Linear Functional Equations Research Articles

УСЛОВИЯ СУЩЕСТВОВАНИЯ И ЕДИНСТВЕННОСТИ РЕШЕНИЙ ЛИНЕЙНЫХ ФУНКЦИОНАЛЬНЫХ УРАВНЕНИЙ В КЛАССАХ ПЕРВООБРАЗНЫХ ОТ ЛЕБЕГОВСКИХ ФУНКЦИЙ НА ПРОСТОЙ ГЛАДКОЙ КРИВОЙ

The article describes linear functional equations on simple smooth curves with a shift function and fixed points only at the ends of the curve. The case when the shift function has a nonzero derivative satisfying the Hölder condition is considered. The objective of the article is to find the conditions of the existence and uniqueness of such equations solution in the classes of Lebesgue functions antiderivatives with a coefficient and the right-hand part belonging to the same classes. These conditions depend on the values of the equation coefficient at the ends of the curve. It is shown that if the coefficient and the right-hand side of a functional equation belong to the class of Lebesgue functions antiderivatives, then its solution also belongs to this class. The indicators of Hölder and of classes of Lebesgue functions antiderivatives are determined for the solutions. The research method is based on F. Riesz’s criterion of a function’s belonging to the class of antiderivatives of Lebesgue integrable functions. The possibilities of applying linear functional equations for studying and solving singular integral equations with logarithmic singularities are shown.

Asymptotic Stability for a Class of Equations of Neutral Type

Under study is the problem of asymptotic but exponential stability for a class of linear autonomous neutral functional differential equations. We demonstrate that the asymptotic stability of an equation of the class takes place for all integrable initial functions if the roots of the characteristic equation lie on the left of and approach the imaginary axis.

ON SUFFICIENT CONDITIONS OF EXPONENTIAL STABILITY FOR SYSTEMS OF LINEAR AUTONOMOUS DIFFERENTIAL EQUATIONS WITH AFTEREFFECT

We consider systems of linear autonomous functional differential equa-tion with aftereffect and propose an approach to obtain effective sufficient conditions of exponential stability for these systems. In the approach we use the positiveness of the fundamental matrix of an auxiliary system (a comparison system) with concentrated and distributed delays.

Predictable Forward Performance Processes: Infrequent Evaluation and Robo-Advising Applications

We study discrete-time predictable forward processes when trading times do not coincide with performance evaluation times in the binomial tree model for the financial market. The key step in the construction of these processes is to solve a linear functional equation of higher order associated with the inverse problem driving the evolution of the predictable forward process. We provide sufficient conditions for the existence and uniqueness and an explicit construction of the predictable forward process under these conditions. Furthermore, we show that these processes are time-monotone in the evaluation period. Finally, we argue that predictable forward preferences are a viable framework to model preferences for robo-advising applications and determine an optimal interaction schedule between client and robo-advisor that balances a tradeoff between increasing uncertainty about the client's beliefs on the financial market and an interaction cost.

On oscillation of solutions for linear autonomous functional differential equations with two delays

On oscillation of solutions for linear autonomous functional differential equations with two delays

D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

Oscillation of damped second-order linear mixed neutral differential equations

This article concerns the oscillatory behavior of solutions to damped second-order linear functional differential equations with a mixed neutral term. The authors present new oscillation criteria that improve and extend some existing ones in the literature. Examples to illustrate the main results are included.

On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Functional Differential Equations of Pointwise Type: Bifurcation

The importance of functional differential equations of pointwise type is determined by the fact that their solutions are used to construct traveling-wave solutions for induced infinite-dimensional ordinary differential equations, and vice versa. Solutions of such equations exhibit bifurcation. A theorem on branching bifurcation is obtained for the solution to a linear homogeneous functional differential equation of pointwise type.

Computer assisted solution of systems of two variable linear functional equations

In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class.

Existence of Nonnegative Solutions of Linear Autonomous Functional Differential Equations

It is shown that if we exclude the existence of nontrivial small solutions, then a linear autonomous functional differential equation has a nontrivial nonnegative solution if and only if it has a nonnegative eigenfunction.

HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Zhang [‘On hyperstability of generalised linear functional equations in several variables’, Bull. Aust. Math. Soc.92 (2015), 259–267] proved a hyperstability result for generalised linear functional equations in several variables by using Brzdęk’s fixed point theorem. We complete and extend Zhang’s result. We illustrate our results for general linear equations in two variables and Fréchet equations.

