Search Results (58)

The paper presents a shared control strategy that allows a human operator to physically guide the end-effector of a robotic manipulator to perform different tasks, possibly in interaction with the environment. To switch among different operational modes referring to a finite state machine algorithm, ElectroMyoGraphic (EMG) signals from the user’s arm are used to detect muscular contractions and to interact with a variable admittance control strategy. Specifically, a Support Vector Machine (SVM) classifier processes the raw EMG data to identify three classes of contractions that trigger the activation of different sets of admittance control parameters corresponding to the envisaged operational modes. The proposed architecture has been experimentally validated using a Kinova Jaco² manipulator, equipped with force/torque sensor at the end-effector, and with a limited group of users wearing Delsys Trigno Avanti EMG sensors on the dominant upper limb, demonstrating promising results. 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

A granular fuzzy fractional financial system (GFFFS) is important for modeling real-world market uncertainties and complexities compared to conventional financial models. Unlike traditional approaches, a GFFFS offers enhanced precision in risk assessment, captures the long-term memory effects with the fractional derivatives, and effectively deals with the uncertainty and granularity in financial data through fuzzy logic. This model overcomes the limitations of the traditional model by accurately representing nonlinear dynamics, extreme volatility, and uncertain behavioral shifts in financial markets. The study of such models can be complex and challenging. However, developing an effective technique for solving such systems analytically and approximately is essential. This article aims to introduce and investigate a GFFFS using granular Caputo fractional derivatives. The behavior of the proposed model is studied using two distinct approaches, including an analytical approach, by applying the fuzzy Laplace transform technique and a numerical approach by employing fuzzy integral equations. Moreover, the existence and uniqueness of the extracted fuzzy solution are determined using the Banach contraction principle. To analyze the nonlinearity of the proposed model, the introduced numerical scheme is employed to illustrate the uncertain behavior of the proposed model graphically. This research provides deeper insights that can help decision-makers make better financial market decisions. 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

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the $\varphi$-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for $\varphi \left(\mathfrak{t}\right)$ = $\mathfrak{t}$, $\mathfrak{a}=1$. Our findings match the Riemann–Liouville fractional operator for $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=0$. They agree with the Hadamard and Caputo–Hadamard fractional operators when $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=0$ and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=1$, respectively. Second, regarding the space, we are make generalizations for the case $\mathfrak{p}=2$. Full article

In this manuscript, we present several novel results in fixed-point theory for a complete controlled metric space. The first presented result is inspired from the Caristi contraction where we explore the existence and uniqueness of fixed points under specific conditions. Furthermore, we propose a graphical form of it by endowing the considered space with a graph and develop a new fixed-point theorem, which is illustrated by two examples. Also, we establish a theorem for the $\alpha$-admissible mapping. To demonstrate its effectiveness, the last theorem proposes an approach to solve a second-order differential equation. Full article

This research presents a new form of iterative technique for Garcia-Falset mapping that outperforms previous iterative methods for contraction mappings. We illustrate this fact through comparison and present the findings graphically. The research also investigates convergence of the new iteration in uniformly convex Banach space and explores its stability. To further support our findings, we present its working on to a BV problem and to a delay DE. Finally, we propose a design of an implicit neural network that can be considered as an extension of a traditional feed forward network. Full article

This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties. Full article

This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and $D^{\ast \ast }$. Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications. Full article

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. 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 paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric space endowed with a directed graph. A non-trivial example is presented to support the results presented herein. Our results improve and extend some recent results in the existing literature. Full article

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory. Full article

This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others. Full article

In this study, we significantly extend the concept of modular metric-like spaces to introduce the notion of b-metric-like spaces. Furthermore, by incorporating a binary relation $\mathfrak{R}$, we develop the framework of $\mathfrak{R}$-modular b-metric-like spaces. We establish a groundbreaking fixed point theorem for certain extensions of Geraghty-type contraction mappings, incorporating both $𝒵$ simulation function and $E$-type contraction within this innovative structure. Moreover, we present several novel outcomes that stem from our newly defined notations. Afterwards, we introduce an unprecedented concept, the graphical modular b-metric-like space, which is derived from the binary relation $\mathfrak{R}$. Finally, we examine the existence of solutions for a class of functional equations that are pivotal in dynamic programming and in solving initial value problems related to the electric current in an RLC parallel circuit. Full article