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This research presents an advancement in fixed point theory by extending the renowned Banach contractive condition and introducing the new convex contractive condition. Central to this work is exploring and developing new convex contraction mappings characterized by rational expressions. Our investigation not only proves fixed point results for these contractions but also highlights their uniqueness compared to well-established contractions like Banach, Kannan, Chatterjea, Jaggi, and other known contraction mappings. A significant aspect of this research is the presentation of new rational-type convex contraction mappings, demonstrating their independence from traditional contractions. Through illustrative examples, we showcase the comprehensive generalization and inherent advantages of our novel results. Furthermore, the practical applications of our findings are discussed, particularly in the context of solving matrix and functional integral equations. This is achieved by applying our theoretical results to establish the existence and uniqueness of solutions for these equations, thereby underscoring our research's practical significance and potential impact in the broader field of mathematical studies. [ABSTRACT FROM AUTHOR]

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 this article, we introduce the new type of contractions for a mapping and a relation and prove certain coincidence point theorems which generalize some known results in this area. Moreover, an application to the existence of a unique solution for the integral equation is also provided. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce the concept of an (F , γ) R -contraction which is a weak idea of an F-contraction in metric spaces endowed with a binary relation. Fixed point results for such new contractions are investigated, and its benefit is showing from an example. As applications, we apply theoretical results together with the Thompson metric for solving nonlinear matrix equations. Finally, we give a numerical example showing the usage of the proposed algorithm to solve nonlinear matrix equations. [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]

Nowadays, the usage of the existence and convergence results of a common fixed point for nonlinear semigroups of various kinds of self-mappings is applied to several problems in mathematics and another branches. Surprisingly, nobody considered the study on the nonlinear semigroups of weak contraction self-mappings in the sense of Berinde. The purpose of this paper is to attempt to complete this direction and so we investigate the existence of the common fixed point for nonlinear semigroups of weak contraction self-mappings in the sense of Berinde on a bounded closed convex subset of a real Banach space with uniformly normal structure. More precisely, we prove the convergence theorem for the common fixed point of a weak contraction semigroups. An illustrative example is presented. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with a particular type of predator–prey model in which a predator has two choices of preys with probabilities x and 1 - x , respectively. Our aims are to analyze the behavior of predator animals in such situation and to construct a general functional equation for it, which covers all the aspects of such type of relationship. The Banach fixed point theorem is being used to investigate the existence and uniqueness results of the proposed predator–prey model; Hyers–Ulam and Hyers–Ulam–Rassias type stability of the functional equation has also been discussed in it. [ABSTRACT FROM AUTHOR]

This paper deals with a specific type of traumatic avoidance for the learning process of normal dogs enclosed into a small compartment with a steel grid floor. Our aim is to analyze the behavior of the dogs in such situations and to construct a suitable mathematical model for it. The existence and uniqueness results of the proposed traumatic avoidance learning model are being investigated by the use of the Banach fixed point theorem. [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]

The purpose of this paper is to improve the notion of a Z-contraction mapping with respect to a b-simulation function and also prove fixed point results on b-metric spaces endowed with only a transitive relation. Our results can reduce to several important results in the literature. As applications, we extend and generalize Ran and Reurings's results to prove nonlinear matrix equations by giving new notions concerning b-simulation functions. Finally, we give some numerical examples to support some results of our applications. [ABSTRACT FROM AUTHOR]

In this work, we improve the Ri’s fixed point theorem in Ri (Indag Math 27:85-93, 2016) by creating the φ-fixed point version of such result in metric spaces. These results generalize and extend several well-known comparable results in the literature. Several applications and some examples of our theorems are also given. [ABSTRACT FROM AUTHOR]

The purpose of this short note is to present some corrections and clarifications concerning the proof of the main result given in the paper "On analytic model for two-choice behavior of the paradise fish based on the fixed point method, J. Fixed Point Theory Appl. 2019, 21:56". [ABSTRACT FROM AUTHOR]

In this work, we introduce the concept of an s-simulation function, where s is a given real number with $$s \ge 1$$ , and use this concept for proving many results in b-metric spaces. One of these results is the existence and uniqueness of a fixed point for nonlinear mappings involving s-simulation functions. Also, we give some examples to illustrate the main results with some numerical results. Our results generalize and improve many results in b-metric spaces and also in metric spaces. [ABSTRACT FROM AUTHOR]

In (Fixed Point Theory Appl 94:6, 2012), the author introduced a new kind of contractions, called F- contractions, that extended the Banach contractions in a newfangled way. In this work, we introduce the notion of $$F_\mathfrak {R}$$ - contraction where $$\mathfrak {R}$$ is a binary relation on its domain that has not to be neither transitive nor a partial order. Consequently, we establish some fixed point results for such contractions in complete metric spaces that improve the Wardowski's original idea and we also give illustrative examples. Furthermore, we show some results to guarantee existence and uniqueness of fixed point of N-order. As an application, we apply our main result to study a class of nonlinear matrix equation. [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]

In this paper, we introduce the concept of generalized cyclic contraction pairs in b-metric spaces. We establish some fixed point results on b-metric spaces and also give some examples to illustrate the main results. As applications, we show the existence of a common solution for the following system of integral equations: where $$a, b \in \mathbb {R}$$ with $$a

