On fixed point theorems in modular spaces characterized by $C^*$-class functions

Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. Banach’s formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in the metric related aspects of the theory. The metric fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances, conditions cast in this framework are more natural and more easily verified than their metric analogs. In this chapter, we study the existence and construction of fixed points for monotone nonexpansive mappings acting in modular functions spaces equipped with a partial order or a graph structure.

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

This article introduces a new type of C*-algebra valued modular G-metric spaces that is more general than both C*-algebra valued modular metric spaces and modular G-metric spaces. Some properties are also discussed with examples. A few common fixed point results in C*-algebra valued modular G-metric spaces are discussed using the “C*-class function”, along with some suitable examples to validate the results. Ulam–Hyers stability is used to check the stability of some fixed point results. As applications, the existence and uniqueness of solutions for a particular problem in dynamical programming and a system of nonlinear integral equations are provided.

Thompson aggregators, Scott continuous Koopmans operators, and Least Fixed Point theory

The study of common fixed point theory with different types of mappings and contractive conditions takes a significant role in present research activity. At first, the great mathematician Brouwer introduced the famous fixed-point theory in 1912. According to Brouwer, “Every continuous function from the closed and bounded subset of Rn into itself has a fixed point”. After that, the great mathematician S. Banach in 1922 invented the well-known fixed point theory, namely Banach Fixed Point Theory. “Huang & Zhang [3] have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space [3]”. The present research article is about some common fixed-point theorems for “self-mappings and commuting mappings” on Cone and Cone Rectangular Metric Space.

In this paper, we announce a beautiful and important application of the results of the paper of two Romanian mathematicians: R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19, 2153-2163 (2017). Namely, the method they gave in the mentioned paper for b-metric spaces can be applied without difficulty to b-metric-like spaces as well. Thus, the results on b-metric spaces established in the papers: Hassen Aydi, Monica-Felicia Bota, Erdal Karapinar and Slobodanka Mitrović, A fixed point theorem for set-valued quasi-contraction in b-metric spaces, Fixed Point Theory and Applications 2012, 2012:88, and Nicola Fabiano, Zoran Kadelburg, Stojan Radenović, and Milica Savatović, Remarks on ”On new orthogonal contractions in b-metric spaces” and ”On orthogonal partial b-metric spaces with an application”, can be extended to b-metric-like spaces as well. Also, using the significant paper at the beginning of the mentioned Romanian professors, a positive answer can be directly given to the question given at the end of the recent paper: M. Gardašević-Filipović, K. Kukić, D. Gardašević, Z. D. Mitrović, Some best proximity point results in the orthogonal 0-complete b-metric-like spaces, J. Contemp. Math. Anal. Armen. Acad. Sci., 58 (2), 105–115 (2023). In the same way, the interval [0,1s ) belonging to the coefficient of some type of contraction within the b-metric-like space can be extended to the entire interval [0,1).

A new generalization of the metric space notion, named $${\mathcal {F}}$$ -metric space, was given in [M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl. 20 (2018), no. 3, Art. 128, 20 pp.]. In this paper, we investigate some properties of $${\mathcal {F}}$$ -metric spaces. A simple proof is given to show that the natural topology induced by an $${\mathcal {F}}$$ -metric is metrizable. We present a method to construct s- $$\hbox {relaxed}_{{p}}$$ spaces and, therefore, $${\mathcal {F}}$$ -metric spaces from bounded metric spaces. We give some results that reveal differences between metric and $${\mathcal {F}}$$ -metric spaces. In particular, we show that the ordinary open and closed balls in $${\mathcal {F}}$$ -metric spaces are not necessarily topological open and closed, respectively. This answers a question posed implicitly in the quoted paper. We also show that $${\mathcal {F}}$$ -metrics are not necessarily jointly continuous functions. Despite some topological differences between metrics and $${\mathcal {F}}$$ -metrics, we show that the Nadler fixed point theorem and, therefore, the Banach contraction principle in the frame of $${\mathcal {F}}$$ -metric spaces can be reduced to their original metric versions. This reduction even happens when the Schauder fixed point theorem is investigated in $${\mathcal {F}}$$ -normed spaces structure. By applying the given technique in this paper, it turns out that some nonlinear $${\mathcal {F}}$$ -metric contractions and, therefore, the related $${\mathcal {F}}$$ -metric fixed point results can naturally be reduced to their metric versions. In addition, the same happens for some topological fixed point results.

Let X and Y be Banach spaces and P and Q be two subsets of X and Y respectively. Let T1 : X ⊗ Y → X and T2 : X ⊗ Y → Y be two mappings and S be a self mapping on P⊗Q. Using T1 and T2 we define a self mapping T on X⊗Y. Different conditions under which T +TS+S has a fixed point in P ⊗Q are established here. Analogous results are also established taking the pair (T1,T2) as (k,k / ) contraction mappings. Again considering X ⊗ Y as a reflexive Banach space. We derive the conditions for 1 m (T + ST + S), m > 2, m ∈ N, for having a fixed point in P ⊗ Q. Some iteration schemes converging to a fixed point of T + ST + S in P ⊗ Q are also presented here.

In this paper, we first establish some new types of fixed point theorem which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and many results in [W.-S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Anal. 73 (2010), 1439-1446] and references therein. Applying those new results, we also present some existence theorems of coincidence point and others.

In this paper we present a fixed point theorem of Banach type in modular spaces. Also, we give some applications of this result to a nonlinear integral equation in Musielak-Orlicz space.

Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class of functions are presented. Ultimately, the solutions to a system of non-linear functional integral equations by the fixed point results obtained are discussed, and non-trivial examples to illustrate the validity of our study are derived.

The aim of this paper is to prove some fixed point theorems which generalize well known basic fixed point principles of nonlinear functional analysis. Moreover, we investigate the class of mappings f: X→ X, where X is a Banach space, for which one of the main conditions in the metric fixed point theory, namely the condition (1), is satisfied. We obtain essential applications of this fact. All our results are illustrated by suitable examples.

Geometrical coefficients and the structure of the fixed-point set of asymptotically regular mappings in Banach spaces

We establish a relation theoretic version of the main result of Aydi et al. [H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler?s fixed point theorem on partial metric space, Topol. Appl. (159), 2012, 3234-3242] and extend the results of Alam and Imdad [A. Alam, M. Imdad, Relation-theoretic contraction priciple, J. Fixed Point Theory Appl., 17(4), 2015, 693-702.] for a set-valued map in a partial Pompeiu-Hausdorff metric space. Numerical examples are presented to validate the theoretical finding and to demonstrate that our results generalize, improve and extend the recent results in different spaces equipped with binary relations to their set-valued variant exploiting weaker conditions. Our results provide a new answer, in the setting of relation theoretic contractions, to the open question posed by Rhoades on continuity at fixed point. We also show that, under the assumption of k-continuity, the solution to the Rhoades? problem given by Bisht and Rakocevic characterizes completeness of the metric space. As an application of our main result, we solve an integral inclusion of Fredholm type.

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.