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Introduction/purpose: In this paper, several critical remarks are presented concerning the paper of Iqbal & Rizwan: Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance from 2020. Methods: Conventional theoretical methods of functional analysis. Results: It is shown that their use of the non-decreasing "control" function F instead of a strictly increasing one in Wardowski-type results usually produces contradictions. Conclusion: It is shown that such results can be obtained in a more general class of metric-like spaces, where strict monotonicity is the only assumption that has to be imposed on the function F. An example is presented showing that the obtained results are stronger than the classic ones.

In this paper, inspired by the idea of Meir–Keeler contractive mappings, we introduce Meir–Keeler expansive mappings, say MKE, in order to obtain Krasnosel’skii-type fixed point theorems in Banach spaces. The idea of the paper is to combine the notion of Meir–Keeler mapping and expansive Krasnosel’skii fixed point theorem. We replace the expansion condition by the weakened MKE condition in some variants of Krasnosel’skii fixed point theorems that appear in the literature, e.g., in [T. Xiang, R. Yuan, A class of expansive-type Krasnosel’skii fixed point theorems, Nonlinear Anal., Theory Methods Appl., 71(7–8):3229–3239, 2009].

New fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. Since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces.

We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

In this paper, we improve some recent coupled fixed point results for single-valued operators in the framework of ordered b-metric spaces established by Bota et al. [M-F. Bota, A. Petrusel, G. Petrusel and B. Samet, Coupled fixed point theorems for single-valued operators in b-metric spaces, Fixed Point Theory Appl. (2015) 2015:231]. Also, we prove that Perov-type fixed point theorem in ordered generalized $b$-metric spaces is equivalent with Ran-Reurings-type theorem in ordered b-metric spaces.

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.

We obtain some fixed point theorems for mappings acting in b-metric spaces. The results extend those obtained in [R.P. Agarwal, E. Karapınar, A.F. Roldán-López-de-Hierro, On an extension of contractivity conditions via auxiliary functions, Fixed Point Theory Appl., 2015:104, 20015] using families of control functions, here also through conditions that involve α-admissibility of type S. We furnish an illustrative example to demonstrate the validity of the hypotheses and the degree of usefulness of our results. As an application, the existence of solution for functional equations arising in dynamic programming is discussed, followed by suitable examples.

We bring in the notion of rational g-ω-weak contractions in metric-like spaces and demonstrate common fixed point results for such mappings in 0-σ-complete metric-like spaces. Examples are given to support the usability of our results and to show that they are improvements of some known ones. An application to second-order differential equations is presented in the final section.

In this paper, we first prove a version of the fixed point theorem obtained in [V. Pata, A fixed point theorem in metric spaces, J. Fixed Point Theory Appl. 10 (2011) 299–305], adjusted for monotone mappings in ordered metric spaces, as well as some generalizations. Then we apply them to obtain results of this type for tripled fixed points in two cases—for monotone and mixed-monotone mappings with three variables. An example is given to show the difference between some of these results.

The purpose of this paper is two fold. Firstly, using the notion of weak reciprocal continuity due to Pant et al. Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(1), 181-190 (2011)], we prove unified common fixed point theorems for various variants of compatible and $R$-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. Secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable subspaces. The realized improvements in our results are also substantiated using appropriate examples.

Recently, E. Karapinar and P. Salimi [Fixed point theorems via auxiliary functions, J. Appl. Math. 2012, Article ID 792174] have obtained fixed point results for increasing mappings in a partially ordered metric space using three auxiliary functions in the contractive condition. In this paper, these results are extended to 0-complete ordered partial metric spaces with a more general contractive condition. Examples are given showing that these extensions are proper.

In this paper, we introduce the class of extended rectangular b-metric spaces as a generalization of both rectangular metric and rectangular b-metric spaces. In addition, some fixed point results connected with certain contractions are obtained and examples are given to illustrate these results.

In this work, the class of b-rectangular metric spaces is introduced, some fixed point results dealing with rational type contractions and almost generalized weakly contractive mappings are obtained. Certain examples are given to support the results.

The aim of this paper is to use common limit range property for a quadruple of non-self mappings for deriving common fixed point results under a generalized Φ-contraction condition in symmetric spaces. Some examples are given to exhibit different types of situations where these conditions can be used and to distinguish our results from the known ones. As an application, an existence result for certain systems of integral equations is presented.

We discuss topological structure of b-metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences. As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results in b-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.

We prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings via common limit range property. Our result improves some results from the existing literature, especially the ones contained in Sintunavarat and Kumam (2009). Some illustrative and interesting examples to highlight the realized improvements are also furnished.

The object of this paper is to emphasize the role of a suitable implicit relation involving altering distance function which covers a multitude of contraction conditions in one go. By using this implicit relation, we prove a new coincidence and common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings in a metric space employing the common limit range property. Our main result improves and generalizes a host of previously known results. We also utilize suitable illustrative examples to substantiate the realized improvements in our results.

We introduce a new variant of cyclic contractive mapping in a metric space and originate existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. After these results, an application to integro-differential equations is given.

In this paper we introduce generalized cyclic contractions in G-metric spaces and establish some fixed point theorems. The presented theorems extend and unify various known fixed point results. Examples are given in the support of these results. Finally, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is given.

Several authors have introduced various conditions which can be used in order to prove common fixed point results. However, it became clear recently that some of these conditions, though formally distinct from each other, actually coincide in the case when the given mappings have a unique point of coincidence. Hence, in fact, new common fixed point results cannot be obtained in this way. We make a review of such connections and results in this paper.

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).

We prove the existence of common fixed points for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space using orbital continuity of one of the involved maps. We furnish a suitable example to demonstrate the validity of the hypotheses of our results.

New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.

We generalize in this paper some results on common fixed points of two, respectively four, contractive-type mappings in abstract metric spaces by removing condition of normality of the cone in their formulations. Further, some results about periodic points of self-maps are extended to the setting of abstract metric spaces.

We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

The aim of this article is to clearly formulate various possible assumptions for a comparison function in contractive conditions and to deduce respective (common) fixed point results in partial metric spaces. Since standard metric spaces are special cases, these results also apply for them. We will show by examples that there exist situations when a partial metric result can be applied, while the standard metric one cannot.

Results on common fixed points of mappings in cone metric spaces under weak contractive conditions (B. S. Choudhury and N. Metiya (2010)) are unified and generalized. Also, cone metric versions of some other related results on weak contractions are proved. Examples show that our results are different than the existing ones.

Fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in the frame of partial metric spaces, ordered partial metric spaces, and metric-type spaces. Examples are given showing that these results are proper extensions of the existing ones.