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A two-sided sequence ${(c ₙ)}_{n \in ℤ}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n + m} + c_{n - m} = 2 c ₙ c ₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${(c ₙ)}_{n \in ℤ}$ is bounded if $s u p_{n \in ℤ} | | c ₙ | | < \infty$ . A (bounded) group decomposition for a cosine sequence $c = {(c ₙ)}_{n \in ℤ}$ is a representation of c as $c ₙ = (b ⁿ + b^{- n}) / 2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $s u p_{n \in ℤ} | | b ⁿ | | < \infty$ , respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called...

On operator-valued solutions of d'Alembert's functional equation, I

J. Kisyński (1971)

Colloquium Mathematicae

On Opial-type inequalities in two varaiables.

Wing-Sum Cheung (1989)

Aequationes mathematicae

On orthogonally additive mappings.

Jürg Rätz (1985)

Aequationes mathematicae

On orthogonally additive mappings, IV.

J. Rätz, Gy. Szabó (1989)

Aequationes mathematicae

On orthogonally additive mappings. (Short Communication).

Jürg Rätz (1982)

Aequationes mathematicae

On principal iteration semigroups in the case of multiplier zero

Dorota Krassowska, Marek Zdun (2013)

Open Mathematics

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ , = φ: ℝ → ℝ|φ is continuous, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

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