O-regularly varying functions and some asymptotic relations

Dragan Djurci\'c

Abstract: We prove that in the class of measurable positive functions defined on the interval $I_a = [\,a,+\infty )$ $(a > 0)$, the class of functions which preserve the strong asymptotic equivalence on the set of functions $\{x \,\colon I_a \mapsto \Bbb R^+ ,\, x(t) \to +\infty, t \to +\infty \}$, is a class of $\Cal O$--regularly varying functions with continuous index function. We also prove a representation theorem for functions from this class, and a morphism-theorem for some asymptotic relations.

Classification (MSC2000): 26A12

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