Search Results (59)

In this paper, we introduce a novel class of quadruple controlled metric-type spaces formulated in a four-dimensional setting. Within this framework, we extend the notion of a $\Theta$-contraction mapping to accommodate such spaces. By employing these concepts, we establish several new fixed-point theorems that significantly generalize and enhance the existing results in the literature. The validity and effectiveness of the proposed approach are demonstrated through carefully constructed illustrative examples within the defined space and for the derived fixed-point results. Finally, an application to a boundary value problem is presented, highlighting the potential of the developed theory. Furthermore, a coupled four-component thermo-chemical system was examined as an additional application, demonstrating the broader applicability of the proposed framework. Full article

Neural ordinary differential equations (NODEs) have emerged as an effective methodology in artificial neural networks (ANNs) and deep learning for capturing unknown or unmodeled dynamics in compartmental and dynamical mathematical models arising from real-life applications, particularly under limited-data conditions, through learned data-driven corrections. Nevertheless, accurate one-step prediction errors do not necessarily guarantee reliable long-horizon rollouts. In this work, we study residual Neural ODE models of the form $\widehat{f}=f+h_{\theta }$ and derive a priori rollout-error estimates showing that long-time prediction behavior is generated by the incremental stability structure of the learned dynamics. Contracting regimes produce uniformly controlled rollout errors, whereas weakly contractive or expansive regimes can amplify persistent approximation errors over long time horizons. The analysis is illustrated on a flow-reactor chemical reaction network (CRN), where the washout parameter controls rollout reliability on the data-supported region. Numerical experiments further demonstrate that models with comparable empirical one-step prediction losses may exhibit substantially different multi-step behaviors. Rollout-error analysis and projected-gradient-descent (PGD) sensitivity directions additionally reveal that locally expansive regions align with worst-case perturbation amplification. Full article

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results. Full article

In this work, we construct a homotopy theory for a class of intersection graphs arising from topological monoids. We introduce the M-intersection graph of a ${\tau }_e$-monoid, where the vertices correspond to proper ${\tau }_e$-submonoids and adjacency is defined by trivial intersection. Several structural properties of the graph, including total disconnectedness, bipartiteness and planarity, are investigated and shown to be closely related to the algebraic structure and decomposition of finite ${\tau }_e$-monoids. Based on this framework, we develop a graphical homotopy theory by introducing graphical ${\tau }_e$-monoids, graphical homomorphisms, and graphical homotopies. We study graphical homotopy equivalence, graphical contractibility, and path monoids, and examine retraction properties through graphical retracts, D-graphical retracts and graphical homotopy extension properties. Furthermore, we present an example of graphical comprehensive monoids and construct a $\theta$-homogeneous topology on the set of graphical path homotopy classes. We show that this topology is compatible with the induced monoid operation, yielding a well-behaved functorial topological monoid structure. Full article

Building upon classical fixed point theory, the concept of enriched contractions introduces a new class of mappings. For a normed linear space $(X,\parallel ·\parallel )$, a mapping $T:X\to X$ is called an enriched contraction if there exist $b\in [0,\infty )$ and $\theta \in [0,b+1)$ such that $\parallel b(x-y)+Tx-Ty\parallel \le \theta \parallel x-y\parallel ,\forall x,y\in X.$ This class of mappings includes both the well-known Picard–Banach contraction and certain nonexpansive mappings. In this paper, we extend the definition by allowing $b\in \mathbb{R}\backslash \{-1\}$ instead of $b\in [0,\infty )$. This extension enables the condition to cover both contraction and certain nonexpansive mappings. We establish results on the existence and uniqueness of fixed points and present the Krasnosel’skii iteration for approximating such points. An example is provided to demonstrate mapping that meets the extended condition but not the original. Full article

After the introduction of the relation-theoretic contraction principle, the branch of metric fixed-point theory has attracted much attention in this direction, and various fixed-point results have been proven in the framework of relational metric space via different approaches. The aim of this article is to establish some fixed-point outcomes in the framework of relational metric space verifying a generalized nonlinear contraction utilizing three test functions $\mathrm{\Phi }$, $\mathrm{\Psi }$ and $\mathrm{\Theta }$ satisfying the appropriate characteristics. The findings obtained herein expand, sharpen, improve, modify and unify a few well-known findings. To demonstrate the utility of our outcomes, several examples are furnished. We utilized our outcomes to investigate a unique solution of second-order ordinary differential equations prescribed with specific boundary conditions. Full article

