Search Results (15)

This paper is inspired by cutting-edge advancements in chaos theory, fractional calculus, and fixed point theory, which together provide a powerful framework for examining the dynamics of complex systems. At the heart of our research is the fractional-order Dadras–Momeni chaotic system, a pivotal model in chaos theory celebrated for its intricate, multi-scroll dynamics. Leveraging the Atangana–Baleanu fractional derivative, we extend fractional computation to chaotic systems, offering deeper insights into their behavior. To fortify the mathematical foundation of our analysis, we employ the relaxed $\theta$ rational contractions in the realm of metric spaces, enabling a more precise exploration of the system’s dynamics. A key goal of this work is to simplify the definition of the function class $\mathsf{\Theta }$ while maintaining the existence and uniqueness of fixed points under $\theta$-relaxed contractions, integrating this framework with the established literature on complete metric spaces. We explore the system’s behavior across six distinct cases by varying $\delta$ with a fixed fractional order of $\hslash =0.98$. In the first case, a single scroll forms, while successive cases lead to increased scrolls—reaching up to four by the sixth case. Phase portraits and time series analyses reveal a progression in complexity and chaos, with denser, intertwined scrolls as $\delta$ increases. This behavior highlights the system’s heightened sensitivity to parameter variations, demonstrating how fractional parameters influence the chaotic dynamics. Our results offer meaningful contributions to both the theoretical foundations and practical applications of chaos theory and fractional calculus, advancing the understanding of chaotic systems in new and impacted ways. Full article

In this manuscript, we present a novel concept termed graphical ${\Theta }_c$-Kannan contraction within the context of graphically controlled metric-type spaces. Unlike traditional Kannan contraction, this novel concept presents a modified method of contraction mapping. We discuss the significance and the existence of fixed point results within the framework of this novel contraction. To strengthen the credibility of our theoretical remarks, we provide a comparison example demonstrating the efficiency of our suggested framework. Our study not only broadens the theoretical foundations inside graphically controlled metric-type spaces by introducing and examining visual ${\Theta }_c$-Kannan contraction, but it also demonstrates the practical significance of our innovations through significant examples. Furthermore, applying our findings to second-order differential equations by constructing integral equations into the domain of Fredholm sheds light on the broader implications of our research in the field of mathematical analysis and contributes to the advancement of this field. Full article

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 $\theta$-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. Full article

Fixed point theory is a versatile mathematical theory that finds applications in a wide range of disciplines, including computer science, engineering, fractals, and even behavioral sciences. In this study, we propose triple controlled metric-like spaces as a generalization of controlled rectangular metric-like spaces. By examining the $\Theta$-contraction mapping within these spaces, we extend and enhance the existing literature to establish significant fixed point results. Utilizing these findings, we demonstrate the existence of solutions to a Fredholm integral equation and provide an example of a numerical iteration method applicable to a specific case of this Fredholm integral equation. Full article

The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction, rational $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction and almost $\left(\mathrm{\Theta },\alpha ,\theta ,\mathrm{\Psi }\right)$ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of ${\left(\mathrm{\Theta },\theta ,\mathrm{\Psi }\right)}_{{\upsilon }_0}$, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated. Full article

The Na⁺, K⁺-ATPase is an integral membrane protein which uses the energy of ATP hydrolysis to pump Na⁺ and K⁺ ions across the plasma membrane of all animal cells. It plays crucial roles in numerous physiological processes, such as cell volume regulation, nutrient reabsorption in the kidneys, nerve impulse transmission, and muscle contraction. Recent data suggest that it is regulated via an electrostatic switch mechanism involving the interaction of its lysine-rich N-terminus with the cytoplasmic surface of its surrounding lipid membrane, which can be modulated through the regulatory phosphorylation of the conserved serine and tyrosine residues on the protein’s N-terminal tail. Prior data indicate that the kinases responsible for phosphorylation belong to the protein kinase C (PKC) and Src kinase families. To provide indications of which particular enzyme of these families might be responsible, we analysed them for evidence of coevolution via the mirror tree method, utilising coevolution as a marker for a functional interaction. The results obtained showed that the most likely kinase isoforms to interact with the Na⁺, K⁺-ATPase were the θ and η isoforms of PKC and the Src kinase itself. These theoretical results will guide the direction of future experimental studies. Full article

