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.
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.
A subset M of X is said to be q-starshaped if there exists a q ∈ M, called the starcenter of M, such that for any x ∈ M and 0 ≤ α ≤ 1, αq + (1 - α) x ∈ M. Shahzad [20] introduced the notion of R-subcommuting maps and proved that this class of maps contains properly the class of commuting maps.
A mapping T : X → X is said to be a quasi-contraction if there exists 0 ≤ q < 1 such that for any x, y ∈ X, d ( T x, T y ) ≤ q max { d ( x, y ), d ( x, T x ), d ( y, T y ), d ( x, T y ), d ( y, T x ) }, In 1974, Ćirić [2] introduced these maps and proved an existence and uniqueness fixed point theorem.
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.
With a motivation to remove this strong condition, in this paper we introduce a new iteration scheme for a pair of hybrid mapping and prove some convergence theorems for generalized nonexpansive mappings.
Later, we introduce the notion of a G - -Meir-Keeler contrandive maprove and prove some fixed point theorems for this class of mappings in the setting of G-metric spaces.
