Functional Equation

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Functional Equation

 

an equation in which the unknown is a function.

Defined in this way, functional equations form a very large class. Although differential, integral, and finite difference equations are, in principle, functional equations, the term “functional equation” is not usually applied to equations of these types. Functional equations in the narrow sense are equations in which the unknown functions are linked to the given functions of one or more variables by means of the operation of composition of functions. A functional equation can also be viewed as the expression of a property that characterizes a certain class of functions. For example, the functional equation f(x) = f(– x) characterizes the class of even functions, and the functional equation f{x + 1) = f(x) characterizes the class of periodic functions of period 1.

One of the simplest functional equations is the equation f{x + y) = f(x) + f(y). Its continuous solutions are of the form f(x) = Cx. In the class of discontinuous functions, however, this equation has additional solutions. Related to this equation are the functional equations f(x + y) = f(x)f(y), f(xy) = f(x) + f(y), and f(xy) = f(x)f(y); their continuous solutions are eCx, C In x, and xα (x > 0), respectively. Thus, these functional equations determine the exponential, logarithmic, and power functions.

In the theory of analytic functions, functional equations are often used to introduce new classes of functions. For example, doubly periodic functions are characterized by the functional equations f(z + a) = f(z) and f(z + b) = f(z), and automorphic functions are characterized by the functional equations f(sα) = f(z), where {sα} is some group of linear fractional transformations. If a function is defined in a certain domain and satisfies a known functional equation, this equation can be used to extend the domain of definition of the function. Thus, the functional equation f(x + 1) = f(x) for a periodic function enables us to define the function for all x provided that we know its values on the interval [0,1]. This fact is often used for analytic continuation of functions of a complex variable. For example, by using the functional equation Γ(z + 1) = zΓ(z), we can define Γ(z) for all zas soon as we know its values in the strip 0 ≤ Re z ≤ 1 (seeGAMMA FUNCTION).

The symmetry conditions in a physical problem determine the transformation laws of the solutions of the problem for various coordinate transformations. The functional equations that must be satisfied by a solution of the problem are thereby determined. In turn, these functional equations often simplify the task of finding solutions.

Solutions of functional equations can be specific functions or classes of functions, the classes being dependent on arbitrary parameters or functions. For some functional equations, it is possible to obtain the general solution as soon as we know one or more particular solutions. For example, the general solution of the functional equation f(x) = f(ax) is φ[ω(x)], where φ(x) is an arbitrary function and ω(x) is a particular solution of the functional equation. In many cases, a functional equation can be solved by reducing it to a differential equation. This method yields only differentiable solution functions.

Another method of solving functional equations is the method of iteration. It is applicable, for example, to Abel’s equation f[α(x)] = f(x) + 1, where α(x) is a given function, and to the related Schroder equation f[α(x)] = Cf(x). A. N. Korkin showed that if α(x) is analytic then Abel’s equation has an analytic solution. These results were applied in the theory of Lie groups (seeTOPOLOGICAL GROUP), and subsequently led to the development of the iteration theory of analytic functions. In some cases it is possible to obtain solutions of Abel’s equation in closed form. Thus, a particular solution of the functional equation

REFERENCE

Atsel’, Ia. “Nekotorye obshchie metody v teorii funktsional’nykh uravnenii odnoi peremennoi: Novye primeneniia funktsional’nykh uravnenii.” Uspekhi matematicheskikh nauk, 1956, vol. II, issue 3, pp. 3–68.
References in periodicals archive ?
Manosas [8] gave the maximum number of polynomial solutions of some integrable polynomial Abel differential equations; Jaume Gine Claudia and Valls [9] studied the center problem for Abel polynomial differential equations of second kind; Jianfeng Huang and Haihua Liang [10] were devoted to the investigation of Abel equation by means of Lagrange interpolation formula; they gave a criterion to estimate the number of limit cycles of the Abel's equations; Berna Bulbul and Mehmet Sezer [11] introduced a numerical power series algorithm which is based on the improved Taylor matrix method for the approximate solution of Abel-type differential equations; Ni et al.
Liang, "Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves," Nonlinear Differential Equations and Applications NoDEA, vol.
Our strategy for proving Theorem 1 will be to transform the system (1) into a scalar Abel equation and to study it.
The study of periodic orbits of equation (1) that surround the origin, for [p.sub.2] [not equal to] 0, reduces to the study of non contractible solutions that satisfy x(0) = x(2[pi]) of the Abel equation
As it is well-known, the two first Lyapunov constants of an Abel equation are given by
Consider the Abel equation (2) and assume that either A([theta]) [not equivalent to] 0 or B([theta]) [not equivalent to] 0 does not change sign.
Condition (a) of Theorem 1 implies that one of the functions A([theta]), B([theta]) of the Abel equation does not change sign.
By Lemma 1, we reduce the study of the periodic orbits of equation (1) to the analysis of the non contractible periodic orbits of the Abel equation (2).
Sezer, "On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials," Applied Mathematics and Computation, vol.
Of course for Abel equations, to note that the results of this paper are valid if f (x) has sufficient continuity, otherwise it would be necessary to develop special starting formula, for this case see [1,4,5].
Abel Equations. Consider the generalized linear Abel integral equations of the first and second kinds, respectively, as [25]
Figures 3 and 4 show the plot of the error of presented method and the exact solution of Abel equations (Examples 1 and 2, resp.).