Sentence examples for by a normal cone from inspiring English sources
Let X be a Banach space ordered by a normal cone (Ksubset X).
From now on, we always suppose that E is a noncommutative Banach space with a partial ordering ≲ induced by a normal cone P with the normal constant N.
We establish a fixed point theorem for decreasing and convex operators in a probabilistic Banach space partially ordered by a normal cone.
Throughout the paper, we assume that E is a real Banach space with a partial order introduced by a normal cone P of E. Take (hin E), (h>theta), (P_{h}) is given as in the introduction.
We assume that E is a real Banach space with a partial order introduced by a normal cone P of E. Take (hin E), (h>theta), (P_{h}) is given as in the preliminaries.
In this paper, unless specified otherwise, X expresses a real ordered Hilbert space with an inner product 〈 ⋅, ⋅ 〉, a norm ∥ ⋅ ∥, a zero element θ, a normal cone P with normal constant N > 0 and a partially ordered relation ≤ defined by a normal cone P. For x, y ∈ X, x and y are said to be comparable to each other if and only if x ≤ y (or y ≤ x ) holds (denoted by x ∝ y for x ≤ y and y ≤ x ) [23].
is a normed linear space with the maximum norm and partially ordered by the cone. is a normal cone in.
Since P is a normal cone and by induction, we get that ∥ y n − x n ∥ ≤ C N N N + 1 ) n ∥ y 0 − x 0 ∥, (3.1).
The following obviously holds in an arbitrary (cone) metric type space: We will sometimes need the continuity of metric-type function D in one variable: or in two variables: The last property always holds in the case of an MTS ( X, D, K ) generated by a CMS ( X, d ) over a normal cone, see Example 2, but not in general, as the following example shows.
Let (e(t)=t^{alpha-n+1}), we define a normal cone of (C[0,1]) by P= bigl{ xin C[0,1]:x(t)geq0leqleq tleq1 bigr}, also define a component of P by Q_{e}= biggl{ xin P: mbox{there exists } D>1, frac{1}{D}e(t)leq x t)leq De(t), tin[0,1] biggr}.
So by renorming the we can suppose is a normal cone with constant one.
