Functional Equation

Functional equations form a modern branch of mathematics. Topics which are covered under functional equation include Cauchy equations, additive functions, functional equations for distance measures and Pexider’s functional equations. In mathematics, a functional equation is any equation that specifies a function in implicit form. Often the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional equations they satisfy. The term functional equation usually refers to equations that cannot be simply reduced to algebraic equations.

A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example the following are functional equations: f (x) + 2f (1/x) =2x, g (x) +4g (x) +4= 8 sinx. An equation of the form f(x, y) =0, where f contains a finite number of independent variables, known functions, unknown functions which are used to solve for. Many properties of functions can be determined by studying the types of functional equations that they satisfy.

Solving functional equations

Solving functional equations can be very difficult but there are some common methods of solving them. For example, in dynamic programming a variety of successive approximation methods are used to solve Bellman’s functional equation, including methods based on fixed point iterations. The main method solving elementary functional equation is substitution. It is often useful to prove surjectivity or injectivity and prove oddness or evenness, if possible.

The following functional equations are as a generalization of the b- parts functional equation for semi groups and groups, even in a binary system,

Associate equations: F (f (xy) z) = f (xf (yz)), f (f (xy) z) = f (xf (yz)) = f (xyz)

Decomposer equations: F (f (xy) z) = f (xf (yz)), f (f (xy) z) = f (xy (yz)) f (xyz)

Strong decomposer equations:  f (f*(x) y) = f (y), f (xf*(y)) = f (x)

Canceler equations: f (f(x) y) = f (xy), f (xf(y)) = f (xy), f (xf(y) z) = f (xyz)

Cauchy’s functional equation:

Cauchy's functional equation is represented by f (x+y) =f(x) + f (y), it was proved by Cauchy in 1821 and the solutions to this equation can be called as additive functions. The Cauchy equation has still control over the real numbers. Constraints on f sometimes preclude other solutions in the following cases if f is continuous, if f is monotonic and bounded on any interval.

Questions:

• What is Functional equation?
• What are the methods used in solving Functional equation?