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Journal: Annales Mathematicae Silesianae

Unified approach to bounded, periodic and almost periodic solutions of differential systems

Jan Andres , Bohumil Krajc

Language: EN | Published: 30-09-1997 | Abstract | pp. 39-53

The criteria for an entirely bounded solution of a quasi-linear differential system are developed via asymptotic boundary value problems. The same principle allows us to deduce at the same time the existence of periodic orbits, when assuming additionally periodicity in time variables of the related right-hand sides. For almost periodicity, the situation is unfortunately not so straightforward. Nevertheless, for the Lipschitzean uniformly almost periodic (in time variables) systems, we are able to show that every bounded solution becomes almost periodic as well.

1997-09-30



Journal: Annales Mathematicae Silesianae

Generalized periodic solutions of ordinary linear differential equations in the Colombeau algebra

Jan Ligęza

Language: EN | Published: 30-09-1997 | Abstract | pp. 67-87

It is shown that from the fact that the unique periodic solution of homogeneous system of equations is the trivial one it follows the existence of periodic solutions of nonhomogeneous systems of equations in the Colombeau algebra.

1997-09-30



Journal: Annales Mathematicae Silesianae

The problem of classification of triangular maps with zero topological entropy

Zdeněk Kočan

Language: EN | Published: 30-09-1999 | Abstract | pp. 181-192

Relations between the following properties of triangular maps F: I2→I2 are studied in this paper: (P1) the period of any cycle is a power of two; (P2) every cycle is simple; (P3) the topological entropy of F restricted to the set of periodic points is 0; (P4) the topological entropy of F is 0; (P5) every ω-limit set contains a unique minimal set; (P6) F has no homoclinic trajectory; (P7) every ω-limit set either is a cycle or contains no cycle; (P8) no infinite ω-limit set contains a cycle. It is known that for continuous maps of the interval these properties are mutually equivalent. In the case of triangular maps of the square we prove that the properties (P1), (P2) and (P3) are mutually equivalent, and that (P7) is equivalent to (P8). Moreover, we show which of the remaining implications are true and which not. The problem is completely solved, with the following exceptions: we conjecture that (P7)⇒(P6) but we do not provide the argument; validity of (P7)⇒(P1) remains open. Our paper gives a partial solution of the problem stated in 1989 by A.N. Sharkovsky.

1999-09-30



Journal: Annales Mathematicae Silesianae

On periodic boundary value problem for a differential equation of n-th order

Tadeusz Dłotko

Language: EN | Published: 30-09-1991 | Abstract | pp. 43-50

The problem (1) is investigated. New sufficient conditions are derived for the existence of at least one solution in the space C_[0,w]^(n-1). The proof is based on the topological degree method in the Banach space of solutions.

1991-09-30