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This paper introduces the notion of k-fuzzy metric spaces, which generalizes and extends the concept of fuzzy metric spaces due to George and Veeramani in [A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399.] for the fuzzy sets involving more than one (k) parameters. It is shown that the topology generated by the k-fuzzy metric is first countable, and the k-fuzzy metric space is Hausdorff. Finally, we prove a fixed point theorem, which generalizes and extends the results of Grabiec [M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389.] into k-fuzzy metric spaces. [ABSTRACT FROM AUTHOR]

In mathematical psychology, themodel of decision practice represents the development ofmoral judgment that deals with the time to decide themeaning of the various choices and selecting one of themfor use. Most animal behavior research classifies such situations as two distinct phenomena. On the other hand, reward plays a big part in this kind of study since, based on the selected side and food location, such circumstances may be classified into four categories. This paper intends to investigate such types of behavior and establish a general functional equation for it. The proposed functional equation can be used to describe several psychological and learning theory models in the existing literature. By using the fixed point theory tools, we obtain the results related to the existence, uniqueness, and stability of a solution to the proposed functional equation. Finally, we give two examples to support our main results. [ABSTRACT FROM AUTHOR]

Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22 (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of the averagedmapping Tλ, where λ ∈ (0, 1], is also a fixed point of the initialmapping T. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set of all fixed points of a weak enriched contraction mapping. Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given. [ABSTRACT FROM AUTHOR]

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C 4 H 10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. [ABSTRACT FROM AUTHOR]

Recently, the idea of φ-fixed point and the elementary results on φ-fixed points were first investigated by Jleli et al. [Jleli M, Samet B, Vetro C (2014) Fixed point theory in partial metric spaces via φ-fixed point's con- cept in metric spaces. Journal of Inequalities and Applications, 2014(1):1- 9.]. Based on this work, Karapinar et al. [Karapinar E, O'Regan D, Samet B (2015) On the existence of fixed points that belong to the zero set of a certain function. Fixed Point Theory and Applications, 2015(1):1-14.] established the new φ-fixed point results, which can be reduced to the famous fixed point result of Boyd and Wong in 1969. However, the main result of Karapinar et al. does not cover the φ-fixed point results of Jleli et al. This paper aims to fulfill this gap by proving φ-fixed point results covering several φ-fixed point results and fixed point results [ABSTRACT FROM AUTHOR]

A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method's range by introducing the concept of an octane graph, which is an essential organic compound having the formula C8H18. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph's edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study. [ABSTRACT FROM AUTHOR]

In this paper, we present a concept of semigroups of enriched Kannan mappings, which is based on enriching contractive type mappings that have been effectively used in many recent studies. We investigate some weak and strong convergence theorems for the Mann iterative process to approximate common fixed points of enriched Kannan semigroups in uniformly convex Banach spaces. In the end, by using MATLAB, we present a numerical example to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form 2 m f (x + m y) + 2 f (m x - y) = (m 3 + m) [ f (x + y) + f (x - y) ] + 2 (m 4 - 1) f (y) for all x , y ∈ X , where m is a fixed scalar such that m ≠ 0 , 1 , and f is a map from a quasi-normed space X to a quasi-Banach space Y over the same field with X by applying the alternative fixed point theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the numerical algorithm for approximation the solutions of nonlinear matrix equations with the relation-theoretic contractions. First, we introduce the new contraction namely (ψ, φ, R)-contraction and prove the fixed point theorem for relationtheoretic (ψ, φ, R)-contractions in a metric space endowed with a T-orbital transitivity. We also give an example to show the benefit of our theorems. As applications, we apply our main results together with the Thompson metric to solve the nonlinear matrix equations Xr = Q+∑i=1m Ai* Gi(X)Ai, Where r ≥1, Q is an n x n positive definite matrix, Ai is an nx, non-singular matrix with its conjugate transpose Ai* and Gi is a continuous order preserving mapping from a set of all n x n positive definite matrices into itself for all i=1,2,3....,m. Finally, we furnish a numerical example to support results of our applications. [ABSTRACT FROM AUTHOR]

