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In this work, we improve and correct some results in the paper by Proinov (J Fixed Point Theory Appl 22:1–27, 2020). [ABSTRACT FROM AUTHOR]

In this work, a counterexample is given to refute a result in the paper by Younis et al. (J Fixed Point Theory Appl 21:1–33, 2019). Furthermore, applying this counterexample, we give negative answers to some of the open problems in their paper. Recently, Baradol et al. give a new theorem to rectify the result of Younis et al. However, we find that there are some gaps in the proofs of their results. Finally, we introduce a fundamental lemma and fill these gaps. [ABSTRACT FROM AUTHOR]

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind. [ABSTRACT FROM AUTHOR]

This paper mainly deals with a four-point boundary value problem for a class of generalized Laplacian equations with singular weight and positive parameter. By applying the fixed point theorem of expansion/compression of a cone and Schauder’s fixed point theorem, we obtain the relations between the number of positive solutions and two different kinds of asymptotic behaviors of the nonlinearity at 0 and ∞ . The conclusions in this paper can also be suitable for the corresponding multi-point boundary value problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain an existence Feng–Liu type fixed point result for multi-valued mappings in relational metric spaces which generalizes, extends and unifies several relevant fixed point results of the existing literature. Suitable illustrative examples are adopted to substantiate the genuineness of our newly proved result. Besides this, we present a result in a Banach space and utilize the same to show the convergence of successive approximation for the modified nonlinear Bernstein operators in C[0, 1]. [ABSTRACT FROM AUTHOR]

This paper presents an overview that aims to discuss a brief historical account of the development through the definitions and comparison of weaker forms of continuity notions in metric fixed point theory. In addition to that, we categorize various contractive definitions which ensure the existence of the fixed point, yet the mappings may be discontinuous at the fixed point with or without some weaker continuity notions. [ABSTRACT FROM AUTHOR]

In this paper, we present new variants of some known fixed point theorems and new fixed point results for cyclic operators on ordered sets, on distance spaces, and on ordered distance spaces. Some results in terms of the fixed point structures are also presented. Our results include existence, uniqueness, convergence of successive approximations, well posedness of fixed point problems and the Ostrowsky property. We also identify some of the main research directions regarding cyclic operators and formulate several open problems for each of them. [ABSTRACT FROM AUTHOR]

In this paper, we present a new fixed point theorem for the sum of two mixed monotone operators of Meir–Keeler type on ordered Banach spaces through projective metric, which extends the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear singular fourth-order elastic beam equations with nonlinear boundary conditions. [ABSTRACT FROM AUTHOR]

An operator T on metric space (X, d) is called a Picard operator if T has a unique fixed point u in X and for any x ∈ X , the sequence { T n x } n ∈ N converge to u. In this paper, we give new results concerning the existence of Picard operators. [ABSTRACT FROM AUTHOR]

The equilibrium problem defined by the Nikaidô–Isoda–Fan inequality contains a number of problems such as optimization, variational inequality, Kakutani fixed point, Nash equilibria, and others as special cases. This paper presents a picture for the relationship between the fixed points of the Moreau proximal mapping and the solutions of the equilibrium problem that satisfies some kinds of monotonicity and Lipschitz-type condition. [ABSTRACT FROM AUTHOR]

Ran and Reurings (Proc Am Math Soc 132(5):1435–1443, 2003) extended the Banach contraction principle to the setting of partially ordered metric spaces and recently Proinov (J Fixed Point Theory Appl 22:21, 2020) extended and unified many earlier fixed point theorems. In this paper we will present analogous results for the significantly wider class of mappings on preordered metric spaces. We give non-trivial examples of Kannan-type mappings. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [ On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L space. [ABSTRACT FROM AUTHOR]

This paper is intended for a general mathematical audience. The examples show how the study of existence of fixed points of averaged mappings T λ = (1 - λ) I + λ T , where 0 < λ < 1 and I is the identity operator, can help in the study of existence of fixed points of mappings T. [ABSTRACT FROM AUTHOR]

This paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to show that the Meir–Keeler contraction principle, as well as some of its generalizations, is, in general, not true in quasi-metric spaces. After that, we suggest a new Meir–Keeler type contraction that guaranties the existence and uniqueness of fixed points in quasi-metric spaces. Finally, to illustrate the wide usability of our findings, we discuss the existence and uniqueness of solutions for an integro-differential equation in Musielak–Orlicz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider evolution problems involving time-dependent maximal monotone operators in Hilbert spaces. Existence and relaxation theorems are proved. [ABSTRACT FROM AUTHOR]

In this paper, we reexamine the concept of firmly nonexpansiveness in the modular sense in the variable exponent sequence spaces ℓ p (·) . In particular, we extend the classical fixed point results for firmly nonexpansive mappings defined in Banach spaces to the modular case within the spaces ℓ p (·) . [ABSTRACT FROM AUTHOR]

In this paper, we will present some fixed point results for the case of non-self operators satisfying a nonlinear graphic contraction condition in complete metric spaces. Properties of the corresponding fixed point equation are established and some applications are given. [ABSTRACT FROM AUTHOR]

In this paper, we give an investigation of the asymptotics for the iterations associated with (c)-mappings acting on unbounded closed convex subsets of Banach spaces. In particular, the almost fixed point property (in short; AFPP) and conditions ensuring an ergodic type theorem for this class of mappings are established. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the notions of extended F-contraction of Hardy–Rogers type, extended F-contraction of Suzuki–Hardy–Rogers type and generalized F-weak contraction of Hardy–Rogers type and to establish some new fixed point results for such kind of mappings in the setting of complete b-metric spaces. These fixed point results improve (and/or) extend those obtained in Vetro (Nonlinear Anal Model Control 21(4):531–546, 2016) and Lukács and Kajántó (Fixed Point Theory 19(1):321–334, 2018) since some conditions made therein are removed or weakened. In addition, some illustrative examples are provided to show the usability of the obtained results. As an application of our results, we obtain the existence and uniqueness of solutions for certain functional, integral and differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of graphical convex metric spaces and some basic properties of the underlying spaces. Different from related literature, we generalize Mann iterative scheme and Agrawal iterative scheme for set-valued mappings to above spaces by introducing the concepts of T-Mann sequences and T-Agrawal sequences. Furthermore, by using the iterative techniques and graph theory, we investigate the existence and uniqueness of fixed points for set-valued G-contractions in a graphical convex metric space. Moreover, we present some notions of well-posedness and G-Mann stability of the fixed point problems in the above space. Additionally, as an application of our main results, we discuss the well-posedness and G-Mann stability of the fixed point problems for set-valued G-contractions in a graphical convex metric space. [ABSTRACT FROM AUTHOR]

The aim of the paper was to prove some new fixed point theorems, coincidence point theorems and common fixed point theorems for multivalued F-contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet). We also prove existence of common solutions to integral inclusions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if the circle acts on an 8-dimensional compact almost complex manifold M with 4 fixed points, all the Chern numbers and the Hirzebruch χ y -genus of M agree with those of S 2 × S 6 . In particular, M is unitary cobordant to S 2 × S 6 . [ABSTRACT FROM AUTHOR]

In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms H 2 2 in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of H 2 2 using two invariants, det f and tr f . [ABSTRACT FROM AUTHOR]

The purpose of this paper is threefold. Firstly, recognizing the concept of Piri and Kumam (Fixed Point Theor Appl 210, 2014), we define generalized F-contractive mappings in the framework of G-metric spaces and by employing this, some fixed point theorems in the structure of G-metric spaces are established that can not be obtained from the existing results in the context of allied metric spaces and do not meet the remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Infact, we utilize the pattern, mentioned in Karapinar and Agrawal (Fixed Point Theor Appl 154, 2013), a counter paper to remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Secondly, in the setting of G-metric spaces, certain fixed point results for integral inequalities under generalized F-contraction are presented. Finally, as an application, our results are utilized to establish the existence and uniqueness of solution the equations arising in Oscillation of a spring. In the sequel, another application is given to set-up the existence and uniqueness of solution of functional equations occurring in dynamic programming. Our investigations are also authenticated with the aid of some appropriate and innovative examples. [ABSTRACT FROM AUTHOR]

Variational relations problems include in a general approach various problems like variational inequalities, equilibrium problems, optimization problems and variational or differential inclusions. In this paper, we study the existence of solutions for a general variational problem, using classical fixed point results for generalized contractions, both for set-valued and single-valued mappings. We particularize then the result for a class of multivalued equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space, and we derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded $${\tau}$$ -compact subset of a Banach space has a common fixed point, and we give a qualitative complement to the Markov-Kakutani theorem. [ABSTRACT FROM AUTHOR]

We study nonoscillation/oscillation of the dynamic equation where $${t_0 \in \mathbb{T}}$$ , $${{\rm sup} \mathbb{T} = \infty}$$ , $${r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}$$ , $${p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}$$ . By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation which is extensively discussed on time scales. [ABSTRACT FROM AUTHOR]

Given an S-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this note is, by using the language of equivariant cohomology, to present a unified method to deal with such problems, of which the related known results are direct corollaries. [ABSTRACT FROM AUTHOR]

Let (X, d) be a complete b-metric space endowed with a partial order relation and f : X → X be a Ćirić type operator. In this paper, an extended study of the fixed point equation x = f (x) , x ∈ X , is considered. As an application, coupled fixed point results are given in the same framework. Our results generalize some recent theorems in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we have some observations on F-weak contractions due to Wardowski and Van Dung (Demonstr Math 47(1):146–155, 2014). Our observations lead us to introduce the notion of F ∗ -weak contractions and utilize the same to prove some fixed point results. The proven results give an affirmative answer to certain open questions raised by Kannan (Bull Calcutta Math Soc 60:71–76, 1968) and Rhoades (Contemp. Math. 72:233–245, 1988) on the existence of contractive definitions not forcing the continuity at the fixed point. Some illustrative examples are also given. As an application, we investigate the existence and uniqueness of the solution of an integral equation of Volterra type. [ABSTRACT FROM AUTHOR]

In this paper, we consider some generalization of the Banach contraction principle, namely cyclic contraction and cyclic φ -contraction. For the application to the fractal, we develop new iterated function systems (IFS) consisting of cyclic contractions and cyclic φ -contractions. Further, we discuss about some special properties of the Hutchinson operator associated with the cyclic (c)-comparison IFS. [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]

In this paper, we study the existence of nontrivial solutions to the following nonlinear differential equation with derivative term: u ′ ′ (t) + a (t) u (t) = f (t , u (t) , u ′ (t)) , t ∈ [ 0 , ω ] , u (0) = u (ω) , u ′ (0) = u ′ (ω) , where a: [ 0 , ω ] → R + ( R + = [ 0 , + ∞)) is a continuous function with a (t) ≢ 0 , f: [ 0 , ω ] × R × R → R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on nonlinearity, using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in C 1 [ 0 , ω ] . Finally, an example is given to demonstrate the validity of our main result. [ABSTRACT FROM AUTHOR]

In this paper, we solve two fixed point problems associated with the class of (c)-mappings. The first one is devoted to obtain the existence of fixed points for such mappings defined on weak ⋆ closed bounded convex subsets of duals of separable Banach spaces when the orthogonality relation ⊥ is uniformly weak ⋆ approximately symmetric. For the second problem, using the idea of R. Smarzewski (On firmly nonexpansive mappings, Proc. Am. Math. Soc 113(3):723–725,1991), we prove the existence of fixed points for such mappings which are defined on a finite union of weakly compact convex subsets of UCED (uniformly convex in every direction) Banach spaces. [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 the present paper, we introduce the concept of F-contraction on vector-valued metric space. Then, we achieved a fixed point result that includes the famous Perov fixed point theorem as properly. We provided a nontrivial and illustrative example showing this fact. Finally, we give an existence result for semilinear operator system in a Banach space. [ABSTRACT FROM AUTHOR]

