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In this paper, we introduce the concepts of rational type and subrational type contractive mappings. We expand and improve some fixed point theorems obtained by Alsulami et al. (Fixed Point Theory Appl., 2015, 2015: 97). Moreover, we give an example to support our results.

We discuss, extend, improve and enrich results on simulation functions established by several authors. Furthermore, by using Lemma 2.1 of Radenovic et al. [Bull. Iran. Math. Soc., 2012, 38, 625],we get much shorter and nicer proofs than the corresponding ones in the existing literature.

In the present paper, coincidence theorems of a crisp mapping and a sequence of L-fuzzy mappings have been produced under graphic contractive conditions in connection with notions of \(D_{\alpha _{L}}\) and \(d_{L}^{\infty }\) distances on the class of L-fuzzy sets. Further, we obtain some fixed point theorems for L-fuzzy set-valued mappings and pull out a variety of current results on fixed points for fuzzy mappings and multivalued mappings in the literature. As applications, we acquire coincidence points of a sequence of multivalued mappings with a self mapping and prove the existence of solution for fuzzy integral equations.

In this paper, we prove some fixed point theorems for new type generalized contractive mappings involving $${ C }$$ -class function in complex valued $$G_b$$ -metric spaces. The obtained results generalize and improve some fixed point results in the literature.

In this paper we consider, discuss and slightly complement recent fixed point results for mappings in b-metric spaces established by Sintunavarat (J Fixed Point Theory Appl 18:397–416, 2016). Thus, all our results are with shorter proofs. In addition, an application to integral equations is given to illustrate the usability of our approach.

The paper shows a new approach to proving the recent fixed point results in ordered b-metric as well as ordered metric spaces, established by several authors, with much shorter and nicer proofs. An example is given to illustrate our results.

We combine some classes of functions with a notion of hybrid $GP_{(\Lambda,\Theta )}$ - $H$ - $F$ - contractive mapping for establishing some fixed point results in the setting of $GP$-metric spaces. An illustrative example supports the new theory.

Using the concepts of a pair upclass, ?-admissible and ?-subadmissible mappings in this paper, are proven a coupled coincidence point results for mappings F : X2 ? X and g : X ? X. In this way, recent papers [27, 33] have been reformed and generalized. Two examples are given to support the theoretical results.

In this paper we consider ( s − q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.

In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results.

We have introduced the new notions of R-weakly graph preserving and R-weakly &alpha, admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as &alpha, admissible mappings of type S, &alpha, * -admissible mappings of type S and &alpha, * - orbital admissible mappings of type S respectively. Some generalized contraction and rational contraction classes are also introduced for a pair of multivalued mappings and common fixed point theorems are proved in a b-metric space endowed with a graph. We have also applied our results to obtain common fixed point theorems for R-weakly &alpha, admissible pair of multivalued mappings in a b-metric space which are the proper extension and generalization of many known results. Proper examples are provided in support of our results. Our main results and its consequences improve, generalize and extend many known fixed point results existing in literature.

In this work, we introduce the new concepts of fuzzy (h, β)-contractive mapping via triangular (h, β)-admissible mappings. Later, we prove some fixed point results for some mappings that provide fuzzy (h, β)-contractibility and triangular (h, β)-admissibility in complete non-Archimedean fuzzy metric spaces. Some examples are supplied in order to support the applicability of our results. Our main results substantially generalize and extend some known results in the existing literature.

A new proper generalization of metric called as &theta, metric is introduced by Khojasteh et al. (Mathematical Problems in Engineering (2013) Article ID 504609). In this paper, first we prove the Caristi type fixed point theorem in an alternative and comparatively new way in the context of &theta, metric. We also investigate two &theta, metrics on CB ( X ) (family of nonempty closed and bounded subsets of a set X). Furthermore, using the obtained &theta, metrics on CB ( X ) , we prove two new fixed point results for multi-functions which generalize the results of Nadler and Lim type in the context of such spaces. In order to illustrate the usability of our results, we equipped them with competent examples.

In this paper, we prove a unique common coupled fixed point theorem for two pairs of w-compatible mappings in S b -metric spaces. We also furnish an example to support our main result.

The main idea in this article is to establish some fixed and common fixed point results for multivalued H + -type contraction mappings in symmetric spaces. New results are accompanied with illustrative examples. An application of the obtained results to probabilistic spaces is presented.

