Generalized Cauchy difference functional equations

The Pompeiu functional equation is defined by Neagu for Schwartz distributions. His method is extended to the four Cauchy functional equations by means of two new operators Q ∗ {Q^*} and R ∗ {R^*} on D ′ ( I ) \mathcal {D}’(I) . The Cauchy equations in distributions reduce to the classical equations when the solutions are regular distributions, i.e. locally integrable functions.

and determine the conditions on a polynomial F (x, y) in order that (1) can have a continuous strictly monotone (say increasing) solution. If F(x, y) is x + y or xy, we have the classical cases of Cauchy's functional equations. The cases when F(x, y) is a general linear function,1 and also the case when it is x + y + nxy, have been treated.2 In this paper the case when F(x, y) is a polynomial of degree greater than unity is settled, as follows:

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.

Equations including unknown functions are called function al equations([1]). Among them, functionalequations containing only one independent variable are cal led iterative functional equations([9]). On theother hand, functional equations, like Cauchy's equation( [10]), containing two or more independent vari-ables for unknown functions of a single variable are called f unctional equations of several variables([2]).Among iterative functional equations, the problem of itera tive roots is important. If the n -th iterateof a function f is equal to g, i.e.f

The paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

Approximate versions of Cauchy's functional equation

A Generalization of Cauchy's Functional Equations

SOME RESULTS IN FUNCTIONAL INTEGRAL EQUATIONS IN A BANACH SPACE

Let X be a real inner product space of dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the recent studies made on the alternative-conditional functional equation $$(x, y) = 0 \Rightarrow f(x + y)^2 = [f(x) + f(y)]^{2}$$ in this paper we study another alternative-conditional equation related to the previous, namely $$(x, y) = 0 \Rightarrow [f(x + y) - f(x)]^2 = f(y)^2$$ in order to describe its solutions. This research follows previous studies concerning alternative functional equations and the Cauchy functional equation on the restricted domain of the orthogonal vectors.

Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.

<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>

In this paper, we consider a fixed point problem related to some contraction mappings and introduce new classes of Picard operators for such mappings in the framework of F-metric space, yielding some interesting and novel results. As application of the obtained results, we investigate the Hyers-Ulam stability of a fixed point problem, a Cauchy functional equation, and an integral equation. Also, we present the well-posedness of the fixed point problem and integral equation. Some illustrative examples are also provided to support the new findings.

In Chap. 17, n-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the n-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most n−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of n-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the n-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for n-th-order functional differential equations.

This paper presents some functional equations that have played an essential role in the characterization of utility and probability functions in decision analysis. We survey some previous results with improvements and derive several new results. We also discuss some simple methods for the solution of these equations and highlight some subtle points about their use. We show that one functional equation can determine several unknown functions within it, and that relaxing differentiability and requiring only continuity of the functions leads to generalizations of many well-known results in utility and Bayesian probability theory.

The time-periodic response of a gas-filled tube to a reciprocating piston is described in terms of the solution to a functional difference equation. When both the gas velocity and acceleration remain small, this functional equation may be approximated by a nonlinear ordinary differential equation. In this paper we demonstrate, through three classes of exact solutions, when and in what sense the differential equation is a good approximation to the functional equation.