Abstract: This research explores equiexpansive mappings of the Hardy-Roger type, focusing on the conditions required for the existence and uniqueness of fixed points. By utilizing foundational results established by Krasnoselskii and Sadovskii, the study develops a robust theoretical framework to support these conditions. To enhance understanding, illustrative examples with graphical representations are provided, showcasing fixed points in mappings that satisfy specific criteria. Furthermore, the study demonstrates the practical relevance of equiexpansive mappings by addressing a fractional differential system governed by the semi-linear Atangana-Baleanu-Caputo model. This application serves to validate the theoretical findings while emphasizing the significance of these mappings in solving complex differential systems. In summary, the work enriches the theoretical landscape and underscores its practical utility, paving the way for further advancements and applications in this domain.

Abstract: In a product space, making the product using a complete metric space induced with a graph, the goal of this study is to define a new type F-contraction called set-valued (α, F)-Geragthy, G-Prešić type contraction and using this new type of contraction mappings we investigate the presence of fixed points. Also, there are some fixed point theorems, on product spaces, proved for different types of set-valued contractions. Our findings extend and generalize certain previously published results in product spaces. There is an application for the nth-order nonlinear difference equation. 

Abstract: In the context of modular function spaces, we propose and investigate the Fibonacci-Ishikawa iteration method applied to non-expansive, asymptotically monotonic operators. We establish both ω−convergence and ω−almost everywhere convergence of the Fibonacci-Ishikawa sequence, thereby extending several existing results in the literature. Furthermore, our findings highlight the convergence behavior of the sequence toward its ω−almost everywhere limit, particularly as it relates to the minimizing sequence of a given function. Also, a comparative analysis with classical Mann and Ishikawa iteration methods reveals that the Fibonacci-Ishikawa iteration exhibits superior convergence properties, rapidly approaching the fixed point in significantly fewer iterations. Finally, we discuss the possible applicability of practical relevance.

Abstract: This research explores fixed points for particularly integral type multivalued mappings, in 𝑀_𝑣^𝑏-metric spaces. Additionally, we study fixed circle problems offering geometric insights into sets of fixed points. This research paper contributes to the evolving field of multivalued mapping results in 𝑀_𝑣^𝑏-spaces, drawing inspiration from the framework of Hausdorff. Further, motivated by the wide applications of differential inclusions as set-valued maps, we explore first-order nonlinear differential inclusions in 𝑀_𝑣^𝑏-metric spaces using established conclusions.

Abstract: We propose an efficient iterative method called Picard-CR for approximating fixed points under weak perturbative contraction conditions in uniformly convex hyperbolic metric spaces. Theoretical analysis establishes both weak and strong convergence, with performance validated against classical methods (CR, Picard-Noor, and Picard-SP) through numerical experiments. We extend our convergence results to non-expansive and contraction mappings, supported by MATLAB-based visualizations. The iterative scheme is shown to be stable and more efficient, with direct application to computing equilibrium concentrations in reversible chemical reactions. Our findings contribute not only to fixed point theory but also provide practical computational tools for chemical and engineering problems.

Abstract: This study delves into the concept of α-fuzzy mappings and their associated α-fuzzy fixed points within the framework of Hausdorff intuitionistic fuzzy metric-like spaces. A general fixed point theorem for α-fuzzy mappings is established in complete HIFMS and its subclasses. The results unify and generalize several classical and fuzzy fixed point theorems, extending them to more complex fuzzy structures. Additionally, recognizing the significant applications of differential inclusions as set-valued mappings, the research explores first-order nonlinear Cauchy differential inclusions within Hausdorff intuitionistic fuzzy metric spaces by leveraging the derived theoretical results. The findings demonstrate the robustness of fuzzy fixed point theory in modeling systems with uncertainty and imprecision, with potential applications in control theory and optimization. This work not only broadens the scope of fuzzy mapping studies but also bridges the gap between fuzzy metric theory and differential inclusions.

Abstract: This study investigates set-valued contractions within the framework of m_b^v−metric spaces, extending classical contraction principles. By introducing and examining the Hausdorff m_b^v−metric, we establish a foundation for set-valued fixed point theorems, thereby contributing significantly to this area of research. Our findings generalize several well-known contraction concepts, including those of Banach, Sehgal, Wardowski, Altun, Bianchini, and Nadler, within the context of m_b^v−metric spaces. These advancements have practical implications, particularly in the study of nonlinear systems and the mathematical model of Fredholm integral inclusions. The results presented here emphasize the growing importance of set-valued fixed points and pave the way for further exploration and application across various scientific and engineering domains.

