On a problem of Th. Davison

Rok, strany: 2006, 229 - 235

A general solution $h:[0,1) o Bbb R$ of the functional equation of the form

$$ h(x) = ((1) / (a))iggl[higgl(((x) / (a))iggr) + higgl(((x+1) / (a))iggr)iggr] ,   x in [ 0,min{a-1,1}) , $$

where $a > 1$ is a given constant, depends on an arbitrary function. Some theorems on continuous solutions for this equation are also given. Answering a problem of Th. Davison: [Problem 3, Aequationes Math. 64, (2002), 186] we determine all functions $h:[0,1) o Bbb R $ satisfying the following functional equation

$$ h(x) = ((1) / (a))iggl[higgl(((x) / (a))iggr) + higgl(((x+1) / (a))iggr)iggr],   x in [0, min{1,a-1}), $$

where $a in (1,∞) setminus {2}$ is a given constant. In the case $ a = 2 $ this equation was considered by many authors (T. Hilberding [A functional equation for the cotangent on the open unit interval, Aequationes Math. 61, (2001), 179–189], Z. Kominek, J. Matkowski [{it On the functional equation $φ(x)=αφ(α x)+{(1-α)φ(1-(1-α)x)}$}, Bull. Math. Soc. Sci. Math. Répub. Soc. Roum. 30(78), (1986), 327–334], W. Walter [Old and new approaches to Euler's trigonometric expansions, Amer. Math. Monthly, 89, 1982, 225–230], for example). Generally, our equation has a large class of solutions, however, in the case $ a >2$ the zero function is a unique continuous solution for our equation.

ISO 690:

Kominek, Z. 2006. On a problem of Th. Davison. In Tatra Mountains Mathematical Publications, vol. 34, no.3, pp. 229-235. 1210-3195.

APA:

Kominek, Z. (2006). On a problem of Th. Davison. Tatra Mountains Mathematical Publications, 34(3), 229-235. 1210-3195.