R C), X.

1. p(x), - X, , :

1) p() = 0;2) p(x) 0 x X;3) p(x) = p(x) x X;4) p(x+ y) p(x) + p(y) x y X;5) tn t p(xn x) 0, p(tnxn tx) 0.

, 6) p(x) = 0 x = .

2. (X, p), X , p , X, (. [1]).

c . . , . . , 2008

• 40 . . , . .

(X, p) - (X, d), - d(x, y) p(x)

d(x, y) = p(x y). (1)

d(x, y), (1), : d(x+ z, y+ z) = d(x, y) x, y, z X.

|p(x) p(y)| p(x y) = d(x, y), x, y X,

p(x). -

. , .

, (. [1]).

X ( X).

2. , X - , p, X, , (X, p) .

3. K X , x K, x 6= x K 0 x / K.

K X : x > y ( y 6 x), x y K. x > ( x K) .

X - u, v = {x : u 6 x 6 v}. .

, - X, . , , - X.

• 41

4. K X ,

x1 6 x2 6 . . . 6 xn 6 . . . , (2)

xn 6 y, (3)

X.

5. K X , - 6 x 6 y p(x) p(y).

6. K X , - {x1, . . . , xn} X sup{x1, . . . , xn}, , - - .

3. , X - K, A, - X, , A , .

7. A, X,

, A(K) K; D X, x, y D x 6 y A(x) 6 A(y).

, -.

A, X, - v0 w0,

v0 6 w0 A(v0) > v0, A(w0) 6 w0. (4)

A v0, w0 . -, v0 6 x 6 w0

v0 6 A(v0) 6 A(x) 6 A(w0) 6 w0.

• 42 . . , . .

vn = A(vn1), wn = A(wn1), n = 1, 2, . . . (5)

, (4), -, . , K . - A , (5) .,

v = A(v), w = A(w),

v w {vn} {wn} -. v w .

, A(x) = x X - A - - v0 w0, (4). , - .

1. K X A - v0, w0, . - A v0, w0, , . (5) A.

2. K X, A :

1) v0, w0 X ;2) ;3) x v0, w0.

yn = A(yn1), n = 1, 2, . . . , (6)

x, y0 v0, w0 X.

. (5) (6) -

vn 6 yn 6 wn, n = 1, 2, . . . (7)

• 43

(5) - A. , (5) . -, - (. [2]). , (6) x. . 2

.

3. ( [3]). K X . A( ), - v0, w0, v0, w0, , - .

ResumeSome fixed point theorems for monotone operators in paranormed spaces

with a cone are proved.

[1] Wilansky A. Modern methods in topological sector spaces / A. Wilansky.New York, 1978.

[2] McArthur C. V. Convergence of monotone nets in ordered topological vectorspaces / C. V. McArthur // Studia Math. 34(1970). P. 116.

[3] . . /. . , . . . : - , 1984.

, ,185910, , . , 33E-mail: shirokov@petrsu.ru

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