Identification of the acoustic signal models of audio exchange systems under conditions of interference and acoustic feedback

In this paper questions of building models of information exchange systems with discrete and distributed delay and with delayed feedback by methods of the theory of linear functional differential equations are investigated. When solving the said equations, it is necessary to consider restrictions caused by the uncertainties in the system under modeling, such as the absence of the exact data on the parameters of the model elements, their natural spread and temporal variations, thus requiring the solution of an identification problem. The models with continuous aftereffect introduced in this work take a fuller account of reflected signal characteristics in closed space, which increases the reliability of modeling results in comparison with the known differential models. At the same time, there is a problem of finding functions that characterize the value distribution of the echo delay. In this work, these functions (kernels) are approximated by a number of exponents, which simplifies the equations and allows the assumption that the aftereffect can be concentrated on both final and infinite intervals. The echo components caused by closed-space resonances are modeled by transfer functions of the corresponding linear links. In numerical modeling, a single-channel model represented by a second-order resonance link and a pulse-shaped kernel described by a sum of two decreasing exponents is considered. The analysis of stability of the models of systems with delayed feedback is conducted by a frequency method. In the paper an approach to estimating the correlation and spectral functions of signals and noise components based on the parametric representation of the latter is considered. The paper also considers issues relating to the practical significance of the research results.

Functional Equations and Identification Problems

We consider identification problems and connections of such problems with linear functional equations. The results are applied to the study of solutions to evolution equations and Navier–Stokes equations.

APr solutions for linear functional difference equations with infinite delay with/without Favard's property

In this paper, we first consider Favard's type theorem that the linear functional difference equation (LFDE) with infinite delay has a unique solution, , if it has at least one bounded solution and the bounded solutions of the homogeneous equation in hull satisfy the separation among bounded solution. If there are bounded solutions which are non-separated, an almost periodic solution does not exist [Ortega and M. Tarallo, Almost periodic linear differential equations with non-separated solutions, J. Funct. Anal. 237 (2006), pp. 402–426]. We second consider without Favard's property that the LFDE with infinite delay has a unique solution, if the every non trivial bounded solution of the homogeneous equation in hull satisfies homoclinic to zero.

A Galerkin-like method for solving linear functional differential equations underinitial conditions

In this paper, we present a weighted residual Galerkin method to solve linear functional differential equations. We consider the problem with variable coefficients under initial conditions. Assuming the exact solution of the problem has a Taylor series expansion convergent in the relevant domain, we seek a solution of the given problem in the form of a polynomial having degree $N$ of our choice. Substituting this polynomial with unknown coefficients in the given equation yields an expression linear in these coefficients. We then proceed as in the weighted residual method and take inner product of this expression with monomials up to degree $N$, resulting in $N+1$ linear algebraic equations. Appropriately incorporating the initial conditions and solving the resulting linear system, we obtain the approximate solution to the given problem. Additionally, we present a way of estimating the absolute error of the obtained approximation, which is then used to improve the original approximation through a method called residual correction. We also show that the upper bound for the error of the proposed method depends on the Taylor truncation error of the exact solution. The proposed scheme and the residual correction technique are illustrated in several example problems.

Ulam-stability of a generalized linear functional equation, a fixed point approach

We introduce the general functional equation $$\begin{aligned} \sum _{i=1}^m A_i( f(\varphi _i (\bar{x}))) + b = 0, \quad \bar{x} := (x_1, x_2, \dots , x_n) \in X^n, \end{aligned}$$ and study its Ulam–Hyers-stability and hyperstability, using a fixed point approach, where m and n are positive integers, f is a mapping from a vector space X into a Banach space $$(Y, \Vert \ \Vert )$$ , and, for every $$i \in \{1, 2, \dots , m\}$$ , $$\varphi _i$$ is a linear mapping from $$X^n$$ into X, $$A_i$$ is a continuous endomorphism of Y and $$b \in Y$$ . Our result covers most of the former ones in the literature concerning the stability and hyperstability of linear functional equations, as well as new situations.

Линейные функциональные уравнения в гельдеровых классах функций на простой гладкой кривой

The article describes linear functional equations on simple smooth curves with a shift function having a non-zero derivative satisfying the Hölder condition, and fixed points only at the ends of the curve. The objective of the article is to find the conditions of the existence and uniqueness of the solution of such equations in the Hölder class functions with the coefficient and the right-hand side satisfying the Hölder conditions. These conditions are obtained depending on the values of the equation coefficient at the ends of the curve. Various specifics at the ends of the curve are considered. The indicators of the Hölder solutions are determined. The possibilities of applying linear functional equations to the study and solution of singular integral equations with logarithmic singularities are shown.

The Generalized Pomeron Functional Equation

This paper investigates the linear functional equation with constant coefficients φt=κφλt+ft, where both κ>0 and 1>λ>0 are constants, f is a given continuous function on ℝ, and φ:ℝ⟶ℝ is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ>1, then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0<κ<1, then the equation has a unique continuous solution; and (iii) if κ=1, then the equation has a continuous solution depending on a single parameter φ0 under a suitable condition on f.