Since g is continuous at y, so y is a fixed point of g.
Let F : X × X ⟶ X and g : X ⟶ X be two mappings such that g is continuous and nondecreasing, F ( X × X ) ⊆ g ( X ), F has the mixed g-monotone property on X and the pair ( g, F ) is compatible.
Let F : X × X ⟶ X and g : X ⟶ X be two mappings such that g is continuous and nondecreasing, F ( X × X ) ⊆ g ( X ), F has the mixed g-monotone property on X and the pair ( g, F ) is commutative.
We show that if G is continuous then there is a unique Q for which the error G - Q has a strong minimality property involving not only the L∞-norm of G - Q but also the suprema of its subsequent singular values.
Since g is continuous at u, we have that u is a fixed point of g.
(3) If (g) is continuous, then (T_g) is continuous.
Moreover, if g is continuous, then f is uniformly continuous.
However, the argument given by the author to prove that g is not G-continuous at ω is not correct: assuming that g is continuous at ω, it is proved that ({ ffx_{n}}) converges to (omega=fomega), but this does not mean that f is G-continuous at ω (this property must be demonstrated for all sequence ({y_{n}}) converging to ω).
In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem {−Δu=f x,u in Ωu=g(x on ∂Ω, where Ω is a bounded symmetric domain in RN, N⩾2, and f:Ω×R→R is a continuous function of class C1 in the second variable, g is continuous and f and g are somehow symmetric in x.
By the continuity of g, we have G is continuous.
Assume that F and g satisfy the following conditions: (1) F ( X × X ) ⊆ g ( X ), (2) F has the mixed g-monotone property, (3) F is continuous, (4) g is continuous and commutes with F. .
