Contractive Research Articles

Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays

<abstract><p>This paper explores multimode function multistability of Cohen-Grossberg neural networks (CGNNs) with Gaussian activation functions and mixed time delays. We start by using the geometrical properties of Gaussian functions. The state space is partitioned into $ 3^\mu $ subspaces, where $ 0\le \mu\le n $. Moreover, through the utilization of Brouwer's fixed point theorem and contraction mapping, some sufficient conditions are acquired to ensure the existence of precisely $ 3^\mu $ equilibria for $ n $-dimensional CGNNs. Meanwhile, there are $ 2^\mu $ and $ 3^\mu-2^\mu $ multimode function stable and unstable equilibrium points, respectively. Ultimately, two illustrative examples are provided to confirm the efficacy of theoretical results.</p></abstract>

RATIONAL TYPE INTUITIONISTIC FUZZY CONTRACTION RESULTS IN INTUITIONISTIC FUZZY METRIC SPACES WITH AN APPLICATION

The purpose of this study is to propose a new idea of rational type intuitionistic fuzzy contraction mappings in intuitionistic fuzzy metric spaces. We show several fixed point results in intuitionistic fuzzy metric spaces under rational type intuitionistic fuzzy contraction conditions, with detailed examples to back up our claims. This revolutionary idea will be crucial in the theory of intuitionistic fuzzy fixed point outcomes and can be applied to other contractive type mappings in the setting of intuitionistic fuzzy metric spaces. Furthermore, to support our work, we offer an application of a nonlinear integral type problem to obtain the current result for a unique solution.

B-metric spaces and the related approximate best proximity pair results using contraction mappings

b-metric spaces and the related approximate best proximity pair results using contraction mappings

A FIXED POINT THEOREM FOR MORE GENERALIZED CONTRACTION IN PARTIAL b-METRIC SPACE

The main aim of this work is to introduce and study the common fixed point for some new contractive mappings in partial b-metric spaces. We also establish fixed point result for contractive mappings introduced by Jain and Kaur in framework of partial b-metric spaces. We present some examples in support of our results.

Adaptive Synaptic Scaling in Spiking Networks for Continual Learning and Enhanced Robustness.

Synaptic plasticity plays a critical role in the expression power of brain neural networks. Among diverse plasticity rules, synaptic scaling presents indispensable effects on homeostasis maintenance and synaptic strength regulation. In the current modeling of brain-inspired spiking neural networks (SNN), backpropagation through time is widely adopted because it can achieve high performance using a small number of time steps. Nevertheless, the synaptic scaling mechanism has not yet been well touched. In this work, we propose an experience-dependent adaptive synaptic scaling mechanism (AS-SNN) for spiking neural networks. The learning process has two stages: First, in the forward path, adaptive short-term potentiation or depression is triggered for each synapse according to afferent stimuli intensity accumulated by presynaptic historical neural activities. Second, in the backward path, long-term consolidation is executed through gradient signals regulated by the corresponding scaling factor. This mechanism shapes the pattern selectivity of synapses and the information transfer they mediate. We theoretically prove that the proposed adaptive synaptic scaling function follows a contraction map and finally converges to an expected fixed point, in accordance with state-of-the-art results in three tasks on perturbation resistance, continual learning, and graph learning. Specifically, for the perturbation resistance and continual learning tasks, our approach improves the accuracy on the N-MNIST benchmark over the baseline by 44% and 25%, respectively. An expected firing rate callback and sparse coding can be observed in graph learning. Extensive experiments on ablation study and cost evaluation evidence the effectiveness and efficiency of our nonparametric adaptive scaling method, which demonstrates the great potential of SNN in continual learning and robust learning.

Common fixed point theorems for (φ,F)-contraction mappings in cone b-metric spaces over Banach algebra with application

Common fixed point theorems for (φ,F)-contraction mappings in cone b-metric spaces over Banach algebra with application

Some fixed point theorems in fuzzy metric-like spaces

This paper aims to establish some fixed point results for a new class of contractive type mappings in fuzzy metric-like spaces. The results of this paper generalize and extend the Banach contraction principle and some other known results in the setting of fuzzy metric-like spaces.

