Sentence examples for mappings with nonnegative from inspiring English sources

Proof Since { T i } i = 1 ∞, { S i } i = 1 ∞ are countable families of closed and quasi-ϕ-nonexpansive mappings, by Remark 1.7(3), { T i } i = 1 ∞, { S i } i = 1 ∞ are countable families of closed and quasi-ϕ-asymptotically nonexpansive mappings with nonnegative sequences k n = 1 and l n = 1, respectively.

Let (T Crightarrow C) be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences ({nu_{n}}), ({mu_{n}}) and a strictly increasing continuous function (zeta:mathbb{R}^cup{0}rightarrowmathbb{R}^cup{0}) such that (nu_{n},mu_{n}rightarrow0) and (zeta(0)=0).

Let T : C → C be a closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences { ν n }, { μ n } and a strictly increasing continuous function ζ : R + ∪ { 0 } → R + ∪ { 0 } such that ν n, μ n → 0 and ζ ( 0 ) = 0.

Let T i : C → C, i = 1, 2,..., m be m total asymptotically quasi-nonexpansive mappings with nonnegative real sequences {ν in }, {μ in } and a strictly increasing continuous function ζ i : ℜ + → ℜ + with ζ i (0) = 0 such that F : = ⋂ i = 1 m F ( T i ) is nonempty and bounded, ∑ n = 1 ∞ ( ν i n  +  μ i n ) < ∞, i = 1, 2,..., m.

Let { T i } i = 1 ∞ : C → C be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences { ν n ( i ) }, { μ n ( i ) } satisfying ν n ( i ) → 0 and μ n ( i ) → 0 (as n → ∞ and for each i ≥ 1 ) and a sequence of strictly increasing and continuous functions { ζ i } : R + ∪ { 0 } → R + ∪ { 0 } satisfying condition (1.1).

More precisely, let ((X,d)) be a metric space and let (S,T Xto X) be two mappings with two nonnegative real numbers λ and μ such that (lambda+mu<1) and d(Sx,TSx lefrac{lambda}{1-mu}d x,Sx,TSx lefrac{lambda}{1-mu}d x}{1-mu}d(x,Sx), for and (xin X).

Let {T i } be a countable family of uniformly asymptotically nonexpansive mappings from C into itself with nonnegative real sequences {ν n } such that F : = ⋂ i = 1 ∞ F ( T i ) is nonempty and bounded and ∑ n = 1 ∞ ν n < ∞.

Let G= V,E,w) be an undirected graph with nonnegative edge length function w and nonnegative vertex weight function r.

Let P : X → C be a nonexpansive retraction from X onto C. Let T1, T2, T3 : C → X be three asymptotically nonexpansive in the intermediate sense nonself mappings with F ≠ ∅ and the nonnegative sequence {r n } satisfy ∑ n = 1 + ∞ r n < + ∞.

Let G be an undirected graph with nonnegative edge lengths.

Here, (pin mathbb{R}_{2n-2}^) is a vector with nonnegative integers coordinates.