Sentence examples for by a nonlinear scalarization from inspiring English sources

Despite these features, some authors demonstrated that the fixed point results proved on cone metric spaces are the straightforward outcome of the corresponding results of usual metric spaces where the real-valued metric function (d^) is defined by a nonlinear scalarization function (xi_{e}) (see [14]) or by a Minkowski functional (q_{e}) (see [5]).

Concretely, the authors showed that any cone metric space ((X, d)) is equivalent to a usual metric space ((X, d^)) if the real-valued metric function (d^) is defined by a nonlinear scalarization function (xi_{e}) (see [13]) or by a Minkowski functional (q_{e}) (see [14]).

Actually, they showed that any cone metric space ( X, d ) is equivalent to a usual metric space ( X, d ∗ ), where the real-valued metric function d ∗ is defined by a nonlinear scalarization function ξ e (see [12]) or by a Minkowski functional q e (see [13]).

Recently, scholars obtained that any cone metric space ( X, d ) is equivalent to the usual metric space ( X, d ∗ ), where the real-valued metric function d ∗ is defined by a nonlinear scalarization function  ξ e. See, for instance, [2, 3] and [4].

Actually, it has been shown that each cone metric space ( X, d ) is equivalent to a usual metric space ( X, d e ), where the real-valued metric function d e is defined by a nonlinear scalarization function [8] or by a Minkowski functional [9].

Li et al. [19] studied some generalized minimax theorems for set-valued mappings by using a nonlinear scalarization function.

In this paper, we establish the bounded rationality model M for generalized vector equilibrium problems by using a nonlinear scalarization technique.

Very recently, Li and Xia [16] investigated Levitin-Polyak well-posedness for vectorial optimization problems by using a nonlinear scalarization function.

By using a nonlinear scalarization technique, we present the sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasi-equilibrium problem.

Miglierina et al. [11] investigated several types of well-posedness concepts for vectorial optimization problems by using a nonlinear scalarization procedure.

Question Can one establish the Hölder continuity of a solution mapping to the parametric generalized vector quasi-equilibrium problem with set-valued mappings by using a nonlinear scalarization method?