Some theorems on expansion mappings and their fixed points

Demonstratio Mathematica

Demonstratio Mathematica, 1994

2007

Fixed point theorems due to Lal et al. (1996) and Jungck (1988) are used to derive two common fixed point theorems involving six mappings in complete and compact metric spaces, respectively.

Fixed Point Theory and Applications, 2010

Coincidence and fixed point theorems for a new class of contractive, nonexpansive and hybrid contractions are proved. Applications regarding the existence of common solutions of certain functional equations are also discussed.

Sedghi et al. [28] introduced í µí±†-metric space and established some fixed point theorems for a self-mapping on a complete S-metric space. In the present paper, we prove some fixed point theorems for surjection satisfying various expansive type conditions in the setting of a í µí±†-metric space. The presented theorems extend, generalize and improve many existing results in the literature.

Journal of Applied Mathematics, 2014

International Journal of Mathematics and Mathematical Sciences, 1990

Jungck [1] obtained a fixed-point theorem for a pair of continuous selfmappings on a complete metric space. Recently, Barada K. Ray [2] extended the theorem of Jungck [1] for three self-mappings on a complete metric space. In the present paper we omit the continuity of the mapping used by Ray [2] and replace his four conditions by a single condition. Our results so obtained generalize and/or unify fixed-point theorems of Jungck [1], Ray [2], Rhoades [3], Ciric [4], Pal and Maiti [5], and Sharma and Yuel [6].

Applied Mathematics and Computation, 2012

Afterward Berinde [12, Theorem 3.4] generalized the above definition and proved the following fixed point result. Theorem 1.3 [12]. Let (X, d) be a complete metric space and T : X ? X a mapping for which there exist a 2 ]0, 1[ and some L P 0 such that for all x, y 2 X dðTx; TyÞ 6 aMðx; yÞ þ L minfdðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg; ð1Þ where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. Then (1) T has a unique fixed point, i.e., F(T) = {x ⁄ }; (2) for any x 0 2 X, the Picard iteration fx n g 1 n¼0 defined by (1.1) converges to some x ⁄ 2 F(T); (3) the prior estimate dðx n ; x Ã Þ 6 a n ð1ÀaÞ 2 dðx 0 ; x 1 Þ holds for n = 1,2,.. .; (4) the rate of convergence of Picard iteration is given by d(x n , x ⁄) 6 hd(x nÀ1 , x ⁄) for n = 1,2,.. .. The contractive condition (1) is termed as generalized condition (B). Recently, Abbas and Ilić in [1] introduced the following definition: Definition 1.4 [1]. Let T and f be two self maps of a metric space (X, d). A map T is called generalized almost f-contraction if there exists d 2 [0, 1[ and L P 0 such that for all x, y 2 X,

Mathematical and Computer Modelling, 2011

a b s t r a c t Recently T. Suzuki showed that the Mizoguchi-Takahashi fixed point theorem is a real generalization of Nadler's fixed point theorem. Taking inspiration from the result of Mizoguchi and Takahashi and using the ideas of Feng and Liu, Klim and Wardowski obtained some fixed point theorems and showed that their results are different from the Reich point theorem and the Mizoguchi-Takahashi fixed point theorem. Very recently, Pathak and Shahzad introduced a class of functions and generalized some fixed point theorems of Klim and Wardowski by altering distances, i.e., via the mapping T (from a complete metric space (X, d) to the class of nonempty closed subsets of X ). In this paper we introduce a new class of functions which is a subclass of the class introduced by Pathak and Shahzad and improve some results of Pathak and Shahzad by allowing T to have values in closed subsets of X .

Abstract and Applied Analysis, 2013

We introduce the concept of the generalized -contraction mappings and establish the existence of fixed point theorem for such mappings by using the properties of -distance and -admissible mappings. We also apply our result to coincidence point and common fixed point theorems in metric spaces. Further, the fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results generalize the result of Kutbi, 2013, and several results in the literature.