Sentence examples for proved a contraction from inspiring English sources

Banach [5], in 1922, proved a contraction principle, this key principle ensures the existence and uniqueness of fixed point theorem for Banach contraction.

Recently, Miheţ [14] enlarged the class of fuzzy contractive mappings of Gregori and Sapena [7] and proved a fuzzy Banach contraction result for complete non-Archimedean fuzzy metric spaces [7] (see also Vetro [15]).

Later, the authors in [15] proved a more reasonable contraction principle in a partial metric space.

In particular, Heilpern introduced in [1] the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework.

Later, the authors in [17] proved a more reasonable contraction principle in the partial metric space in which they used self-distance terms.

Later on, Wardowski and Van Dung [18] gave the idea of F-weak contraction and proved a theorem concerning F-weak contraction.

In this regard, starting from a function F : ( 0, ∞ ) → R satisfying some suitable properties, Wardowski provided in [11] a new type of such a contractive mapping, namely F-contraction, and proved a fixed point theorem concerning F-contractions on a complete metric space.

Recently, Wardowski [24] introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle [1].

On the other hand, in 2012, Wardowski [7] introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle.

In 2012, Wardowski [23] introduced a different type of contraction called an F-contraction and proved a new fixed point theorem, which generalizes the Banach contraction principle in a new way with supporting examples.

Wardowski and Van Dung [3] introduced the notion of a F-weak contraction and proved a fixed point theorem for F-weak contractions.