Sentence examples for mappings by introducing from inspiring English sources

In 1996 Jungck again generalized the notion of compatible mappings by introducing weakly compatible mappings [4].

In this continuation, Lakshmikantham and Ćirić [22] generalized these results for nonlinear contraction mappings by introducing two ideas namely: coupled coincidence point and mixed g-monotone property.

In this continuation, Lakshmikantham and Ćirić [17] generalized these results for nonlinear ϕ-contraction mappings by introducing two ideas namely: coupled coincidence point and mixed g-monotone property.

In 2009, Lakshmikantham and Ćirić [20] generalized these results for nonlinear contraction mappings by introducing the notions of coupled coincidence point and mixed g-monotone property.

Recently, Aliouche and Fisher [13] proved two new fixed point theorems for complex non-self mappings by introducing the notion of the real functions ℱ satisfying an implicit relation in two metric spaces, which is a generalization of the Telci fixed point theorem in [12].

Salimi et al. [8] modified the notions of α-ψ-contractive and α-admissible self-mappings by introducing another function η and established some fixed-point theorems for such mappings in complete metric spaces.

On the other hand, Sessa [22] introduced the notion of weakly commuting mappings, which are a generalization of commuting mappings, while Jungck [23] generalized the notion of weak commutativity by introducing compatible mappings and then weakly compatible mappings [24].

Cho and Lan [23] considered and studied a class of generalized nonlinear random (A, η -accretive equations with random relaxed cocoercive mappings in Banach spaces and by introducing some random iterative algorithms, they proved the convergence of iterative sequences generated by proposed algorithms.

In [12], by introducing (C^{2}) mappings (alpha(x)= alpha_{1}(x),ldots,alpha_{m}(x))in R^{n times m}) and (beta(x)=(beta_{1}(x),ldots,beta_{l}(x))in R^{n times l}), we further extended the results in [11] to more general nonconvex sets with both inequality and equality constraint functions.

This is corrected by introducing a mapping-rate parameter f, defined as the fraction of reads that are mapped unambiguously to the reference.

Rhoades [28] extended Banach's principle by introducing weakly contractive mappings in complete metric spaces.