Sentence examples for map we say from inspiring English sources

If h = i X (the identity map), we say that f has the mixed monotone property.

Given a smooth map, we say that is holomorphic to first order if is holomorphic and (4.6).

For a multivalued map, we say. is contraction [1] if there exists a constant, such that for all, (1.2).

Considering a multivalued map, we say (c) is weakly -contractive if there exist - function on and constants,, such that for any, there is satisfying (1.6)  . is weakly -contractive if there exist - function on and constants,, such that for any, there is satisfying (1.6).

Given a (family of) map s),, we say that is harmonic to first order if (4.1).

If g : X → X is an identity mapping, we say that F has the mixed monotone property.

In turn, if (Gamma:Xrightarrow2^{Y}) is a set-valued mapping, we say that Γ is upper order-preserving if (xsucccurlyeq_{X}y) implies that (Gamma y)=emptyset), or for every (y'inGamma y)) there is (x'inGamma(x)) such that (x'succcurlyeq_{Y}y').

Suppose W is a Banach space and A : W → W ∗ is a mapping, we say that A is a type ( S ) + if for every sequence { x n } n = 1 ∞ such that x n ⇀ x ∈ W and lim sup n → ∞ 〈 A ( x n ), x n − x 〉 ≤ 0. Considering the nonlinear mapping A : W n 1, p ( Z ) → W n 1, p ( Z ) ∗ defined for all x ∈ W n 1, p ( Z ) by 〈 A ( x ), y 〉 = ∫ Z | D x ( z ) | p − 2 D x ( z ) ⋅ D y ( z ) d z. (2.1).

If p 1 : M ̃ 1 → M 1 and p 2 : M ̃ 2 → M 2 are covering maps, then we say that a continuous map f ̃ : M ̃ 1 → M ̃ 2 is a homotopy lift of a continuous map f : M1 → M2 when f ̃ is the lift of a map homotopic to f. Definition 2.2.

If g is the identity mapping, then we say that F has the mixed monotone property.

Let X be a nonempty set, T : X → 2 X and α : X × X → [ 0, ∞ ) be a given mapping. We say that T is an α-admissible whenever, for each x ∈ X and y ∈ T ( x ) with α ( x, y ) ≥ 1, we have α ( y, z ) ≥ 1 for all z ∈ T ( y ).