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On operator-valued cosine sequences on UMD spaces

Wojciech Chojnacki (2010)

Studia Mathematica

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

On oscillation and nonoscillation properties of Emden-Fowler difference equations

Mariella Cecchi, Zuzana Došlá, Mauro Marini (2009)

Open Mathematics

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation Δ ( a n Δ x n α s g n Δ x n ) + b n x n + 1 β s g n x n + 1 = 0 is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

On oscillatory nonlinear fourth-order difference equations with delays

Arun K. Tripathy (2018)

Mathematica Bohemica

In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form Δ 2 ( r ( n ) Δ 2 ( y ( n ) + p ( n ) y ( n - m ) ) ) + q ( n ) G ( y ( n - k ) ) = 0 is studied under the assumption n = 0 n r ( n ) < . New oscillation criteria have been established which generalize some of the existing results in the literature.

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.

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