Abstract: In this paper, we introduce the concepts of α − θ − E−Geraghty Pata proximal contractions and their weak φ−variants, which generalize and unify several well-known contraction conditions in fixed point theory. By integrating the ideas of Geraghty and Pata contractions with the framework of α − θ−proximal admissibility, we establish best proximity point theorems for both single-valued and multivalued non-self mappings in metric spaces. Furthermore, we extend our results to the context of coupled fixed points and partially ordered metric spaces, thus broadening their applicability. The presented theorems generalize a range of existing results and offer a more flexible setting for analyzing nonlinear problems. We also include a concrete application to fractional differential equations and provide illustrative examples to demonstrate the effectiveness of our results.

Abstract: Our research analyzes a refined iterative method within a new framework involving enriched non-expansive mappings in uniformly convex Banach spaces. We establish weak convergence using the well-known Opial’s condition and prove strong convergence under various domain or mapping assumptions. Through a detailed example of an enriched non-expansive mapping that is not non-expansive and graphical analysis, we demonstrate that the convergence rate of our iteration scheme is more effective than those proposed by Thakur, Ali, and D∗∗ in a new setting. Furthermore, we investigate the impact of various parameters on the convergence behavior of our proposed method. Finally, we apply our fundamental findings to practical problems by estimating solutions for a partial differential equation, which is a model of various physical and biological processes such as chemical reactions, population dynamics, heat transfer, and the spread of diseases. These applications demonstrate the practical utility of our method and highlight its potential for addressing complex challenges in applied mathematics. 

Abstract: This paper introduces a new iterative method to find fixed points of contractive mappings in uniformly convex Banach spaces. The method is proven to be stable and shows faster convergence than existing methods by Thakur, Piri, Ullah, and Harmouchi, as demonstrated through numerical examples and MATLAB plots. A detailed comparison highlights how different parameters affect convergence. We also establish a new result on data dependence using an approximate operator. Finally, the method is applied to solve a Caputo-type fractional differential equation, relevant to real-world problems in viscoelasticity, anomalous diffusion, and electrical circuits.

Abstract: Our study introduces a novel iterative approach for approximating common fixed points of general contractive mappings. We establish theorems that demonstrate the convergence and stability of this iteration process, supported by examples and graphical representations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem illustrating the escape criterion for the transcendental sine function Tα,β(u) = sin(um) + αu + β, for u, α, β ∈ C and m ≥ 2. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Graphical illustrations are provided to show the impact of different parameters on the color appearance and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set's structure. 

Abstract: In this work, we introduce extended 𝑀_𝑣^𝑏-metric space and extended 𝑀_𝑣^𝑏-metric space with a focus on their topological notions including relations between m_𝑣-metric, extended b_𝑣(s)-metric and extended 𝑀_𝑣^𝑏-metric. By challenging the conventional assumption of zero self-distance, we investigate novel postulates to establish the existence and uniqueness of fixed point theorems in these domains and pave the way for more accurate mathematical models applicable to real-world scenarios. Our findings are supported by illustrative examples. Additionally, using our result we explore the problem that indicates the ascending motion of a rocket. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains.

Abstract: Our study presents a novel orbit with s-convexity, for illustration of the behavior shift in the fractals. We provide a theorem to demonstrate the escape criterion for transcendental cosine functions of the type Tα,β(u) = cos(um) + αu + β, for u, α, β ∈ C and m ≥ 2. We also demonstrate the impact of the parameters on the formatted fractals with numerical examples and graphical illustrations using the MATHEMATICA software, algorithm, and colormap. Moreover, we observe that the Julia set appears when we widen the Mandelbrot set at its petal edges, suggesting that each Mandelbrot set point contains a sizable quantity of Julia set picture data. It is commonly known that fractal geometry may capture the complexity of many intricate structures that exist in our surroundings.

Abstract: In this work, we introduce novel postulates to establish fixed figure theorems with a focus on their extension to the domain of 𝑀_𝑣^𝑏-metric spaces. Consequently, we dene conditions ensuring the existence and uniqueness of fixed circles, fixed ellipses, fixed Apollonius circles, fixed Cassini curves, fixed hyperbola, and so on for self mapping. We also partially address an open problem demonstrating that a JS-contraction possesses a fixed elliptic disc. This property extends to smaller discs and ellipses within a complete 𝑀_𝑣^𝑏-metric space. By challenging the conventional assumption of zero self-distance, we pave the way for more accurate mathematical models applicable to real-world scenarios. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains. Our findings are supported by illustrative examples. Additionally, we explore the concept of fixed figures in the context of Rectied Linear Unit (ReLU), a widely-used activation function in neural networks and machine learning models. Our exploration of fixed figures in the context of Rectified Linear Units (ReLU) further deepens our understanding of nonlinear systems and their relationship to neural network behavior. 

