Sentence examples for mappings to generalize from inspiring English sources

Jungck [4] coined the term compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting maps are compatible but the converse is not true.

In this paper, we introduce the notion of -contractive multivalued mappings to generalize and extend the notion of α-ψ-contractive mappings to closed valued multifunctions.

Jungck [15] defined the notion of compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true [15].

The purpose of this paper is to introduce the notion of -contractive multivalued mappings to generalize and extend the notion of α-ψ-contractive mappings to closed valued multifunctions and to provide fixed-point theorems for -contractive multivalued mappings in complete metric spaces.

In this work, we extend the concept of generalized α-ϕ-Geraghty contraction type mappings to generalized α-ϕ-Geraghty proximal contraction mappings to the case of non-self mappings.

In this paper, we exploit this concept for contractive mappings [36] to generalize, extend and improve some classical fixed point results for two, three and four mappings in the framework of an ordered complete dislocated metric space X.

Afterward, Jungck [13] introduced the concept of compatible single-valued mappings in order to generalize the concept of weak commutativity by Sessa [12] and showed that weakly commuting mappings are compatible, but the converse is not true.

In 2008 Suzuki [8] introduced a new type of mappings in order to generalize the well-known Banach contraction principle.

Many mathematicians (e.g., [1, 2, 3, 4, 5, 6]) proved several fixed point theorems to explore some new contraction-type mappings in order to generalize the classical Banach Contraction Principle.

In the sequel, we shall denote by ℕ the set of nonnegative integers, that is, (mathbb{N}={0,1,2,ldots}). We next define the concept of α-admissible mappings which have been recently introduced by Samet [5] and used by many authors to generalize contraction mappings of various types; see [6 8] for details.

Recently, Imdad and Soliman [15] introduced fixed point theorems for an asymptotically regular semigroup of uniformly generalized Lipschitzian mappings which generalize the results due to Jen-Chih Yao and Lu-Chuan Zeng [14].