A FIXED POINT RESULT IN GENERALIZED METRIC SPACES WITH GRAPHS

Study on existence of fixed points of contraction and contractive (type) mappings in topological spaces is a challenging task. The main goal of this article is to deal with this challenging task. To achieve our goal, we define two new contractive type mappings, namely, h-A-contractive and h-A1-contractive mappings on a topological space X, where h : X ? X ? R+ is a function and A, A1 are two collections of implicit functions. Then, we obtain some fixed point results concerning such contractive type mappings. Finally, as an application of one of the above mentioned fixed point results, we obtain a newer version of the implicit function theorem in topological spaces.

In 2012, the notion of --contractive type mappings was introduced by Samet, C. Vetro, and P. Vetro. By using a simple method, we give some coupled fixed point results for --contractive type mappings.

We obtain a characterization of Hausdorff left K-complete quasi-metric spaces by means of α – ψ -contractive mappings, from which we deduce the somewhat surprising fact that one the main fixed point theorems of Samet, Vetro, and Vetro (see “Fixed point theorems for α – ψ -contractive type mappings”, Nonlinear Anal. 2012, 75, 2154–2165), characterizes the metric completeness.

A new, simple and unified approach in the theory of contractive mappings was recently given by Samet \emph{et al.} (Nonlinear Anal. 75, 2012, 2154-2165) by using the concepts of $\alpha$-$\psi$-contractive type mappings and $\alpha$-admissible mappings in metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called generalized $\alpha$-$\psi$ contractive pair of mappings and study various fixed point theorems for such mappings in complete metric spaces. For this, we introduce a new notion of $\alpha$-admissible w.r.t $g$ mapping which in turn generalizes the concept of $g$-monotone mapping recently introduced by Ćirić et al. (Fixed Point Theory Appl. 2008(2008), Article ID 131294, 11 pages). As an application of our main results, we further establish common fixed point theorems for metric spaces endowed with a partial order as well as in respect of cyclic contractive mappings. The presented theorems extend and subsumes various known comparable results from the current literature. Some illustrative examples are provided to demonstrate the main results and to show the genuineness of our results.

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of a saturated class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed-point results. We illustrate the concept by surveying some significant fixed-point results obtained recently for five remarkable unsaturated classes of contractive mappings. In the second part of the paper, we also identify two important classes of saturated contractive mappings, whose main feature is that they cannot be enlarged by enriching the contractive mappings.

Samet et. al. (Nonlinear Anal. 75, 2012, 2154-2165) introduced the concept of alpha-psi-contractive type mappings in metric spaces. In 2013, Alghamdi et. al. [2] introduced the concept of G-β--contractive type mappings in G-metric spaces. Our aim is to introduce new concept of generalized G-η-χ-contractive pair of mappings. Further, we study some fixed point theorems for such mappings in complete G-metric spaces. As an application, we further establish common fixed point theorems for G-metric spaces for cyclic contractive mappings.

In this paper, we prove some fixed point theorems for generalized (?-?)-contractive mappings in uniform spaces and apply them to study the existences-uniqueness problem for a class of nonlinear integral equations with unbounded deviations. We also give some examples to show that our results are effective.

In this paper, we introduce the concept of ?-?-?-contractive mappings and ?-?-?-contractive multifunctions and give some fixed point results for such mappings and multifunctions. We show that our fixed point result of ?-?-?-contractive mappings is different from that of ?-?-contractive mappings which has been proved recently by Samet, Vetro and Vetro.

Aims/ objectives: In this research, we initially present a novel mapping within the framework of cone metric spaces associated with Banach algebra, referred to as generalized (α-Ψ)-contractive type mappings. Subsequently, a number of fixed point theorems related to (α-Ψ)-contractive type mappings in these cone metric spaces over Banach algebra are generalized and expanded.

In this paper we introduce some new types of contractive mappings by combining Caristi contraction, Ćirić-quasi contraction and weak contraction in the framework of a metric space. We prove some fixed point theorems for such type of mappings over complete metric spaces with the help of φ-diminishing property. Some examples are given in strengthening the hypothesis of our established theorems.

In this paper, we introduce a ρ- Z- contraction mapping and obtained some fixed point results for such class of contractions the setting of triangular -admissible mapping in the framework of b-metric-like spaces. Our results generalize and extend some theorems in the literature. An example is given to support these results.

We establish fixed point theorems for a new class of contractive mappings. As consequences of our main results, we obtain fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. Various examples are presented to illustrate our obtained results.

In this article, we introduce some generalized contractive mappings over a metric space as extensions of various contractive mappings given by Kannan, Ciric, Proinov and G?rnicki. Some fixed point theorems have been proved for such new contractive type mappings via asymptotic regularity and some weaker versions of continuity. Supporting examples have been given in strengthening the hypothesis of our established theorems. As a by-product we explore some new answers to the open question posed by Rhoades.

We investigate in this manuscript, we study a new type of mappings so called F_s −contractive, in additionto we establish some fixed point results related to F_s −contractive type mappings in controlled type metricspaces. Also, examples are provided to illustrate our results.

Samet et al. in (Nonlinear Anal. 75:2154-2165, 2012) introduced the concepts of α-ψ-contractive type mappings and α-admissible mappings in metric spaces. The purpose of this paper is to present a new class of almost contractive mappings called almost generalized $(\alpha\mbox{-}\psi\mbox{-}\varphi\mbox {-}\theta)$ -contractive mappings and to establish some fixed and common fixed point results for this class of mappings in complete ordered b-metric spaces. Our results improve and generalize several known results from the current literature and its extension. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.