Sentence examples for be a retraction map from inspiring English sources

Let (rcolonmathcal{U}to X) be a retraction map.

Let X be a compact ANR-space and (rcolon Xto B) be a retraction map.

In view of the arguments presented in the foregoing section for (varphiin A(B,B)), we denote by (tilde{varphi}in A X,X)) the map defined by the formula (tilde{varphi}=i circvarphicirc r), where r is a retraction map and i is an inclusion.

Let be a retraction and the normalized duality mapping on.

Let (gamma: Xrightarrow C) be a retraction, that is, a continuous mapping such that (gamma(x)=x) for all (xin C).

As a result, the coincidence equation L x = N x is equivalent to x = Ψ x, where Ψ = P + J Q N + K P ( I − Q ) N. Let ρ : X → C be a retraction, that is, a continuous mapping such that ρ ( x ) = x for all x ∈ C. Put Ψ ρ = Ψ ∘ ρ.

Then for every u ∈ P ∖ , there exists a positive number σ ( u ) such that ∥ x + u ∥ ≥ σ ( u ) ∥ x ∥ for all x ∈ P. Let γ : X → P be a retraction, that is, a continuous mapping such that γ ( x ) = x for all x ∈ P. Set Ψ = P + J Q N + K P, Q N. and Ψ γ = Ψ ∘ γ.

Then for every u ∈ C ∖ { 0 }, there exists a positive number σ ( u ) such that ∥ x + u ∥ ≥ σ ( u ) ∥ x ∥, for all x ∈ C. Let γ : X → C be a retraction, i.e., a continuous mapping such that γ ( x ) = x for all x ∈ C. Denote Ψ : = P + J Q N + K P ( I − Q ) N, and Ψ γ : = Ψ ∘ γ.

Let D be a nonempty subset of C. Let (Q Crightarrow D) be a retraction and J be the normalized duality mapping on X.

Let E be a smooth Banach space and C be a nonempty subset of E. Let Q : E → C be a retraction and j be the normalized duality mapping on E. Then the following are equivalent: (1) Q is sunny and nonexpansive;   (2) ∥ Q x − Q y ∥ 2 ≤ 〈 x − y, j ( Q x − Q y ) 〉, ∀ x, y ∈ E ;   (3) 〈 x − Q x, j ( y − Q x ) 〉 ≤ 0, ∀ x ∈ E, y ∈ C.  .

Let E be a smooth Banach space and let C be a nonempty subset of E. Let Proj C : E → C be a retraction and J φ be the duality mapping on E. Then the following are equivalent: (1) Proj C is sunny and nonexpansive;   (2) 〈 x − Proj C x, J φ ( y − Proj C x ) 〉 ≤ 0, ∀ x ∈ E, y ∈ C ;   (3) ∥ Proj C x − Proj C y ∥ 2 ≤ 〈 x − y, J φ ( Proj C x − Proj C y ) 〉, ∀ x, y ∈ E.  .