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In 1994, S.G. Matthews introduced the notions of partial metric and weighted quasi-metric and, in addition, he showed that both are dual. In fact, he developed a method for generating weightable quasi-metrics from partial metrics and conversely. Recently, J.J. Mi~nana and O. Valero introduced a general method with the aim of inducing quasi-metrics from partial metrics via real-valued functions in such a way that the original Matthews method can be retrieved as a special case when an appropriate function is chosen. However, they showed that the aforementioned method does not induce in general weighted quasi-metrics and, thus, they posed the question about what properties must be ful_lled by this type of functions in order to guarantee that the generated quasi-metric is weighted. In this paper, we answer such a question providing a characterization of such functions through the use of what we have called weight difference function and a new type of quadrangular triplets. Finally, a new characterization of weightable quasi-metric spaces is also given and a few properties of weightable quasi-metric spaces are discussed.

In this article, we present a study of two iterative schemes to approximate the fixed points of enriched non-expansive maps and enriched generalized non-expansive maps. The schemes introduced in this article generalize those given by Thakur et al. in (“A new iterative scheme for approximating fixed points of nonexpansive mappings,” Filomat, vol. 30, no. 10, pp. 2711–2720, 2016.) and Ali et al. in (“Approximation of Fixed points for Suzuki’s generalized nonexpansive mappings,” Mathematics, vol. 7, no. 6, pp. 522–532, 2019.) in a sense that our schemes work for larger classes of enriched mappings and the schemes given by Thakur et al. and Ali et al. reduce to a particular case of our iterative techniques. Taking inspiration from Berinde (“Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,” Fixed Point Theory Appl., vol. 2004, no. 2, pp. 97–105, 2004.) and Maniu (“On a three-step iteration process for Suzuki mappings with qualitative study,” Numer. Funct. Anal. Optim., 2020.), we also give stability results of the our procedures for enriched contractions (introduced by Berinde in 2019). Lastly, we compare the rate of convergence of our schemes with each other and the conventional Krasnoselskii iteration process used for approximating fixed points of enriched contractions along with some examples. As an application to the proposed iterative schemes, we give a few results on the solutions of linear system of equations.

In this paper we prove a theorem which ensures the existence of a unique fixed point and is applicable to contractive type mappings as well as mappings which do not satisfy any contractive type condition. Our theorem contains the well known fixed point theorems respectively due to Banach, Kannan, Chatterjea, Ciric and Suzuki as particular cases; and is independent of Caristi?s fixed point theorem. Moreover, our theorem provides new solutions to Rhoades problem on discontinuity at the fixed point as it admits contractive mappings which are discontinuous at the fixed point. It is also shown that the weaker form of continuity employed by us is a necessary and sufficient condition for the existence of the fixed point.

The purpose of the paper is to propose some new probabilistic α-minimum Ciric type contraction results. Our results are established on probabilistic generalization of metric spaces or probabilistic metric spaces. The use of class of control fucntion φ which was introduced by Choudhury et al. in 2008 helped us to deduce the result. We also get a corollary. Some illustrative examples are given here. Our results are supported by those examples. Lastly an application of integral equation is given. An important conclusion is also made at the end of the results.

We generalize the Caristi’s fixed point theorem for single valued as well as multivalued mappings defined on ametric space endowed with a graph andw-distance. Particularly, we modify the concept of the (OSC)-propertydue to Alfuraidan and Khamsi (Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metric spaceswith graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5.) which enable us to reformulated their stated graphtheory version theorem (Theorem 3.2 in Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metricspaces with graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5. ) to the case ofw-distance. Consequently,we extend and improve some recent works concerning extension of Banach Contraction Theorem tow-distancewith graph e.g. (Jachymski, J.,The contraction principle for mappings on a metric space with graph, Proc. Amer. Math.Soc.,136(2008), No. 4, 1359–1373; Nieto, J. J., Pouso, R. L. and Rodriguez-Lopez R.,Fixed point theorems in orderedabstract spaces, Proc. Amer. Math. Soc.,135(2007), 2505–2517 and Petrusel, A. and Rus, I.,Fixed point theorems inorderedL−spaces endowed with graph, Proc. Amer, Math. Soc.,134(2006), 411–418.