On AI-iteration process for finding fixed points of enriched contraction and enriched nonexpansive mappings with application to fractional BVPs

On AI-iteration process for finding fixed points of enriched contraction and enriched nonexpansive mappings with application to fractional BVPs

Stability Analysis and Simulation of Diffusive Vaccinated Models

This paper begins by analyzing the key mathematical properties of diffusive vaccinated models, including existence, uniqueness, positivity, and boundedness. Equilibria are identified, and the basic reproductive number is calculated. The Banach contraction mapping principle is applied to rigorously establish the solution existence and uniqueness. In order to understand the disease’s time transmission, it is important to examine the global stability of the equilibrium points. Disease‐free equilibrium and endemic equilibrium are the two equilibria in this model. Here, we demonstrate that the endemic equilibrium is worldwide asymptotic stable when the basic reproductive number is greater than 1, and the disease‐free equilibrium is globally asymptotic stable whenever the basic reproductive number is less than 1. Moreover, based on the Caputo fractional derivative of order and the implicit Euler’s approximation, we offered an unconditionally stable numerical solution for the resultant system. This work explores the solution of some significant population models of noninteger order using an approach known as the iterative Laplace transform. The proposed methodology is developed by effectively combining Laplace transformation with an iterative procedure. A series form solution that exhibits some convergent behavior towards the precise solution can be attained. It is noted that there is a close contact between the obtained and precise solutions. Moreover, the suggested method can handle a variety of fractional order derivative problems because it involves minimal computations. This information will be helpful in further studies to determine the ideal strategy of action for preventing or stopping the spread disease transmission.

The $ AA $-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales

<p>We explored the $ AA $-iterative algorithm within the hyperbolic spaces (HSs), aiming to unveil a stability outcome for contraction maps and convergence outcomes for generalized $ (\alpha, \beta) $-nonexpansive ($ G\alpha \beta N $) maps in such spaces. Through this algorithm, we derived compelling outcomes for both strong and $ \Delta $-convergence and weak $ w^2 $-stability. Furthermore, we provided an illustrative example of $ G\alpha \beta N $ maps and conducted a comparative analysis of convergence rates against alternative iterative methods. Additionally, we demonstrated the practical relevance of our findings by applying them to solve the linear Fredholm integral equations (FIEs) and nonlinear Fredholm-Hammerstein integral equations (FHIEs) on time scales.</p>

Mittag-Leffler projective synchronization of uncertain fractional-order fuzzy complex valued neural networks with distributed and time-varying delays

<p>To study the Mittag-Leffler projective synchronization (MLPS) problem of fractional-order fuzzy neural networks (FOFNNs), in this work we introduced the FOFNNs model. On this basis, we discussed the MLPS of uncertain fractional-order fuzzy complex valued neural networks (FOFCVNNs) with distributed and time-varying delays. Utilizing Banach contraction mapping principle, we proved the existence and uniqueness of the model solution. Moreover, employing the construction of a new hybrid controller, an adaptive hybrid controller, and the fractional-order Razumikhin theorem, algebraic criteria was obtained for implementing MLPS. The algebraic inequality criterion obtained in this article improves and extends the previously published results on MLPS, making it easy to prove and greatly reducing the computational complexity. Finally, different Caputo derivatives of different orders were given, and four numerical examples were provided to fully verify the accuracy of the modified criterion.</p>

Stability Analysis of Equilibrium of DC Microgrid Under Distributed Control

Constant power loads (CPLs) often lead to system instability and voltage collapse. This paper analyzes the existence and stability of equilibrium of meshed DC microgrids with multiple CPLs under distributed control, which aims to realize current sharing and voltage regulation. Firstly, the power-flow equation of the system is derived. To analyze the solvability of power-flow equation, we transform the problem of the multidimensional quadratic equation solvability into the existence of a fixed-point for a contraction mapping. Then, the sufficient solvability condition is derived based on Banach's fixed-point theorem. Secondly, we build the small-signal model to determine the system qualitative behavior around equilibrium. By analyzing the eigenvalue of system Jacobian matrix based on inertia theorem, the analytical stability conditions are obtained. Further, under the load uncertainty, a robust stability condition which only depends on the maximum load information instead of the real-time load information is derived. Simulation results verify the feasibility of the proposed theorems.

