Sentence examples for has nonempty values from inspiring English sources

Obviously, has nonempty values.

From assumption (1) we know that T has nonempty values.

As F ˜ is upper semicontinuous with closed values, there exists integrable selection of F ˜. Thus T has nonempty values.

Thus the multivalued operator U is defined by U ( t ) = S F, y ∩ K ( t ), where K ( t ) = { w ∈ R | | v 1 ( t ) − w ( t ) | ≤ k ( t ) ∥ x − y ∥ }. has nonempty values and is measurable.

A subset D ⊂ X is said to be approximative if the set-valued mapping P D ( x ) = { y ∈ D : M ( x, y, t ) = M ( x, D, t ), ∀ t > 0 }, ∀ x ∈ X. has nonempty values.

For A ⊂ X, let F ( A ) : = ⋃ { F ( x ) ∣ x ∈ A }. Throughout this paper, we assume that multimaps have nonempty values otherwise explicitly stated or obvious from the context.

First we observe that under the hypotheses of Theorem 14, the mappings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_k$$\end{document} S k have nonempty values.

If has the limit ordinal property, is increasing and has nonempty closed values.

Assume that is increasing, has nonempty closed values, and satisfies one of the following hypotheses, then has maximal and minimal fixed points on.

Let ; we say that the multivalued map is a contraction if has nonempty, closed values, and there exists such that (1.2).

By what precedes, the multifunction Q has nonempty closed values and, for each fixed (tin[a,b]), the multifunction (Q t,cdot,cdot )) is lower semicontinuous in Ω.