Sentence examples for mappings were first from inspiring English sources

In fact, the asymptotically nonexpansive and asymptotically pseudocontractive mappings were first introduced by Goebel-Kirk [1] and Schu [2], respectively.

Scalar minimax theorems and set minimax theorems for non-continuous set-valued mappings were first proposed by Lin et al. [1].

Remark 1.2 The class of relatively asymptotically nonexpansive mappings were first considered in [4]; see also, [5] and the references therein.

The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings were first considered in Zhou et al. [24] (see also Qin and Agarwal [18], Qin et al. [20], Qin et al. [21], Qin et al. [41]).

A map (A:Hrightarrow2^{H}) is called monotone if for each (x,yin H), langleeta-nu,x- y ranglegeq0quad foralletain Ax, nu in Ay. (1.1) Monotone mappings were first studied in Hilbert spaces by Zarantonello [1], Minty [2], Kačurovskii [3] and a host of other authors.

(I2) dually, for each v ∈ T ( y ) there is u ∈ T ( x ) such that u ⪯ v. Isotone mappings were first studied by Smithson [16], and it is well known that one of these conditions may be satisfied while the other is not.

Remark 1.2 The class of relatively nonexpansive mappings was first considered in Butnariu et al. [3].

The class of κ-strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [35] in Hilbert spaces.

The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3].

The class of k-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3].

The existence of endpoints for a special kind of contractive mappings was first studied by Aubin and Siegel [1].