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This paper combines the concept of an arbitrary binary connection with the widely recognized principle of θ -contraction to investigate the innovative features of vector-valued metric spaces. This methodology demonstrates the existence of fixed points for both single- and multi-valued mappings within complete vector-valued metric spaces. Through the utilization of binary relations and θ -contraction, this study advances and refines the Perov-type fixed-point results in the literature. Furthermore, this article furnishes examples to substantiate the validity of the presented results. Additionally, we establish an application for finding the existence of solutions to a system of matrix equations. [ABSTRACT FROM AUTHOR]

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β -enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature. [ABSTRACT FROM AUTHOR]

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p ; q) -difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p ; q) -operators in this regard. Application of the fixed points of the ρ - θ -contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p ; q) -difference Navier problems. [ABSTRACT FROM AUTHOR]

The aim of this research is to introduce two new notions, Θ -(Ξ ,h)-contraction and rational (α , η) - ψ -interpolative contraction, in the setting of F -metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to generalize the concept of classical fuzzy set to vector-valued fuzzy set which can attend values not only in the real interval [ 0 ,   1 ] , but in an ordered interval of a Banach algebra as well. This notion allows us to introduce the concept of vector-valued fuzzy metric space which generalizes, extends and unifies the notion of classical fuzzy metric space and complex-valued fuzzy metric space and permits us to consider the fuzzy sets and metrics in a larger domain. Some topological properties of such spaces are discussed and some fixed point results in this new setting are proved. Multifarious examples are presented which clarify and justify our claims and results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the novel concept of generalized distance denoted as J θ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance J θ (u , H ★) , a generalized Hausdorff type distance and a P J θ -property of a pair (H ★ , K ★) of nonempty subsets of extended b-metric space (U ★ , ρ θ). Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α -nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki's theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki's theorem, which was applied to discuss types of difference equations and recurrence equations. [ABSTRACT FROM AUTHOR]

This article introduces a novel iterative process, denoted as F ★ , designed for the class of generalized α -Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F * for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology. [ABSTRACT FROM AUTHOR]

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper. [ABSTRACT FROM AUTHOR]

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce a modified Halpern iterative technique combined with an inertial approach and Tseng methods for finding a common solution of the fixed-point problem of Bregman strongly nonexpansive mapping and the pseudomonotone variational inequality problem. Using our iterative approach, we develop a strong convergence result for approximating the solution of the aforementioned problems. We also discuss some consequences of our major finding. The results presented in this paper complement and build upon many relevant discoveries in the literature. [ABSTRACT FROM AUTHOR]

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in S -metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, quasi-contractions, Ćirić–Reich–Rus-type contractions, Hardy–Rogers-type contractions, Reich-type contractions, interpolative Kannan contractions, interpolative Chatterjee contractions, among others, with their generalizations in S -metric spaces. We offer significant examples to substantiate the reliability of our results. This study also establishes consequential fixed-point results and applies them to nonlinear fractional differential equations and the boundary value problem for homogeneous transverse bars. At the end of the manuscript, we present an important open problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence and uniqueness result for a class of Dirichlet boundary value problems whose model is − Δ p u = β | ∇ u | q + c | u | p − 2 u + f in   Ω , u = 0 on   ∂ Ω , where Ω is an open bounded subset of R N , N ≥ 2 , 1 < p < N , Δ p u is the so-called p-Laplace operator, and p − 1 < q < p . We assume that β is a positive constant, c and f are measurable functions belonging to suitable Lorentz spaces. Our approach is based on Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence { u n } , which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result. [ABSTRACT FROM AUTHOR]

In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors. [ABSTRACT FROM AUTHOR]

In this paper, the concepts of Wardowski-type set-valued contractions and Işik-type set-valued contractions are introduced and fixed point theorems for such contractions are established. A positive answer to the open Question is given. Examples to support main theorems and an application to integral inclusion are given. [ABSTRACT FROM AUTHOR]

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H - (ϑ , φ) -contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems. [ABSTRACT FROM AUTHOR]

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki's theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki's theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results. [ABSTRACT FROM AUTHOR]

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t-strong, and we establish an alternative version of Miheţ's theorem for p-complete t-strong fuzzy metrics (Theorem 2). In addition, an example of t-strong fuzzy metric spaces that are not non-Archimedean is provided. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) Δ and strong (metric) convergence of Picard iterates of the α -averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of orthogonal α –F–convex contraction mapping and prove some fixed-point theorems for self-mapping in orthogonal complete metric spaces. The proven results generalize and extend some of the well-known results in the literature. Following the derivation of these fixed-point results, we propose a solution for the fractional integro-differential equation, utilizing the fixed-point technique within the context of orthogonal complete metric spaces. [ABSTRACT FROM AUTHOR]

