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Level sets of continuous functions increasing with respect to each variable

Katarzyna Sajbura (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We are going to prove that level sets of continuous functions increasing with respect to each variable are arcwise connected (Theorem 3) and characterize those of them which are arcs (Theorem 2). In [3], we will apply the second result to the classical linear functional equation φ∘f = gφ + h (cf., for instance, [1] and [2]) in a case not studied yet, where f is given as a pair of means, that is so-called mean-type mapping.

Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments

Özkan Öcalan (2009)

Archivum Mathematicum

This paper is concerned with the nonlinear advanced difference equation with constant coefficients x n + 1 - x n + ∑ i = 1 m p i f i ( x n - k i ) = 0 , n = 0 , 1 , ⋯ where p i ∈ ( - ∞ , 0 ) and k i ∈ { ⋯ , - 2 , - 1 } for i = 1 , 2 , ⋯ , m . We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients x n + 1 - x n + ∑ i = 1 m p i n f i ( x n - k i ) = 0 , n = 0 , 1 , ⋯ where p i n ≤ 0 and k i ∈ { ⋯ , - 2 , - 1 } for i = 1 , 2 , ⋯ , m .

Lineární posloupnosti

Miroslav Laitoch (1968)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica-Physica-Chemica

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