We introduce the notion of a JS 𝕽 quasi-contraction mapping, where 𝕽 is a binary relation on its domain. Also, we prove some fixed point results for such contractions in complete metric spaces endowed with a transitive relation. An example is given to substantiate our obtained theorems. In addition, we introduce a JS 𝕽N -quasicontraction and also establish fixed point of N-order theorems for such contractions. [ABSTRACT FROM AUTHOR]

In recent times, fixed point theory has undergone a great development, especially through the use of new auxiliary functions and also by weakening the axioms that characterize a metric space. In this work, in the setting of b-metric spaces, the notions of a modified R-function and an R s -contraction are introduced and some new fixed point results for such contractions involving modified R-functions are established. Also, we give some example for supporting our results. Finally, we prove some application by using the main results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a special type of functional equation that describes the relationship between the predator animals and their two choices of prey with their corresponding probabilities. Our aim is to find the existence and uniqueness results of the proposed functional equation using the Banach fixed point theorem. Finally, we give an illustrative example that supports our main results. [ABSTRACT FROM AUTHOR]

In this paper, the new modified Picard-FB iterative algorithm based on the idea of the iterative algorithm of Fukhar-ud-din and Berinde in [H. Fukhar-ud-dina, V. Berinde, Iterative methods for the class of quasicontractive type operators and comparsion of their rate of convergence in convex metric spaces, Filomat 30 (2016), 223-230.] is defined. We also introduce the class of conditional quasi-contractive mappings in convex metric spaces and prove the convergence result of the purposed iterative algorithm for approximating the fixed points for conditional quasi-contractive mappings in convex metric spaces. Furthermore, a comparison result of the rate of convergence of the new purposed iterative algorithm, the iterative algorithm of Fukhar-ud-din and Berinde and many iterative algorithms in convex metric spaces is investigated. [ABSTRACT FROM AUTHOR]

Recently, Asadi et al. [M. Asadi, E. Karapinar, P. Salimi, New extension of p-metric spaces with some fixed-point results on M-metric spaces, Journal of Inequalities and Applications, 18 (2014)] introduced and studied the concept of an M-metric space and proved Banach and Kannan fixed point results in complete M-metric spaces. In this work, we establish some fixed point results for nonlinear contractive mappings by utilizing the control functions which were initiated by Sintunavarat et al. [W. Sintunavarat, Y. J. Cho, P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Computers and Mathematics with Applications, 62 (2011) 1969-1978]. Presented theorems extend and generalize many existing results in the literature. We also give some illustrative examples. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce notions of a modification of the common limit in the range property and two generalized multi-valued F-contraction mappings in metric spaces. Fixed point results with two concepts mentioned above are established. Also, we give an example to illustrate our main results. As applications, we prove the existence of a common solution for the system of functional equations in dynamic programming. [ABSTRACT FROM AUTHOR]

The purpose of this work is to introduce the new contractive condition which involves mw-distances in metric spaces. The fixed point theorems for mappings satisfying such contractive condition are established via the idea of an adapting-ceiling distance. Two illustrative examples are also given to confirm that our results rightly generalize many results in the literatures. As an application, the existence and uniqueness of the solution for nonlinear Fredholm integral equations and nonlinear Volterra integral equations are investigated via our main result. [ABSTRACT FROM AUTHOR]

Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result. [ABSTRACT FROM AUTHOR]

In this work, we establish new fixed point theorems for w-generalized weak contraction mappings with respect to w-distances in complete metric spaces by using the concept of an altering distance function. As an application, we use the obtained results to aggregate the existence and uniqueness of the solution for nonlinear Fredholm integral equations and Volterra integral equations together with nonlinear fractional differential equations of Caputo type. [ABSTRACT FROM AUTHOR]

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation $$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$ where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results. [ABSTRACT FROM PUBLISHER]

