Functional Equations Research Articles

Pth Moment Exponential Stability of Impulsive Stochastic Differential Equations with Unbounded Delays

Stochastic impulsive differential equations with Markovian switching have been applied extensively in various areas including ecology systems, neural networks, and control systems, and stability analysis is one fundamental premise of their applications. For two categories of Markovian switched impulsive stochastic differential functional equations with unbounded delays, this paper investigates the pth moment exponential stability by adopting stochastic Lyapunov stability theory and stochastic analysis approach. Several criteria on pth moment exponential stability have been acquired. In the proposed model, the time-varying coefficients and the hybrid impulsive effects are considered simultaneously. It can be seen that the criteria derived in this paper are more concise and the conditions are easier to verify compared with those existing results based on Razumikhin theory. Finally, two examples are illustrated to show the effectiveness of the theoretical findings.

Hyperstability for a Generalized Class of Pexiderized Functional Equations on Monoids via Páles’ Approach

Hyperstability for a Generalized Class of Pexiderized Functional Equations on Monoids via Páles’ Approach

A modified parameterization method for invariant Lagrangian tori for partially integrable Hamiltonian systems

In this paper we present an a-posteriori KAM theorem for the existence of an (n−d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω∈Rd, of type (γ,τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n−d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω,ωp), with ωp∈Rn−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width ρ, and the corresponding error in the functional equation is ɛ. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ−2ρ−2τ−1ɛ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ−4ρ−4τɛ is small enough. The approach is suitable to perform computer assisted proofs.

Asymptotic properties for the parameter estimation in stochastic (functional) differential equations with Hölder drift

In this paper, by taking Zvonkin's transformation, we investigate parameter estimation for a class of multidimensional stochastic differential equations with small perturbation parameters in diffusion coefficients, where the drift coefficients not only have an unknown parameter θ but also are Hölder continuous. These processes may enhance the applicability of our results to considerable practical models. Due to the irregular drift, the primary challenge is dealing with the mean square error between an accurate and numerical solution. Under these settings, we demonstrate the consistency and asymptotic normality of error concerning the least squares estimator in probability when stepsize δ → 0 and small parameter ε → 0 simultaneously. Moreover, we extend the results to the case of stochastic functional differential equations.

Hydromechanical Transmission IC2OC: Component Sizing and Optimization

The IC2OC transmission is a continuous transmission whose layout can change from simple IC to simple OC configuration and vice versa. It was proposed to cover a wider range of vehicle speeds without adding gears. Its sizing can lead to higher efficiencies than those of the IC and OC layouts. Therefore, this work deals with the sizing methodologies of this transmission. Two methodologies are proposed and discussed: the first uses the functional and constitutive equations of the transmission; the second is based on a mathematical programming problem. Both methodologies start from the choice of the full mechanical point speeds. The comparison between the two methods is carried out on the transmission of a commercially available 230 kW reach stacker. The comparison shows that the functional method, leaner and faster, can provide results very close to those obtained with the heavy and time-consuming optimization, provided that the values of the two full mechanical point speeds are the optimal ones for the two basic transmissions taken individually.

Stabilization by discrete‐time feedback control for highly nonlinear hybrid neutral stochastic functional differential equations with infinite delay

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example.

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

A functional equation related to Wigner’s theorem

A functional equation related to Wigner’s theorem

Index of braking behaviour in two dimensions within risk perception

Time To Collision (TTC) and other physical indices are widely used to predict drivers' braking behaviour. However, these indices often overlook the psychological aspects of how drivers perceive risk. Moreover, such indices are predominantly one-dimensional and do not account for the shared space, where vehicles can move in two dimensions. In this study, we introduce an index designed to predict drivers' braking behaviour in the shared space, taking into account the drivers' risk perception. In Experiment 1, a functional equation was derived to estimate the probability that drivers would brake in a given situation. The results confirmed that the index provides an adequate reflection of drivers' risk perception. In Experiment 2, the braking rate estimated with the index provided a better fit to a third-person risk evaluation when compared to other physical indices.

