A Short Survey on the Recent Fixed Point Results on $b$-Metric Spaces
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paper_title: Contraction mappings in $b$-metric spaces

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Some generalizations of well known Banach's fixed point theorem in so-called b-metric spaces are presented. 1991 Mathematics Subject Classification: 47H10 1. Some problems, particurarly the problem of the convergence of measurable functions with respects to measure lead to a generalization of notion of metric. Using this idea we shall present generalization of some fixed point theorems of Banach type. Lex X be a spece and let R+ denotes the set of all nonnegative numbers. A function d : X x X —> R+ is said to be an b-metric iff for all x, y, z 6 X and all r > 0 the following conditions are satisfacted: d{x, y) = 0iffx = y (1) d{x, y) = d{y, x) (2) d{x, y) 0. Let us consider the following condition: d{y, z) R+ an b-metric iff the conditions (1) (2) and (5) are satisfied. For T : X -> X we denote by T then n-th iterate of T. 2. Now we present following Theorem 1. Let {X, d) be e a complete b-metric space and let T : X —> X satisfy d[T(x), T(y)} R+ is increasing function such that lirnn-+co P{t) = 0 for each fi.xed > 0. Then T has exactly one fixed point u and limn-+00d[T {x), u] = 0
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paper_title: Remarks on Some Recent Fixed Point Results on Quaternion-Valued Metric Spaces

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Very recently, Ahmed et al. introduced the notion of quaternion-valued metric as a generalization of metric and proved a common fixed point theorem in the context of quaternion-valued metric space. In this paper, we will show that the quaternion-valued metric spaces are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be derived as a consequence of the corresponding existing fixed point result in the setting cone metric spaces.
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paper_title: Comparison Functions and Fixed Point Results in Partial Metric Spaces

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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.
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paper_title: On generalized α-ψ-Geraghty contractions on b-metric spaces

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Abstract In this paper, we consider generalized α - ψ {\alpha-\psi} -Geraghty contractive type mappings and investigate the existence and uniqueness of a fixed point for mappings involving such contractions. In particular, we extend, improve and generalize some earlier results in the literature on this topic. An application concerning the existence of an integral equation is also considered to illustrate the novelty of the main result.
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paper_title: Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces

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We generalize the results obtained in Cho et al. (Fixed Point Theory Appl. 2013:329, 2013) and give other conditions to prove the existence and uniqueness of a fixed point of α-Geraghty contraction type maps in the context of a complete metric space.
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paper_title: Fixed point theorems for α–ψα–ψ-contractive type mappings

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In this paper, we introduce a new concept of α–ψα–ψ-contractive type mappings and establish fixed point theorems for such mappings in complete metric spaces. Starting from the Banach contraction principle, the presented theorems extend, generalize and improve many existing results in the literature. Moreover, some examples and applications to ordinary differential equations are given here to illustrate the usability of the obtained results.
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paper_title: Fixed point theorems for α–ψα–ψ-contractive type mappings

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In this paper, we introduce a new concept of α–ψα–ψ-contractive type mappings and establish fixed point theorems for such mappings in complete metric spaces. Starting from the Banach contraction principle, the presented theorems extend, generalize and improve many existing results in the literature. Moreover, some examples and applications to ordinary differential equations are given here to illustrate the usability of the obtained results.
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paper_title: ULAM-HYERS STABILITY FOR OPERATORIAL EQUATIONS AND INCLUSIONS VIA NONSELF OPERATORS

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Using the weakly Picard operator technique, we present some abstract Ulam-Hyers stability results for operatorial equations and inclusions involving nonself single-valued and multivalued operators.
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paper_title: Ulam-Hyers stability for partial differential inclusions

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Using the weakly Picard operator technique, we will present UlamHyers stability results for integral inclusions of Fredholm and Volterra type and for the Darboux problem associated to a partial differential inclusion.
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paper_title: Ulam-Hyers Stability for Operatorial Equations

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Ulam-Hyers Stability for Operatorial Equations Using the weakly Picard operator technique, we present some Ulam-Hyers stability results for coincidence point problems for multivalued operators.
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paper_title: Some Nonunique Fixed Point Theorems of Ćirić Type on Cone Metric Spaces

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Some results of (Ciric, 1974) on a nonunique fixed point theorem on the class of metric spaces are extended to the class of cone metric spaces. Namely, nonunique fixed point theorem is proved in orbitally 𝑇 complete cone metric spaces under the assumption that the cone is strongly minihedral. Regarding the scalar weight of cone metric, we are able to remove the assumption of strongly minihedral.
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paper_title: Some Nonunique Fixed Point Theorems of Ćirić Type on Cone Metric Spaces

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Some results of (Ciric, 1974) on a nonunique fixed point theorem on the class of metric spaces are extended to the class of cone metric spaces. Namely, nonunique fixed point theorem is proved in orbitally 𝑇 complete cone metric spaces under the assumption that the cone is strongly minihedral. Regarding the scalar weight of cone metric, we are able to remove the assumption of strongly minihedral.
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paper_title: Data dependence of the fixed point set of some multivalued weakly Picard operators

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In this paper we study data dependence of the fixed point set for a special class of multivalued weakly Picard operators. Existence and data dependence of the common fixed points for some Reich-type multivalued operators are also proved. Finally, an application to a Fredholm integral inclusion is given.
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paper_title: Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations

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We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.
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paper_title: A fixed point theorem in partially ordered sets and some applications to matrix equations

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An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.
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Title: A Short Survey on the Recent Fixed Point Results on $b$-Metric Spaces

Section 1: INTRODUCTION AND PRELIMINARIES
Description 1: This section provides a foundational understanding of metric spaces, generalizing the Euclidean distance concept, and introduces the notion of $b$-metric spaces with relevant examples.

Section 2: FIXED POINT OF α-ψ-CONTRACTIVE MAPPINGS
Description 2: This section discusses the concept of α-ψ-contractive mappings and related fixed point results on $b$-metric spaces.

Section 3: ULAM-HYERS STABILITY RESULTS THROUGH THE FIXED POINT PROBLEMS
Description 3: This section explores the Ulam-Hyers stability in the context of fixed point problems within $b$-metric spaces and presents significant results.

Section 4: NON UNIQUE FIXED POINTS ON b-METRIC SPACES
Description 4: This section examines the conditions for the existence of non-unique fixed points in $b$-metric spaces, inspired by the non-unique fixed point theorem of Ćirić.

Section 5: CONSEQUENCES
Description 5: This section demonstrates various standard fixed point theorems and their applications on $b$-metric spaces endowed with partial orders, deriving several existing results from the literature.