In this work, we introduce the new concept of a generalization of ( F , φ ) -contraction mapping and establish some existence results of φ -fixed point for such mappings. We also state some illustrative examples to support our results and give numerical experiment for approximating the φ -fixed point in this example. As an application, the obtained results are used to deduce some fixed point theorems in partial metric spaces and the application to nonlinear integral equations is given in order to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate some fixed-point theorems in modular metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative algorithm to approximate fixed points of total asymptotically nonexpansive mappings in $\operatorname{CAT}(0)$ spaces. We also provide two examples to illustrate the convergence behavior of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to obtain some new common fixed point theorems for weakly compatible mappings in fuzzy metric spaces (in the sense of Kramosil and Michálek) satisfying new contractive conditions employing the common limit in the range property. Our results generalize and extend some recent results contained in Jain et al. (2012) [12] to generalized contractive conditions under some control functions. In the process, some results about multidimensional common fixed points as coupled/tripled/quadrupled common fixed point results are derived from our main results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to obtain some new common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics and to give some examples to illustrate the main results. Further, by using our main results, we prove some results about multidimensional common fixed points. Our results generalize and extend some recent results given by Kadelburg et al. (Fixed Point Theory Appl. 2015:27, 2015) and Choudhurya and Kundu (J. Nonlinear Sci. Appl. 5:259-270, 2012). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized cyclic contraction pairs in b-metric spaces and establish some fixed point theorems for such pairs. Also, we give some examples to illustrate the main results which properly generalizes some results given by some authors in literature. Further, by using the main results, we prove some common fixed point results for generalized contraction pairs in partially ordered b-metric spaces. Our results generalize and improve the result of Shatanawi and Postolache (Fixed Point Theory Appl. 2013:60, 2013) and several well-known results given by some authors in metric and b-metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of $(\alpha,\theta,k)$-contraction multi-valued mappings and establish some fixed point results for such mappings by using some control functions due to Jleli et al. (J. Inequal. Appl. 2014:439, 2014) in metric spaces and furnish some interesting examples to illustrate our main results. Also, we give some fixed point results in metric spaces endowed with a graph. Our results generalize and extend recent results given by some authors. [ABSTRACT FROM AUTHOR]

We prove some common fixed point results for two mappings satisfying generalized contractive condition in b-metric space. Note that b-metric of main results in this work are not necessarily continuous. So our results extend and improve several previous works. We also present one example that shows the applicability and usefulness of our results. [ABSTRACT FROM AUTHOR]

The object of this paper is to extend the concept of β-admissible for multivalued mappings due to Mohammadi et al. [B. Mohammadi, S. Rezapour and N. Shahzad, Some results on fixed points of α-ψ-Ciric generalized multifunctions, Fixed Point Theory and Applications 2013 (2013), 24] to fuzzy mappings and utilize this concept to prove a common α-fuzzy fixed point theorem for a pair of fuzzy mappings. We also give illustrative example which demonstrate the validity of the hypotheses of our results. Some related results to common fixed point for multivalued mappings are also derived from the main result. [ABSTRACT FROM AUTHOR]

We introduce the new type of nonself mapping and study sufficient conditions for the existence of past-present-future (for short PPF) dependent fixed point for such mapping in the Razumikhin class. Also, we apply our result to prove the PPF dependent coincidence point theorems. Finally, we use PPF dependence techniques to obtain solution for a nonlinear integral problem with delay. [ABSTRACT FROM AUTHOR]

We study Ulam-Hyers stability and the well-posedness of the fixed point problem for new type of generalized contraction mapping, so called α-λ-contraction mapping. The results in this paper generalize and unify several results in the literature such as the Banach contraction principle. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the new notion which is more general than the previous works on the weakly isotone mappings in the case of single valued mappings and multivalued mappings in an ordered Banach space. We also extend some common fixed point theorems of Dhage et al. [4] and give some applications of the main results. Furthermore, at the end of this paper, we give an open problem for further investigation. [Copyright &y& Elsevier]

We introduce the class of mixed equilibriumproblemswith theweakly relaxed α-monotone bi-function in Banach spaces.Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxed α-monotone bi-function in Banach spaces.The results presented in this paper extend and improve the corresponding results in the existing literature. [ABSTRACT FROM AUTHOR]

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which aremore general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate ourmain results. [ABSTRACT FROM AUTHOR]

We introduce two new concepts of weakly relaxed η-α monotone mappings and weakly relaxed η-α semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed η-α monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed η-α semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem. [ABSTRACT FROM AUTHOR]

We introduce the concept of the generalized wα-contraction mappings and establish the existence of fixed point theorem for such mappings by using the properties of w-distance and α-admissible mappings. We also apply our result to coincidence point and common fixed point theorems in metric spaces. Further, the fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results generalize the result of Kutbi, 2013, and several results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend a multi-valued contraction mapping to a cyclic multi-valued contraction mapping. We also establish the existence of common fixed point theorem for a cyclic multi-valued contraction mapping. Our results extend, generalize and unify Nadler’s multi-valued contraction mapping and many fixed point theorems for multi-valued mappings. [Copyright &y& Elsevier]

