Abstract
The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.
Highlights
In 1940, Ulam [24] conjectured the stability problem of the functional equation
In 1983, Cholewa [10] investigated the superstability of the sine functional equation f (x)f (y) = f x + y 2 − f x − y 2. His result was improved by Kim ([18, 20]) in the following generalized sine functional equation g(x)h(y) = f x + y 2 − f x − y 2
The present work continues the study for the stability and solution of the sine function and Wilson equation from the following Pexider type functional equation
Summary
In 1940, Ulam [24] conjectured the stability problem of the functional equation. year, Hyers [13] obtained partial answer for the case of additive mapping in this problem.Thereafter this problem was improved by Bourgin [9] in 1949, Aoki [4] in 1950, Th. His result was improved by Kim ([18, 20]) in the following generalized sine functional equation g(x)h(y) = f x + y 2 − f x − y 2. The present work continues the study for the stability and solution of the sine function and Wilson equation from the following Pexider type functional equation
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have