Abstract: Our research introduces an innovative iterative method for approximating fixed points of contraction mappings in uniformly convex Banach spaces. To validate the stability of this iterative process, we provide a comprehensive theorem. Through detailed examples and graphical analysis, we demonstrate that our method outperforms previous approaches for contraction mappings, including those developed by Agarwal, Gursoy, Thakur, Ali, and D∗∗, using MATLAB software for implementation and comparison. Moreover, we investigate the effect of various parameters on the convergence behavior of our proposed method. By comparing it with existing iterative schemes through a specific example, we highlight the efficiency and robustness of our approach. Additionally, we establish a significant result concerning data dependence for an approximate operator, utilizing our iterative process to show how small changes in data can affect the outcome. Finally, we apply our fundamental findings to a practical problem by estimating solutions for a fractional Volterra-Fredholm integro-differential equation. This application not only illustrates the practical utility of our method but also underscores its potential for solving complex problems in applied mathematics.

Abstract: The study focuses on establishing a fixed point for a class of interpolative contractions by introducing novel forms of interpolative generalized Gupta-Saxena-Reich-type contraction and generalized Gupta-Saxena-Kannan-type contraction within the setting of 𝑀_𝑣^𝑏-metric spaces. We provide non-trivial examples to support the obtained conclusions which update and expand upon the previous works. The potential applications of non-continuous maps in addressing real-world nonlinear problems inspire the research. We solve nonlinear matrix equations and fourth-order differential equations related to beam theory. Our outcomes advance the theory of interpolative contraction maps and associated fixed point theorems, encouraging further exploration and application of 𝑀_𝑣^𝑏-metric space.

Abstract: In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces and propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces respectively. We also propose the idea of (α, F)-Geraghty-type generalized F-contraction mappings utilizing both fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study we consider an application to dynamic market equilibrium.

Abstract: This study explores the realm of metric spaces, advancing beyond conventional boundaries by introducing two innovative types of metrics known as generalized Branciari-type metrics. Through exacting examination and exemplification, we shed light on the intricacies of these newly defined metric spaces and their extended versions. By drawing parallels with established theorems such as Banach and Kannan, we unveil corollaries that establish necessary symmetric conditions for the existence and uniqueness of fixed points concerning self-operators within these spaces. The inclusion of illustrative examples not only bolsters our theoretical framework but also underscores the practical relevance of our findings. Furthermore, we utilize our research to address real-world applications, showcasing how our results can be employed to determine the existence of unique solutions for algebraic systems of linear equations, thereby bridging the theoretical and applied aspects of mathematical exploration. Through these interventions, our study significantly contributes to the comprehensive understanding and utilization of all the properties in metric spaces within diverse mathematical contexts.

Abstract: This study aims to provide a new class of Ćirić 𝛼 +(θ, ϕ)-proximal contraction, a non-self generalized proximal contraction mapping on a non-empty closed subset of any metric space. Also, we prove that such contractions satisfying some conditions must have a unique best proximity point if we take the base space as complete. For some particular values of the constants, that we have used to generalize the proximal contraction, we conclude different types of proximal contractions. We substantiate the deduced findings with examples. An application to show the solution of an integral equation in the context of proximity point results over the space of all real-valued continuous functions is discussed. By presenting these findings, we aim to encourage a new wave of scholars to keep exploring this exciting topic that has so many applications ahead of it.

Abstract: In uniformly convex Banach spaces, we study within this research Fibonacci-Ishikawa iteration for monotone asymptotically non-expansive mappings. In addition to demonstrating strong convergence, we establish weak convergence results of the Fibonacci-Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure we approximate the solution of the Caputo-type nonlinear fractional differential equation.

Abstract: A new variety of non-self generalized proximal contraction, called Hardy-Rogers 𝛼 + F−proximal contraction, is shown in this work. Also, we prove, with an example, that such contractions satisfying some conditions must have a unique best proximity point. For some particular values of the constants, that we have used to generalize the proximal contraction, we conclude different 𝛼 + F−proximal contraction results of the types ´Ciri´c, Chatterjea, Reich, Kannan, and Banach with proof, that all such types of contractions must have unique best proximity point. We also apply our result to solve a functional equation.