In this paper, we introduce the notion of fuzzy (F, φ,β-ψ)-contractive mappings in fuzzy metric spaces and utilize the same to prove some existence and uniqueness fuzzy φ-fixed point results in both M-complete and G-complete fuzzy metric spaces. The obtained results extend, generalize and improve some relevant results of the existing literature. An illustrative example is utilized to demonstrate the usefulness and effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this article the concept of coupled best proximity point and cyclic contraction pair are introduced and then we study the existence and convergence of these points in metric spaces. We also establish new results on the existence and convergence in a uniformly convex Banach spaces. Furthermore, we give new results of coupled fixed points in metric spaces and give some illustrative examples. An open problems are also given at the end for further investigation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the notion of a hybrid generalized multi-valued contraction mapping and establish the common fixed point theorem for this mapping. Our results generalize, unify, extend and complement several common fixed point theorems of many authors in the literature. [Copyright &y& Elsevier]

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems. [ABSTRACT FROM AUTHOR]

In this article, we define a tangential property which can be used not only for single-valued mappings but also for multi-valued mappings, and used it in the prove for the existence of a common fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type in metric spaces. Our theorems generalize and unify main results of Pathak and Shahzad (Bull. Belg. Math. Soc. Simon Stevin 16, 277-288, 2009) and several known fixed point results. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Cho et al. [Y.J. Cho, R. Saadati, S.H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. 61 (2011) 1254–1260] introduced the concept of the -distance in a cone metric space and established some fixed point theorems on -distance. The aim of this paper is to extend and generalize the main results of Cho et al.  and also show some examples to validate our main results. [Copyright &y& Elsevier]

Abstract: In 2007, Kamran [T. Kamran, Multivalued -weakly Picard mappings, Nonlinear Anal. 67 (2007) 2289–2296] extended the notion of multi-valued mapping from weak contraction and generalized -weak contraction to -weak contraction and generalized multi-valued -weak contraction. He also obtained some common fixed point theorems with the notion of -weakly commuting at a coincidence point of a hybrid pair. In this paper, we can drop the condition of -weakly commuting in Theorems 2.9 and 3.5 in . We further extend the notion of generalized multi-valued -weak contraction and introduce the notion of generalized multi-valued -weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued -weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors. [Copyright &y& Elsevier]

Abstract: In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “ is -weakly commuting” for a hybrid pair of multivalued maps in Theorem 3.10 of T. Kamran (2004) . As an application, an invariant approximation result is derived. [Copyright &y& Elsevier]

This article is written due to a small gap in our published paper. In this erratum, we point out and fix the problem to set our existed results at the best of their perfection. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the new notion of a specific point in the space of the bounded real-valued functions on a given non-empty set and present a result based on the existence and uniqueness of such points. As a consequence of our results, we discuss the existence of a unique common solution to coupled systems of functional equations arising in dynamic programming. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this work is to unify different classes of fuzzy contractive mappings by introducing a new class of fuzzy contractive mappings called fuzzy Z -contractive mappings. For this new class of mappings, suitable conditions are framed to ensure the existence of fixed point in M -complete fuzzy metric spaces (in the sense of George and Veeramani). A comprehensive set of examples are presented to support the claim. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce some new notions of generalized convex contraction mappings and establish some approximate fixed point theorems for such mappings in the setting of complete metric spaces. Examples and application to approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the Boyd-Wong type fixed point result in the F-metric context and apply it to obtain some existence and uniqueness criteria of solution(s) to a second order initial value problem and a Caputo fractional differential equation. We substantiate our obtained result by finding a suitable non-trivial example. [ABSTRACT FROM AUTHOR]

The objective of this paper is to emphasize the role of "common limit range property" to ascertain the existence of common fixed point in fuzzy metric spaces. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. We derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. As an application to ourmain result, we prove an integraltype fixed point theorem in fuzzy metric space. Our results improve and extend a host of previously known results including the ones contained in Imdad et al. (2012). [ABSTRACT FROM AUTHOR]