The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of β−G, ψ−G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points.

In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional common fixed points as coupled/tripled common fixed point results are derived from our main results.

The purpose of this paper is to utilize the property (E.A.) and the common property (E.A.) to prove some existence results on common fixed point for contractive mappings in fuzzy metric spaces which include fuzzy metric spaces of two types, namely, Kramosil and Michalek fuzzy metric spaces along with George and Veeramani fuzzy metric spaces. Our results generalize and extend several relevant common fixed point theorems from the literature. We also furnish an illustrative example. © 2011 Elsevier Ltd. All rights reserved.

In this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we define α- ψ and β- ψ condensing operators and using them we propose new fixed point results. Our results generalize and extend some comparable results from the literature. Additionally, as an application, we apply the obtained fixed point theorems to study the nonlinear functional integral equations.

In 1992 Matthews [11] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. The main difference comparing to the standard metric is that the self-distance of an arbitrary point need not be equal to zero. This notion was originally motivated by the experience of computer science. The authors showed how a mathematics of nonzero self-distance for metric spaces has been established, and leads to interesting research into the foundations of topology [5], [9]. Many authors studied partial metric spaces and their topological properties, as well as fixed point resultsWe begin with the definition of a partial metric space by Matthews [11] as follows

This chapter deals with a concise study of fixed point theorems concerning fuzzy contractive type mappings in fuzzy metric spaces. The results presented in detail were selected to illustrate the direction of research in the field from the past six decades up to most recent contribution in the subject.

In this paper, we establish some new fixed point theorems for generalized ϕ–ψ-contractive mappings satisfying an admissibility-type condition in a Hausdorff rectangular metric space with the help of C-functions. In this process, we rectify the proof of Theorem 3.2 due to Budhia et al. [New fixed point results in rectangular metric space and application to fractional calculus, Tbil. Math. J., 10(1):91–104, 2017]. Some examples are given to illustrate the theorems. Finally, we apply our result (Corollary 3.6) to establish the existence of a solution for an initial value problem of a fractional-order functional differential equation with infinite delay.

Science & Technology Asia, 25, 4, 1-11

We observe that all the results involving $$\alpha$$ -type F-contractions are not correct in their present forms. In this article, we prove some fixed point results for extended F-weak contraction mappings in metric and ordered-metric spaces. Our observations and the usability of our results are substantiated by using suitable examples. As an application, we prove an existence and uniqueness result for the solution of a first-order ordinary differential equation satisfying periodic boundary conditions in the presence of either its lower or upper solution.

We observe that the assumption of set-valued F-contractions (Sgroi and Vetro [13]) is actually very strong for the existence of fixed point and can be weakened. In this connection, we introduce the notion of set-valued ?-F-contractions and prove a corresponding fixed point theorem in complete metric spaces. Consequently, we derive several fixed point theorems in metric spaces. Some examples are given to illustrate the new theory. Then we apply our results to establishing the existence and uniqueness of solutions for a certain type of non-linear integral equations.

In this paper, we present some results on the existence of coincidence and periodic point of F-contractive mappings in the framework of modular metric spaces endowed with a graph. We also present an application to partial differential equations in order to support the theoretical results

Very recently, in order to unify the notions of fuzzy metric space and metric-like space, Shukla and Abbas introduced the concept of fuzzy metric-like space and proved some fixed-point results in this setting. In this article, we modify the notion of Cauchy sequence and completeness to generalize their results. Thus, we extend their theorems to a more general framework, which is also appropriate to generalize some recent, well-known results in this line of research. Furthermore, several examples are presented to illustrate the significance of our results.

In this manuscript, we initiated to study a notion of graphical rectangular metric space and established some fixed point theorems in the aforesaid space. We also discuss the structure of an open ball and Hausdorffness of the new metric. Suitable examples and graphs are presented to demonstrate the novelty of the results established. In the end, we apply the obtained result to show the existence of the solution of a particular type of integral equation.