This article includes some fixed point results for $(\phi ,\psi ,\theta )$-contractions in the context of metric space endowed with a locally $\mathcal{H}$-transitive relation. We constructed an example for attesting to the credibility of our results. We also discussed the existence and uniqueness of the solution of a Fredholm integral equation using our results. Full article

In this paper, the notion of ${\theta }^{*}$-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well. Full article

In this manuscript, we introduce the notions of various types of ${\mathsf{\Theta }}_c$-contractions for which we establish some fixed circle and fixed disc theorems in the setting of metric spaces. Some illustrative examples are also provided to support our results. Moreover, we present some fixed circle and fixed disc results of integral type contractive self-mappings, which generalize many results of invariance and transformations in the literature. Full article

In this paper, inspired by Jleli and Samet (Journal of Inequalities and Applications 38 (2014) 1–8), we introduce two new classes of auxiliary functions and utilize the same to define ${(\theta ,\psi )}_{\mathcal{R}}$ -weak contractions. Utilizing ${(\theta ,\psi )}_{\mathcal{R}}$ -weak contractions, we prove some fixed point theorems in the setting of relational metric spaces. We employ some examples to substantiate the utility of our newly proven results. Finally, we apply one of our newly proven results to ensure the existence and uniqueness of the solution of a Volterra-type integral equation. Full article

The aim of this article is to introduce a new class of contraction-like mappings, called the almost multivalued ( $\mathsf{\Theta },{\delta }_b$ )-contraction mappings in the setting of b-metric spaces to obtain some generalized fixed point theorems. As an application of our main result, we present the sufficient conditions for the existence of solutions of Fredholm integral inclusions. An example is also provided to verify the effectiveness and applicability of our main results. Full article

In this article, we introduce a relatively new concept of multi-valued $(\theta ,\mathcal{R})$ -contractions and utilize the same to prove some fixed point results for a special class of multi-valued mappings in metric spaces endowed with an amorphous binary relation. Illustrative examples are also provided to exhibit the utility of our results proved herein. Finally, we utilize some of our results to investigate the existence and uniqueness of a positive solution for the integral equation of Volterra type. Full article

In this paper, an extension of Darbo’s fixed point theorem via $\theta$ -F-contractions in a Banach space has been presented. Measure of noncompactness approach is the main tool in the presentation of our proofs. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results. Full article

The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion $\phi \left(t\right)\in \left[f\left(t\right)+{\int }_0^{1}K(t,s,\phi \left(s\right))\varrho s\right]$ for $t\in [0,1],$ where $f\in C[0,1]$ is a given real-valued function and $K:[0,1]$ $\times [0,1]$ $\times \mathbb{R}\to K_{cv}\left(\mathbb{R}\right)$ a given multivalued operator, where $K_{cv}$ represents the family of non-empty compact and convex subsets of $\mathbb{R}$ , $\phi \in C[0,1]$ is the unknown function and $\varrho$ is a metric defined on $C[0,1]$ . To attain this target, we take advantage of fixed point theorems for $\alpha$ -fuzzy mappings satisfying a new class of contractive conditions in the context of complete metric spaces. We derive new fixed point results which extend and improve the well-known results of Banach, Kannan, Chatterjea, Reich, Hardy-Rogers, Berinde and Ćirić by means of this new class of contractions. We also give a significantly non-trivial example to support our new results. Full article

The present paper aims to define three new notions: ${\mathsf{\Theta }}_e$ -contraction, a Hardy–Rogers-type $\mathsf{\Theta }$ -contraction, and an interpolative $\mathsf{\Theta }$ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique. Full article