In this work, we extend the concept of an almost generalized weakly contractive mapping in rectangular b-metric spaces. The establish of fixed point of such mappings is also provided. We give an example to support our 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 introduce the notion of $$\alpha $$ -integral type G-contraction mappings to generalize the notions of Banach G-contraction and integral G-contraction mappings. We also prove some fixed point theorems for $$\alpha $$ -integral type G-contraction mappings. By providing some example, we show that our results are real generalization of several results in literature. [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]

In this paper, the notion of α-ψ-contractive mappings in the setting of w-distance is introduced and some new fixed point theorems for such mappings are established. Presented fixed point theorems generalize recent results of Samet et al. [Nonlinear Anal. 75 (2012), 2154-2165] and others. Moreover, some examples and an application to nonlinear fractional differential equations are given to illustrate the usability of the new theory. [ABSTRACT FROM AUTHOR]

In this paper, we aim to introduce new types of α-admissibility in the framework of b-metric spaces. Some examples to show the independently of each type of α-admissibility are given. Using these concepts, fixed point theorems satisfying generalized weak contractive condition in the setting of b-metric spaces are established. We furnish an illustrative example to demonstrate the validity of the hypotheses and the degree of utility of our results. As an application, we discuss the existence of a solution for the following nonlinear integral equation: where $${a, b \in {\mathbb{R}}}$$ such that $${a < b, x \in C[a, b]}$$ (the set of all continuous functions from [ a, b] into $${{\mathbb{R}}}$$ ), $${\phi : [a, b] \rightarrow {\mathbb{R}}}$$ and $${K : [a, b] \times [a, b] \times {\mathbb{R}} \rightarrow {\mathbb{R}}}$$ are given mappings. [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]

In this work we present a new type of stability results for generalized Cauchy functional equation of the form where $${a, b \in \mathbb{N}}$$ and $${f}$$ is a mapping from a commutative semigroup ( $${G_1, \ast}$$ ) to a commutative group ( $${G_2, \diamond}$$ ). Using this form we generalize, extend and complement some earlier classical results concerning the stability of additive Cauchy functional equations. Our results are improvement and generalization of main results of Brzdȩk [Fixed Point Theory Appl. 2013 (), doi::285] and many results in literature. Some of the stability results for many types of functional equations are given here to illustrate the usability of the obtained 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 work, we establish some fixed point theorems by revisiting the notion of ψ-contractive mapping in Menger PM-spaces. One of our results (namely, Theorem 2.3) may be viewed as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings in M-complete Menger PM-spaces under arbitrary t-norm. Some examples are furnished to demonstrate the validity of the obtained results. [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 prove common coupled fixed point theorems under a Geraghty-type condition using monotone instead of the often used mixed monotone property. Also we give some sufficient conditions for the uniqueness of a common coupled fixed point. An example illustrating our results is provided. [ABSTRACT FROM AUTHOR]

We prove a fixed point result in orbitally $$0$$ -complete partial metric spaces, using a variant of almost contractive condition that involves a rational expression. Also, a common fixed point result in $$0$$ -complete partial metric spaces is obtained. Several consequences are deduced and examples are presented, showing that the given result can be used for proving the existence of (common) fixed points when some known results fail. [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]

In this paper, we prove some fixed point theorems for weakly compatible mappings in fuzzy metric spaces employing common limit range property with implicit relations. We also furnish some illustrative examples to support our main results. As an application to our main result, 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. Our results improve and extend a host of previously known results including the ones contained in the paper of Gopal et al. (2011). [ABSTRACT FROM AUTHOR]

In this paper, we prove a new common fixed point theorem for R-weakly commuting mappings of types $$(A_{g})$$, R-weakly commuting mappings of type $$(A_{f})$$ and R-weakly commuting mappings of types ( P) in modified intuitionistic fuzzy metric spaces using the common limit in the range property. An example is furnished which demonstrates the validity of the hypotheses and degree of generality of our main result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and Ćirić’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations. [Copyright &y& Elsevier]

In this paper, we show that the mixed $$g$$-monotone property in coupled coincidence point theorems can be replaced by generalized property. Hence, these results can be applied in a much wider class of problems. We also study the condition for the uniqueness of a common coupled fixed point and give some example of nonlinear contraction mappings where the existence of the common coupled fixed point cannot be obtained by the mixed monotone property, but it follows by our results. At the end of this paper, we give an open problems for further investigation. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of R-weakly commuting of type ( A) of Pathak et al. (Bull Korean Math Soc 34:247-257, ) in fuzzy metric spaces. We also establish the existence of common fixed point theorems by using the common limit in the range property and give an example to validate our the main 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]

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]

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]

In this work, we improve the concept of the Banach contractive condition in the setting of D -generalized metric spaces and obtain φ -fixed point results in such spaces. Our φ -fixed point results improve and extend some recently obtained results in several abstract metric spaces. Moreover, some example and numerical results are also presented. [ABSTRACT FROM AUTHOR]

In this work, we present the viscosity method for the implicit double midpoint rule for finding the fixed points of nonexpansive mappings in real Hilbert spaces. The strong convergence theorem of this method is proved under some specific assumption imposed on the control parameters. Also, we present some numerical results which are presented to illustrate the proposed method and convergence results. Moreover, we provide the applications to the variational inequality problems, constrained convex minimization problems, nonlinear evolution equations, and Fredholm integral equations. [ABSTRACT FROM AUTHOR]