In this paper we give new and much shorter proofs of Boyd-Wong and Meir-Keeler type results. Also, we define a new form of Boyd-Wong type contraction mappings using simulation functions and obtain some sufficient conditions for the existence and uniqueness of a fixed point for such class of mappings in the setting of metric spaces. Also, we present some results regarding the property $P$ for these well-known types of self-mappings.

In this paper, we establish some fixed point theorems for convex contraction mappings in F-metric spaces. Also, we introduce the concept of (\alpha,\beta)-convex contraction mapping in F-metric spaces and give some fixed point results for such contractions. Moreover, some examples are given to support our theoretical results.

The third author was supported in part by the Serbian Ministry of Science and Technological Develop- ments (Project: Methods of Numerical and Nonlinear Analysis with Applications, grant number #174002).

The aim of this paper is to present coincidence and common fixed point results of set-valued mappings satisfying certain generalized graphic F-contractive conditions on a family of sets endowed with a graph. It is worth mentioning that these results are obtained without appealing to any form of continuity of mappings involved herein. Some examples are presented to support the results proved in this paper. Our results unify, generalize and extend various comparable results in the existing literature.

In this paper we consider, discuss, improve and generalize recent fixed point results for mappings in b -metric, rectangular metric and bb-rectangular metric spaces established by Đukic et al. (2011), George and Rajagopalan (2013) and Roshan et al. (2015). Also, we prove a Geraghty type theorem in the setting of bb-metric spaces as well as a Boyd–Wong type theorem in the framework of bb-rectangular metric spaces, in both cases, without using Hausdorff assumption. One example is given to support the results.

This paper attempts to prove fixed and coincidence point results in fuzzy metric space using multivalued mappings. Altering distance function and multivalued strong {bn}-fuzzy contraction are used in order to do that. Presented theorems are generalization of some well known single valued results. Two examples are given to support the theoretical results.

In this paper, fixed point theorems for Ciric-Matkowski-type contractions in \(\nu \)-generalized metric spaces are presented. Then, by replacing the distance function d(x, y) with function of the form \(m(x,y)=d(x,y)+\gamma \bigl (d(x,Tx)+d(y,Ty)\bigr )\), where \(\gamma >0\), results analogue to those due to Proinov (Nonlinear Anal 64:546–557, 2006) are obtained. An example is provided to demonstrate a possible usage of these results.

In this paper, we prove a generalization of Chatterjea’s fixed point theorem, based on a recent result of Pata. Also, we establish common fixed point results of Pata-type for two maps, as well as a coupled fixed point result in ordered metric spaces. An example is given to show that new results are different from the known ones.

In this paper we obtain some equivalences between cyclic contractions and non-cyclic contractions in the framework of b-metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b -metric spaces established by George et?al. (2015) and Nashine et?al. (2014). Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results.

Inspired by the recent result of N. Hussain et al. [N. Hussain, E. Karapinar, P. Salimi, F. Akbar, α-admissible mappings and related fixed point theorems, Journal of Ineaqualities and Applications 2013, 2013:114] we introduced two new type functions, which we denoted as upclass of type II and αβ -contractive mappings. Specifically, all results of the mentioned paper is possible to generalize using this two types of functions.

We show that most fixed point results obtained so far in cone metric spaces over solid non-normal cones can be easily reduced to the case of solid normal cones and, hence, their proofs can be made much simpler. Also, cone tvs-valued spaces over solid cones are not an essential generalization of cone metric spaces. These results are consequences of the simple fact that each solid cone in a topological vector space is in fact normal under a suitably defined norm. The proof follows by using the technique of Minkowski functional. As an application of these results, we prove an extension of the classical Nemytzki-Edelstein fixed point result to (tvs)-(b)-cone metric spaces over solid cones.

In this manuscript we investigate the existence of start-points for the generalized weakly contractive multi-valued mappings in the setting of left K-complete quasi-pseudo metric space. We provide an example to support the given result.

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.

In this manuscript we discuss, consider, generalize, improve and unify several recent results for so-called F-contraction-type mappings in the framework of complete metric spaces. We also introduce ( &phi, F ) -weak contraction and establish the corresponding fixed point result. Using our new approach for the proof that a Picard sequence is a Cauchy in metric space, our obtained results complement and enrich several methods in the existing literature. At the end we give one open question for F-contraction of Ćirić-type mapping.