The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as "Kannan-graph-fuzzy contraction". This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit. [ABSTRACT FROM AUTHOR]

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a G q -implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng–Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side. [ABSTRACT FROM AUTHOR]

It is known that a strict contraction on a complete metric space with a graph possesses a fixed point. In the present paper, we show that this property holds for single valued and set-valued self-mappings of metric spaces with graphs that are of the contractive type. We also show the convergence of iterates of these mappings to fixed points. In particular, our results are true for metric spaces with symmetric graphs. [ABSTRACT FROM AUTHOR]

We introduce the new idea of (α − θ σ) -contraction in quasi-metric spaces in this paper. For these kinds of mappings, we then prove new fixed-point theorems on left K, left M, and left S m y t h -complete quasi-metric spaces. We also apply our results to infer the existence of a solution to a second-order boundary-value problem. [ABSTRACT FROM AUTHOR]

In this paper, our main aim is to present innovative fixed-point theorems that provide solutions to the fixed-circle problem with symmetric contractions. We accomplish this by employing operator enrichment techniques within the context of Banach spaces. Furthermore, we demonstrate the practical application of these theorems by showcasing their relevance to the rectified linear unit (ReLU) activation function. By exploring the connection between fixed points and activation functions, our work contributes to a deeper understanding of the behavior and properties of these fundamental mathematical concepts. [ABSTRACT FROM AUTHOR]

In this study, the authors verify fixed-point results for Geraghty contractions with a restricted co-domain of the auxiliary function in the context of generalized metric structure, namely the S b -metric space. This new idea of defining Geraghty contraction for self-operators generalizes a large number of previously published, closely related works on the presence and uniqueness of a fixed point in S b -metric space. Also, the outcomes are achieved by removing the continuity constraint of self-operators. We also provide examples to elaborate on the obtained results and an application to the integral equation to illustrate the significance in the literature. [ABSTRACT FROM AUTHOR]

In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions. [ABSTRACT FROM AUTHOR]

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ , Ω - Ϝ) -contraction has been introduced, which is called (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction. In the end, the applications of an extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required. [ABSTRACT FROM AUTHOR]

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton's method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric. [ABSTRACT FROM AUTHOR]

This article introduces the novel concept of an extended parametric S b -metric space, which is a generalization of both S b -metric spaces and parametric S b -metric spaces. Within this extended framework, we first establish an analog version of the Banach fixed-point theorem for self-maps. We then prove an improved version of the Banach contraction principle for symmetric extended parametric S b -metric spaces, using an auxiliary function to establish the desired result. Finally, we provide illustrative examples and an application for determining solutions to Fredholm integral equations, demonstrating the practical implications of our work. [ABSTRACT FROM AUTHOR]

The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique fixed point for this class of operators. Thirdly, we study Ulam-Hyers stability, well-posedness, and periodic point properties. Finally, an application of the main results to the variational inequality problem is given. [ABSTRACT FROM AUTHOR]

In an article published in 2015, Hussain et al. introduced a notion of a fuzzy b-metric space and obtained some fixed point theorems for this kind of space. Shortly thereafter, Nădăban presented a notion of a fuzzy b-metric space that is slightly different from the one given by Hussain et al., and explored some of its topological properties. Related to Nădăban's study, Sedghi and Shobe, Saadati, and Šostak independently conducted investigations in articles published in 2012, 2015, and 2018, respectively, about another class of spaces that Sedgi and Shobe called b-fuzzy metric spaces, Saadati, fuzzy metric type spaces, and Šostak, fuzzy k-metric spaces. The main contributions of our paper are the following: First, we propose a notion of fuzzy b-metric space that encompasses and unifies the aforementioned types of spaces. Our approach, which is based on Gabriec's notion of a fuzzy metric space, allows us to simultaneously cover two interesting classes of spaces, namely, the 01-fuzzy b-metric spaces and the K-stationary fuzzy b-metric spaces. Second, we show that each fuzzy b-metric space, in our sense, admits uniformity with a countable base. From this fact, we derive, among other consequences, that the topology induced by means of its "open" balls is metrizable. Finally, we obtain a characterization of complete fuzzy b-metric spaces with the help of a fixed point result which is also proved here. In support of our approach, several examples, including an application to a type of difference equations, are discussed. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to investigate the existence and uniqueness of a solution to the integral equation and homotopy result. To achieve our objective, we introduce the notion of ( α , η , ψ )-contraction in the framework of F -bipolar metric space and prove some fixed point results for covariant and contravariant mappings. Some coupled fixed point results in F -bipolar metric space are derived as outcomes of our principal theorems. A non-trivial example is also provided to validate the authenticity of the established results. [ABSTRACT FROM AUTHOR]