Functional matching and renormalization group equations at two-loop order

We present a systematic method for determining the two-loop effective Lagrangian resulting from integrating out a set of heavy particles in an ultraviolet scalar theory. We prove that the matching coefficients are entirely determined from the (double-)hard region of the loop integrals and present a master formula for matching, applicable to both diagrammatic and functional approaches. We further employ functional methods to determine compact expressions for the effective Lagrangian that do not rely on any previous knowledge of its structure or symmetries. The same methods are also applicable to the computation of renormalization group equations. We demonstrate the application of the functional approach by computing the two-loop matching coefficients and renormalization group equations in a scalar toy model.

Stability of impulsive stochastic functional differential equations with delays

In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results.

Exact Solutions for a Class of Variable Coefficients Fractional Differential Equations Using Mellin Transform and the Invariant Subspace Method

AbstractIn this paper, we propose a class of variable coefficients fractional ordinary differential equations (FODEs). Using Mellin transform (MT), we have transformed this class into a functional equation which can’t be solved in general. So, we have selected many special cases of this functional equation that can be solved exactly. After solving these special cases of the functional equation and using the inverse MT, we obtained some exact solutions for the proposed class. The obtained solutions are given in the form of $$H$$ H -function and the Wright function. The results, as special cases, contain some special forms given in the literature. Also, the invariant subspace method (ISM) is utilized for solving a class of nonlinear fractional diffusion equations with variable coefficients. The solutions of this class of nonlinear fractional diffusion equations depend upon the solutions of the proposed class of FODEs.

Eberlein-Weakly Almost Periodic Solutions for Some Partial Functional Differential Equation with Infinite Delay

Eberlein-Weakly Almost Periodic Solutions for Some Partial Functional Differential Equation with Infinite Delay

On Suzuki-Proinov Type Contractions in Modular $b$-Metric Spaces with an Application

In this paper, by taking ${{\mathcal C}_\mathcal{A}}-$simulation function and Proinov type function into account, we set up a new contraction mapping called Suzuki$-$Proinov $\mathpzc{Z^*}_{\aE^*}^{\aR}(\alpha)-$contraction, including both rational expressions that possess quadratic terms and $\aE-$type contractions. Furthermore, we demonstrate a common fixed point theorem through the mappings endowed with triangular $\alpha-$admissibility in the setting of modular $b-$metric spaces. Besides that, we achieve some new outcomes that contribute to the current ones in the literature through the main theorem, and, as an application, we examine the existence of solutions to a class of functional equations emerging in dynamic programming.

Speed of Light or Composition of Velocities

Abstract We analyze in our paper questions of the theory of relativity. We approach this theory from the point of view of velocities and their composition. This is where the functional equations appear. Solving them leads to a world where velocities are bounded from above, the upper bound being exactly the “speed of light”.

On a new approach of enriched operators

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.

An investigation of Fractional Mixed Functional Integro-Differential Equations with Impulsive Conditions

An investigation of Fractional Mixed Functional Integro-Differential Equations with Impulsive Conditions

P-Laplacian functional differential equations with neutral delay arguments: new oscillation theorems

Inthisstudy,weareinterestedininvestigatingtheoscillatorybehaviorofsolutionstoageneralclassoffunctionaldifferential equations. We consider a neutral-type equation with multiple delays. We first test some monotonic properties of positive solutions to the studied equation. Then, we use some techniques to obtain criteria that guarantee the oscillation of all solutions. We obtain three different forms of oscillation criteria and compare them in terms of efficiency by applying them to a special case of the studied equation. The results obtained are an extension and generalization of previous results in the literature.

Stability of Stochastic Functional Differential Equations With Past-Dependent Random Switching Involving Countably Infinite States

Stability of Stochastic Functional Differential Equations With Past-Dependent Random Switching Involving Countably Infinite States

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods.