𝝐 -Compatible Map and New Approach for Common Fixed Point Theorems in Partial Metric Space Endowed with Graph (original) (raw)
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IEEE Access, 2018
Fixed point theory is a very important tool in mathematics and applied sciences. Latterly, many application examples have been presented for communication network and computer science fields. The proposed schema can be considered as a theoretical foundation for such a type of applications. In this paper, we introduce the notion of the G m-contraction to generalize and extend the notion of G-contraction. We investigate the existence and uniqueness of the fixed point for such contractions in M-metric space endowed with a graph. Our results extend and generalize various results in the existing literature, in particular the results of Jachymski. Some examples are included, which illustrate the results proved herein. INDEX TERMS Fixed point, M-metric spaces, G m-contraction, connected graph.
A Common Fixed Point Theorem in Metric Space under General Contractive Condition
Journal of Applied Mathematics, 2013
We prove a common fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition in a metric space. Some illustrative examples are furnished to highlight the realized improvements. Our result improves the main result of Sedghi and Shobe (2007).
On some generalized contractive maps in partial metric space and related fixed point theorems
Filomat
In this paper we generalize some results of fixed point theory to partial metric spaces by using metric methods in the context of a new extension of Ekeland?s variational principle. We provide some corollaries to unify our results with other existing results in the literature along the same vein. Then, we show that our results require existence assumptions weaker than those for some well-known contractive maps, including the ones in the sense of Banach, Ciric, Song, Kannan and Hardy Rogers. Also, we provide some estimates for the distance to the fixed point set in partial metric space. In order to illustrate the strength of our fixed point theorem, we use it in order to derive a new result on coupled fixed points.
Some fixed point theorems for generalized contractive mappings in complete metric spaces
Fixed Point Theory and Applications, 2015
We introduce new concepts of generalized contractive and generalized α-Suzuki type contractive mappings. Then, we obtain sufficient conditions for the existence of a fixed point of these classes of mappings on complete metric spaces and b-complete b-metric spaces. Our results extend the theorems ofĆirić, Chatterjea, Kannan and Reich.
On fixed point theory in partial metric spaces
Fixed Point Theory and Applications, 2012
In this paper, we continue the study of contractive conditions for mappings in complete partial metric spaces. Concretely, we present fixed point results for weakly contractive and weakly Kannan mappings in such a way that the classical metric counterpart results are retrieved as a particular case. Special attention to the cyclical case is paid. Moreover, the well-posedness of the fixed point problem associated to weakly (cyclic) contractive and weakly (cyclic) Kannan mappings is discussed, and it is shown that these contractive mappings are both good Picard operators and special good Picard operators. 1 Introduction Throughout this paper, the letters R, R + , N and Z + will denote the set of real numbers, the set of nonnegative real numbers, the set of positive integer numbers and the set of nonnegative integer numbers, respectively. The celebrated fixed point theorem of Banach asserts the following. Theorem If (X, d) is a complete metric space and f : X → X is a mapping such that d f (x), f (y) ≤ αd(x, y) (.) for all x, y ∈ X and some α ∈ [, [, then f has a unique fixed point x * ∈ X. Moreover, the Picard sequence of iterates {f n (x)} n∈N converges, for every x ∈ X, to x *. In [], Kannan obtained the following extension of the aforementioned fixed point theorem of Banach to a larger class of mappings, now known as Kannan mappings. Theorem Let (X, d) be a complete metric space and let f : X → X be a mapping such that d f (x), f (y) ≤ α d x, f (x) + d y, f (y) (.) for all x, y ∈ X and some α ∈ [, [, then f has a unique fixed point x * ∈ X. Moreover, the Picard sequence of iterates {f n (x)} n∈N converges, for every x ∈ X, to x *. Another extensions of Banach's fixed point theorem were given by Kirk, Srinivasan and Veeramani in []. They obtained general fixed point theorems for mappings satisfying cyclical contractive conditions. Among other results, the following one was proven in [].
2016
In this paper, the notion of strictly (α, η, ψ, ξ)-contractive multi-valued mappings is introduced where the continuity of ξ is relaxed. The existence of fixed point theorems for such mappings in the setting of α-η-complete partial metric spaces are provided. The results of the paper can be viewed as the extension of the recent results obtained in the literature. Furthermore, we assure the fixed point theorems in partial complete metric spaces endowed with an arbitrary binary relation and with a graph using our obtained results.