# НЕЛИНЕЙНЫЕ ОПЕРАТОРЫ В SF-ПРОСТРАНСТВАХ С КОНУСОМ

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• 15, 2009

517

. .

SF -

SF - .

1. X ( =C R), . - p(x), x X, :

1) p(x) 0, p(x) = 0 x = ;2) p(nxn x) 0 {n} {xn} X , n p(xn x) 0;

3) p(xn + yn x y) 0 {xn}, {yn}, p(xn x) 0 p(yn y) 0;

4) p(x) = p(x) x X , || = 1;5) p(xn) p(x) {xn} X ,

p(xn x) 0.

p SF - [1]., , X -

, {S(x, r)} (x X, r > 0), S(x, r) = {y X, p(y x) < r}, (X, p) SF -.

p(1x) p(2x) x X |1| |2|, p - SF -.

SF - - . F - SF - [1].

c . . , 2009

• SF - 13

2. , X SF -.

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

K X SF -X -: x > y, xy K. x > ( K) . K : {xn}, {yn} ( x y) X, K, xn x, yn y n, xn 6 yn (n = 1, 2, . . .), x 6 y.

X -, , .

M X , - , . - X

v, w = {x X|v 6 x 6 w}. (1)

v, w . , -

SF - X, . -, , X (. [3]).

1. K X , - :

x = sup{x1, x2, . . . , xn},

, .

2. K X ,

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

xn 6 y (n = 1, 2, . . .), (3)

• 14 . .

X.

3. K X -, xn 6 yn 6 zn xn u, zn u , yn u.

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

4. A, SF - X,

, A(K) K; M X, x, y M x 6 y

A(x) 6 A(y).

, -.

A, SF -- 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.

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 .

• SF - 15

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

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

2. K SF -X , - A :

1) v0, w0 ;2) ;3) 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)

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

.

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

• 16 . .

ResumeSome fixed point theorems for nonlinear monotone operators in SF -space

with a cone are proved.

[1] Michenrda B. Bernsteins "lethargy"theorems in SF -spaces / B. Michendra// J. for Analysis and its Applications. V. 22 (2003). No 1. P. 316.

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

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

, ,185910, , . , 33