In this paper we obtain a best approximations theorem for multi-valued mappings in G -convex spaces. As applications, we derive results on the best approximations in hyperconvex and normed spaces. The obtain results generalize many existing results in the literature.

In this article, we establish some new results on Jungck-Ćirić-Wardowski type mappings in complete metric spaces. Using our new approach, the conditions (W2) and (W3) of Wardowski do not apply for the proof of Cauchyness of the Picard-Jungck sequence. Our results generalize, improve and complement several results in the existing literature.

The main purpose of this paper is to generalize, improve and complement several famous results in b-metric spaces. Moreover, an essential estimation of b-quasi-contraction in b-metric spaces is given. We also establish some new results for multi-valued mappings in the metric and b-metric concept. We explore some different proof techniques which provide short proofs of the results.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

The purpose of this paper is to improve some recent fixed point results from Refs. Parvaneh et al. (Sahand Commun Math Anal 7(1):85–104, 2017), Paunovic et al. (J Nonlinear Sci Appl 10(8):4162–4174, 2017) and Faraji and Nourouzi (Sahand Commun Math Anal 7(1):49–62, 2017) in the context of b-metric spaces. The techniques used in our proofs are quite different and most remarkably some of the proofs become simpler and more straightforward. Some examples are used to show that the improvements are indeed genuine. In addition, by using our results, we obtain the existence and uniqueness of a solution to a class of nonlinear integral equations. Furthermore, we provide the precise mathematical expression of the solution.

In this paper, we present some fixed point theorems for a class of contractive mappings in b-metric spaces. We verify the T-stability of Picard’s iteration and the P property for such mappings. We also give an example to support our assertions. In addition, by using our results, we obtain the existence and uniqueness of solution to some ordinary differential equations with initial value conditions. Further, we provide the precise mathematical expressions of solutions to such equations.

In 2000, Branciari replaced the triangle inequality by a more general one which today is known as the rectangular inequality and introduced the notion of generalized metric space or rectangular metric space. Subsequently Azam, Arshad, and Beg introduced the concept of rectangular cone metric space and proved fixed point results for Banach-type contractions in rectangular cone metric spaces. In this paper, we establish fixed point results for mappings that satisfy a contractive condition of Perov type in rectangular cone metric spaces.

In the publication of this article [Fixed Point Theory Appl. 2017:26, 2017], there is an error in Section 3. The error: Corollary 3.22 Let $( X,\sigma_{b} ) $ be a complete b-metric-like space with parameter $s \ge 1$ , and let f, g be two self-maps of X with $\psi \in \Psi $ , $\varphi \in \Phi $ satisfying the condition $$ \psi \bigl( \alpha_{qs^{p}}\sigma_{b} ( fx,fy ) \bigr) \le \lambda \psi \bigl( M ( x,y ) \bigr) $$ for all $x,y \in X$ , where $M ( x,y ) $ is defined as in (3.15) and $q > 1$ . Then f and g have a unique common fixed point in X. Should instead read: Corollary 3.22 Let $( X,\sigma_{b} ) $ be a complete b-metric-like space with parameter $s \ge 1$ , $f:X \to X$ be a self-mapping, and $\alpha :X \times X \to \mathopen[ 0,\infty \mathclose) $ . Suppose that the following conditions are satisfied: Then f has a fixed point $x \in X$ . Moreover, f has a unique fixed point if property $U_{qs^{p}}$ is satisfied. This has now been included in this erratum.

In this paper, we introduce the notions of α - ? ? -contractive and cyclic α - ? ? -contractive mappings and establish the existence and uniqueness of fixed points for such mappings in complete metric-like spaces. The results presented here substantially generalize and extend several comparable results in the existing literature, in particular those of Karapinar and Salimi (2013). As an application, we prove new fixed point results for ? ? -graphic and cyclic ? ? -graphic contractive mappings. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

In this article we introduce the notion of the multivalued fuzzy mappings satisfying w.l.b property and l.b properties and prove some results for multivalued generalize contractive fuzzy mappings in ordered-cone metric spaces without the assumption of normality on cones. We generalize many results in literature.