In this study, we introduce two new classes of contractions, namely enriched $(\mathfrak{I},\rho ,\chi )$-contractions and generalized enriched $(\mathfrak{I},\rho ,\chi )$-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including $\chi$-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and $(\rho ,\chi )$-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched $(\mathfrak{I},\rho ,\chi )$-contractions, employing a suitable binary relation, or control function $\chi \in \Theta$, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article

In this article, we will introduce a new generalized proximal $\theta$-contraction for multivalued and single-valued mappings named ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction and ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction. Using these newly constructed proximal contractions, we will establish new results for the coincidence best proximity point, best proximity point, and fixed point for multivalued mappings in the context of rectangular metric space. Also, we will reduce these contractions for single-valued mappings, named ${\left({\theta }_{\kappa }\right)}_{C_P}$-proximal contraction and ${\left({\theta }_{\kappa }\right)}_{B_P}$-proximal contraction, to establish results for the coincidence proximity point, best proximity point, and fixed point results. We will give some illustrated examples for our newly generated results with graphical representations. In the last section, we will also find the solution to the equation of motion by using our defined results. Full article

This paper investigates the existence of common attractors for generalized $\theta$-Hutchinson operators within the framework of partial metric spaces. Utilizing a finite iterated function system composed of $\theta$-contractive mappings, we establish theoretical results on common attractors, generalizing numerous existing results in the literature. Additionally, to enhance understanding, we present intuitive and easily comprehensible examples in one-, two-, and three-dimensional Euclidean spaces. These examples are accompanied by graphical representations of attractor images for various iterated function systems. As a practical application, we demonstrate how our findings contribute to solving a functional equation arising in a dynamical system, emphasizing the broader implications of the proposed approach. Full article

In this short note, we consider fractal interpolation in the Banach space $V^{\theta }\left(I\right)$ of convex Lipschitz functions defined on a compact interval $I\subset \mathbb{R}$. To this end, we define an appropriate iterated function system and exhibit the associated Read–Bajraktarević operator T. We derive conditions for which T becomes a Ratkotch contraction on a closed subspace of $V^{\theta }\left(I\right)$, thus establishing the existence of fractal functions of class $V^{\theta }\left(I\right)$. An example illustrates the theoretical findings. Full article

The axial-flow pump system has been widely applied to coastal drainage pump stations, but the hydraulic performance optimization based on the contraction angles of the inlet passage has not been studied. This paper combined the computational fluid dynamics (CFD) method, machine learning (ML) algorithms and genetic algorithm (GA) to find the optimal contraction angles of the inlet passage. The 125 sets of comprehensive objective function were obtained by the CFD method. Three contraction angles and comprehensive objective function values were regressed by three ML algorithms. After hyperparameter optimization, the Gaussian process regression (GPR) model had the highest R² = 0.958 in the test set and had the strongest generalization ability among the three models. The impact degree of the three contraction angles on the objective function of the GPR model was investigated by the Sobol sensitivity analysis method; the results indicated that the order of impact degree from high to low was ${\theta }_3>{\theta }_2>{\theta }_1$. The optimal objective function values of the GPR model and corresponding contraction angles were searched through GA; the maximum objective function value was 0.963 and corresponding contraction angles were ${\theta }_1=13.34°$, ${\theta }_2=28.36°$ and ${\theta }_3=3.64°$, respectively. The results of this study can provide reference for the optimization of inlet passages in coastal drainage pump systems. Full article

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

This paper combines the concept of an arbitrary binary connection with the widely recognized principle of $\theta$-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 $\theta$-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. Full article

The aim of this research is to introduce two new notions, $\mathsf{\Theta }$-($\mathsf{\Xi }$,h)-contraction and rational ($\alpha$,$\eta )$-$\psi$-interpolative contraction, in the setting of $\mathfrak{F}$-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. Full article