## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 3, Number 1 (2009), 28-35.

### Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation

#### Abstract

We study the connection between conjugations of a special kind of dynamical
systems, called *P-configurations*, and solutions to homogeneous Cauchy
type functional equations. We find that any two *regular* P-configurations
are conjugate by a homeomorphism, but cannot be conjugate by a diffeomorphism.
This leads us to the following conclusion (answering an open question posed by
Paneah): *there exist continuous nonlinear solutions to the functional
equation:* $$ f(t) = f\left(\frac{t+1}{2}\right) + f\left(\frac{t-1}{2}\right)
\,\, , \,\, t \in [-1,1] . $$

#### Article information

**Source**

Banach J. Math. Anal. Volume 3, Number 1 (2009), 28-35.

**Dates**

First available: 21 April 2009

**Permanent link to this document**

http://projecteuclid.org/euclid.bjma/1240336420

**Mathematical Reviews number (MathSciNet)**

MR2461743

**Zentralblatt MATH identifier**

1157.39013

**Digital Object Identifier**

doi:10.15352/bjma/1240336420

**Subjects**

Primary: 39B22: Equations for real functions [See also 26A51, 26B25]

Secondary: 37B99: None of the above, but in this section

**Keywords**

conditional functional equation Cauchy type functional equation P-configuration guided dynamical system

#### Citation

Shalit, Orr Moshe. Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation. Banach Journal of Mathematical Analysis 3 (2009), no. 1, 28--35. doi:10.15352/bjma/1240336420. http://projecteuclid.org/euclid.bjma/1240336420.