Sentence examples for maps and proved from inspiring English sources

Avila and his co-authors considered a wide class of dynamical systems – namely, those arising from maps with a parabolic shape, known as unimodal maps – and proved that, if one chooses such a map at random, the map will be either regular or stochastic.

In 1974, C'iric' [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

Using this generalized distance, Suzuki and Takahashi [9] have introduced notions of single-valued and multivalued weakly contractive maps and proved fixed point results for such maps.

In [27], Gordji et al. have introduced the concept of a mixed weakly monotone pair of maps and proved some coupled common fixed point theorems for a contractive-type maps with the mixed weakly monotone property in partially ordered metric spaces.

Using the concept of Hausdorff metric, Nadler [11] introduced the notion of multivalued contraction maps and proved a multivalued version of the well-known Banach contraction principle, which states that each closed bounded valued contraction map on a complete metric space has a fixed point.

Recently, Wardowski [7] introduced a new type of contraction called F-contraction in his studies of contractive maps and proved a new fixed point theorem concerning F-contractions, for which the Banach contraction principle and some other known contractive conditions in the literature can be obtained as special cases.

It was a matter of constantly creating new configurations, or maps, and proving that, for them, the four-colour conjecture was true.By 1925 there were 22 configurations in the set; by 1968 there were 40.

In 1981, Heilpern [2] used the concept of fuzzy sets and introduced a class of fuzzy mappings, which is a generalization of the set-valued mapping, and proved a fixed point theorem for fuzzy contraction mappings in metric linear spaces.

Most recently, Samet et al. [12, 15] defined an α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.

Most recently, Samet et al. [36] have defined α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.

On the other hand, Bhaskar and Lakshmikantham have introduced the concept called mixed monotone mapping and proved coupled fixed point theorems for mappings satisfying the mixed monotone property, which is used to investigate a large class of problems, and they discussed the existence and uniqueness of a solution for a periodic boundary value problem.