This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ -contraction and orthogonal ( α , Θ) -contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results. [ABSTRACT FROM AUTHOR]

The present investigation aims to establish the existence and uniqueness of solutions for a system containing sequential fractional differential equations. Furthermore, boundary conditions that include the Riemann–Liouville fractional integral are taken into consideration. The existence of unknown functions, fractional derivatives, and fractional integrals at lower orders are necessary for the nonlinearity to exist. In order to provide proofs for the results presented in this study, the Leray–Schauder alternative and the Banach fixed-point theorem are utilised. Finally, examples are used to support the main results. [ABSTRACT FROM AUTHOR]

Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D * * iteration technique, for approximating fixed points in generalized α -nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D * * iteration approach to the fixed points of generalized α -nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results. [ABSTRACT FROM AUTHOR]

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the F * iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the F * iterative algorithm converges strongly and faster than other current iterative schemes in the literature and extends to the fixed point of a strong pseudocontractive mapping. Finally, under a nonlinear quadratic Volterra integral equation, the application of our findings is shown. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for the Fredholm integral equation Φ (ϑ) = G (ϑ , Φ (ϑ)) + ∫ 0 1 F (ϑ , ζ , Φ (ζ)) d ζ , for ϑ ∈ [ 0 , 1 ] , in the setting of the modular function spaces L ρ . We also derive an application of this research within the framework of variable exponent Lebesgue spaces L p (·) subject to specific conditions imposed on the exponent function p (·) and the functions F and G. [ABSTRACT FROM AUTHOR]

We establish a fixed point property for the Lebesgue spaces with variable exponents L p (·) , focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on ρ -non-expansive mappings defined on convex subsets of L p (·) , satisfying the "condition of uniform decrease" that we define subsequently. [ABSTRACT FROM AUTHOR]

We prove the modular convexity of the mixed norm L p (ℓ 2) on the Sobolev space W 1 , p (Ω) in a domain Ω ⊂ R n under the sole assumption that the exponent p (x) is bounded away from 1, i.e., we include the case sup x ∈ Ω p (x) = ∞ . In particular, the mixed Sobolev norm is uniformly convex if 1 < inf x ∈ Ω p (x) ≤ sup x ∈ Ω p (x) < ∞ and W 0 1 , p (Ω) is uniformly convex. [ABSTRACT FROM AUTHOR]

The purpose of this research is to examine the uniqueness and existence of the (ω , ρ) -BVP solution for a particular solution to a class of Hadamard fractional differential equations with impulsive boundary value requirements on Banach spaces. The notion of Banach contraction and Schaefer's theorem are used to prove the study's key findings. In addition, we offer the prerequisites for the set of solutions to the investigated boundary value with impulsive fractional differential issue to be convex. To enhance the comprehension and practical application of our findings, we offer two illustrative examples at the end of the paper to show how the results can be applied. [ABSTRACT FROM AUTHOR]

The aim of this article is to investigate the relationship between integral-type contractions and the generalized dynamic process. The fixed-point results for multivalued mappings that satisfy both the integral Khan-type contraction and the integral θ -contraction are established in a complete metric space. Furthermore, some corollaries are derived based on our main contribution. To demonstrate the novelty of our findings, several examples are provided. Finally, we look into whether nonlinear fractional differential equations have solutions utilizing the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the b-triangular property in fuzzy b-metric space. Furthermore, we give some new fixed point results in fuzzy b-metric space for non-continuous mappings. Our results generalize and expand some results from the related literature. Two applications of our results, to solving Fredholm integral equation and in dynamic programming, are also given. [ABSTRACT FROM AUTHOR]

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction. [ABSTRACT FROM AUTHOR]

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d > 2 . This solution family is equivalent to a fractal curve in complex space C d with random steps parametrized by N Ising variables σ i = ± 1 , in addition to a rational number p q and an integer winding number r, related by ∑ σ i = q r . This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in R d space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N → ∞ or its chemical potential μ → 0 . The system with fixed N has different asymptotics at odd and even N → ∞ , but the limit μ → 0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν / μ 2 in the continuum limit μ → 0 , leading to anomalous dissipation at μ ∝ ν → 0 . The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t − λ , with a continuous spectrum of indexes λ in the local limit μ → 0 . The spectrum is determined by a resolvent, which is represented as an infinite product of 3 ⊗ 3 matrices depending of the element of the Euler ensemble. [ABSTRACT FROM AUTHOR]

In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors. [ABSTRACT FROM AUTHOR]