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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