Abstract: In the context of hyperbolic spaces, our study presents a novel iterative approach for approximating common fixed points satisfying general contractive condition involving a pair of mappings with weak compatibility. Also, we notice that our iterative procedure approximates to a point of coincidence if the weak compatibility condition is violated. We provide theorems to demonstrate the Δ−convergence, stability, and efficiency of this iteration process. Additionally, we provided some immediate corollaries that involve mappings with contractive condition, instead of general contractive condition. Furthermore, we demonstrate with examples and graphs that our iteration process is faster than all previous procedures, including those of Jungck-SP, Jungck-CR, and Jungck-DK, utilizing MATLAB software. Also, we compare the impact of the initial values and the parameters on the convergence behavior of the proposed iterative process with existing iterative schemes using an example. Finally, we focus on using our iterative technique to approximate the solution of a non-linear integral equation with two delays.

Abstract: It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of 𝐶∗-algebra valued metric spaces. Here, we present the definitions of the gauge function, the Bianchini-Grandolfi gauge function, 𝛼-admissibility, and (𝛼,𝛽)-admissible Geraghty contractive mapping in the sense of 𝐶∗-algebra. Using these definitions, we define (𝛼,𝛽)-Bianchini-Grandolfi gauge contraction of type I and type II. Next, we prove our primary results that the function satisfying our contraction condition has to have a unique fixed point. We also explain our results using examples. Additionally, we discuss some consequent results that can be easily obtained from our primary outcomes. Finally, there is a useful application to integral calculus. 

Abstract: This work aims to prove new results in an 𝑀_𝑣^𝑏-metric space for a noncontinuous single-valued self-map. As a result, we extend, generalize, and unify various fixed-point conclusions for a single-valued map and come up with examples to exhibit the theoretical conclusions. Further, we solve a mathematical model of the spread of specific infectious diseases as an application of one of the conclusions. In the sequel, we explain the significance of 𝑀_𝑣^𝑏-metric space because the underlying map is not necessarily continuous even at a fixed point in 𝑀_𝑣^𝑏-metric space thereby adding a new answer to the question concerning continuity at a fixed point posed by Rhoades’. Consequently, we may conclude that the results via 𝑀_𝑣^𝑏-metric are very inspiring, and underlying contraction via 𝑀_𝑣^𝑏-metric does not compel the single-valued self-map to be continuous even at the fixed point. Our research is greatly inspired by the exciting possibilities of using noncontinuous maps to solve real-world nonlinear problems. 

Abstract: In this study, the authors verify fixed point results for Geraghty contraction with a restricted co-domain of the auxiliary function in the context of generalized metric structure, namely S_b-metric space. This new idea of defining Geraghty contraction for self operators generalizes a large number of previously published, closely related works for 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: This study aims to provide some new classes of (α, β, F*)-weak Geraghty contraction and (α, β, F**)-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we find that the mappings satisfying the definition of such contractions have a unique fixed point if the underlying space is complete. In addition, we provide an application showing the uniqueness of the solution of the two-point boundary value problem.

Abstract: We introduce the Ciric type-contraction principle in a vector-valued b-metric space (G, A, s) that generalizes Perov’s contraction principle. We investigate the possible conditions on four mappings W, E, D, T, for which these mappings admit a unique common fixed point in a vector-valued bmetric space subject to a nonlinear operator F: Pm → Rm. We illustrate the hypothesis of our findings with examples. We consider an Infectious Disease Model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.

Abstract: In this article, we introduce two new classes of single-valued contraction mappings within the framework of S-metric spaces. For the first class, called S-polynomial type contraction mappings, we establish two fixed-point theorems that demonstrate the existence and uniqueness of fixed points. We begin by considering the case where the mapping is continuous and then relax this continuity requirement. Importantly, we also derive Banach’s fixed-point theorem as a special case within the context of S-metric spaces. The second class, termed almost S-polynomial type contraction mappings, builds on the concept of almost contraction mappings introduced by Berinde. We establish two fixed-point theorems that generalize Berinde’s results in S-metric spaces. Additionally, we introduce a condition for uniqueness and prove a theorem for a unique fixed point. Several examples are provided to illustrate the significance of our generalizations. Finally, we demonstrate an application of the main result by describing the solution of the integral equation.

Abstract: In this study, we review various outcomes of contraction mapping in the S_b−metric structure. Through the use of compelling examples and insightful commentary, we compile all key findings in this direction. This work’s primary goals include assisting young researchers by providing a framework for outcomes in S_b−metric space and demonstrating that there is still an opportunity for many more researchers to explore this intriguing area with its vast potential for applications.