Generalized $ (\alpha_s, \xi, \hbar, \tau) $-Geraghty contractive mappings and common fixed point results in partial $ b $-metric spaces

<abstract><p>In this paper, we introduce two new classes of mixed $ (\mathcal{S, T}) $-$ \alpha $-admissible mappings and interspersed $ (\mathcal{S}, \mathfrak{g}, \mathcal{T}) $-$ \alpha $-admissible mappings and study the sufficient conditions for the existence and uniqueness of a common fixed point of generalized $ (\alpha_s, \xi, \hbar, \tau) $-Geraghty contractive mapping in the framework of partial $ b $-metric spaces. We also provide two examples to show the applicability and validity of our results. Moreover, we present an application to the existence of solutions to an integral equation by means of one of our results.</p></abstract>

A NEW FIXED POINT RESULT IN GRAPHICAL bv(s)-METRIC SPACE WITH APPLICATION TO DIFFERENTIAL EQUATIONS

In the present paper, motivated by [13,15], first we give a notion of graphical bv(s)-metric space, which is a graphical version of bv(s)-metric space. Utilizing the graphical Banach contraction mapping we prove fixed point results in graphical bv(s)-metric space. Appropriate examples are also presented to support our results. In the end, the main result ensures the existence of a solution for an ordinary differential equation along with its boundary conditions by using the fixed point result in graphical bv(s)-metric space.

A generalized Gronwall inequality via $ \psi $-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system

<p>This paper establishes a novel generalized Gronwall inequality concerning the $ \psi $-Hilfer proportional fractional operators. Before proving the main results, the solution of the linear nonlocal coupled $ \psi $-Hilfer proportional Cauchy-type system with constant coefficients under the Mittag-Leffler kernel is created. The uniqueness result for the proposed coupled system is established using Banach's contraction mapping principle. Furthermore, a variety of the Mittag-Leffler-Ulam-Hyers stability of the solutions for the proposed coupled system is investigated. Finally, a numerical example is given to show the effectiveness and applicability of the obtained results, and graphical simulations in the case of linear systems are shown.</p>

Best proximity point theorems for tricyclic diametrically contractive mappings

Best proximity point theorems for tricyclic diametrically contractive mappings

Existence and blow up for viscoelastic hyperbolic equations with variable exponents

<p>In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the Faedo$ - $Galerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.</p>

Multivariate fixed point and multivariate common fixed point theorems for $(\psi,\varphi)$-weakly contractive mappings with applications

Multivariate fixed point and multivariate common fixed point theorems for $(\psi,\varphi)$-weakly contractive mappings with applications

On fixed point and an application of $ C^* $-algebra valued $ (\alpha, \beta) $-Bianchini-Grandolfi gauge contractions

<abstract><p>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, $ \alpha $-admissibility, and $ (\alpha, \beta) $-admissible Geraghty contractive mapping in the sense of $ C^* $-algebra. Using these definitions, we define $ (\alpha, \beta) $-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.</p></abstract>

A new class of hybrid contractions with higher-order iterative Kirk's method for reckoning fixed points

<p>The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched $ F $-contraction and enriched $ F^{\prime } $-contraction associated with $ u $-fold averaged mapping, which are involved with Kirk's iterative technique with order $ u $. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, $ u $-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the $ u $-fold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the $ u $-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a $ u $-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the $ u $-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature.</p>