We introduce a large class of contractive mappings, called enriched contractions, a class which includes, amongst many other contractive type mappings, the Picard–Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Krasnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results of this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia-type enriched contractions. Examples to illustrate the generality of our new concepts and the corresponding fixed point theorems are also given. [ABSTRACT FROM AUTHOR]

Let T be a self-mapping on a complete metric space (X, d). In this paper, we obtain new fixed point theorems assuming that T satisfies a contractive-type condition of the following form: ψ (d (T x , T y)) ≤ φ (d (x , y)) or T satisfies a generalized contractive-type condition of the form ψ (d (T x , T y)) ≤ φ (m (x , y)) , where ψ , φ : (0 , ∞) → R and m(x, y) is defined by m (x , y) = max d (x , y) , d (x , T x) , d (y , T y) , [ d (x , T y) + d (y , T x) ] / 2. In both cases, the results extend and unify many earlier results. Among the other results, we prove that recent fixed point theorems of Wardowski (2012) and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof (1977). [ABSTRACT FROM AUTHOR]

In this paper, we prove a common fixed point theorem for commuting mappings and some common fixed point theorems for compatible mappings and its variants in digital metric spaces. Furthermore, we give some examples in support of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider variational relation problems involving a binary relation. The framework presented is more general than that in [J. Optim. Theory Appl. 138 (2008) , 65-76] and in other recent papers which deal with this subject. [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 the present paper, considering the Wardowski's technique, we give a new approach to the Assad-Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad-Kirk fixed point theorem. [ABSTRACT FROM AUTHOR]

The study of non-Newtonian calculi was started in 1972 by Grossman and Katz. These calculi provide an alternative to the classical calculus and they include the geometric, anageometric and bigeometric calculi, etc. Recently, Çakmak and Başar (2002) have studied the concept of non-Newtonian metric. Also they have given the triangle and Minkowski's inequalities in the sense of non-Newtonian calculus. In this paper, we introduce a fixed point theory by defining some topological structures of the relevant non-Newtonian metric space. [ABSTRACT FROM AUTHOR]

We prove a generalization of Ekeland’s variational principal using the notion of P sets. Using this result, we give proofs for fixed point theorems on partially ordered sets. Furthermore, one can obtain theorems for coupled fixed points using this technique. We demonstrate the procedure for proving such theorems. [ABSTRACT FROM AUTHOR]

We introduce the novel concept of integral Read–Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read–Bajraktarević (RB) operator and we derive conditions for the fixed point of the iRB operator to belong to certain function spaces. [ABSTRACT FROM AUTHOR]

In this paper we will point out an error in proving some results on finite iterated function systems consisting of Kannan maps, Reich maps and Chatterjea type maps. In this respect, some counter-examples are given. We also answer some open questions on iterated function systems consisting of contractive maps of Reich type and we present revisions of some theorems on iterated function systems consisting of Kannan, Reich and Chatterjea type maps, by adding a commutativity assumption on the maps. [ABSTRACT FROM AUTHOR]

In this paper, in the setting of complete metric spaces we establish some fixed point theorems for non-self mappings of contractive type satisfying either the Reich-Zaslavski property or the approximate fixed point property. As applications, we obtain some results in endpoint theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a set-valued cubic functional equation and a set-valued quartic functional equation and prove the Hyers-Ulam stability of the set-valued cubic functional equation and the set-valued quartic functional equation by using the fixed point method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of Suzuki-type contractive multivalued mappings and establish the existence of fixed points of such mappings in the setup of b-metric spaces. Some examples are presented to support the results proved herein. An estimate of Hausdorff distance between the fixed point sets of two Suzuki-type contractive multivalued mappings is obtained. As an application of results thus obtained, existence and uniqueness of periodic solution of delay differential equations are shown. [ABSTRACT FROM AUTHOR]