On a Problem of Krasnosel'skii and Rutickii

In \cite[p. 30]{5}, M. A. Krasnosel'skii and Ya. B. Rutickii proposed a problem, which can be reformulated as follows. Let $f$ be an $N$-function such that $f(ts)\leq f(t)f(s)$, $s,t\geq 1$. Is there another $N$-function $F$ such that $F(st)\leq F(t)F(s)$, $s,t>0$ and equivalent to $f$ on $[1,\infty)?$. We give a necessary and sufficient condition for a positive and constructive solution.

On a Problem of Krasnosel'skii and Rutickii Sobre un problema de Krasnosel'skii y Rutickii DIOMEDES BÁRCENAS1, CARLOS FINOL2

1Universidad de Los Ándes, Mérida, Venezuela. Email: null 
2Universidad Central de Venezuela, Caracas, Venezuela. Email:carlos.finol@ciens.ucv.ve 

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Abstract

In [5,p. 30], M. A. Krasnosel'skii and Ya. B. Rutickii proposed a problem, which can be reformulated as follows. Let f be an N-function such thatf(ts)≤ f(t)f(s), s,t≥ 1. Is there another N-function F such that F(st)≤ F(t)F(s), s,t>0 and equivalent to f on [1,∞)?. We give a necessary and sufficient condition for a positive and constructive solution.

Key words: Orlicz Functions, N-Functions, Submultiplicative Functions, Matuszewska--Orlicz indices.

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2000 Mathematics Subject Classification: 39B62, 26B25, 26A51.

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Resumen

En [5,p. 30], M. A. Krasnosel'skii y Ya. B. Rutickii proponen un problema, el cual puede ser reformulado de la siguiente manera: Sea f una N-función tal que f(ts)≤ f(t)f(s) para todo t,s≥ 1. ¿Existe alguna función F tal queF(st)≤ F(s)F(t), s,t≥ 0, que sea equivalente a f en [1,∞)?. En este articulo, damos una condición necesaria y suficiente para una solución positiva y constructiva.

Palabras clave: Funciones de Orlicz, N-funciones, funciones submultiplicativas, indices de Matuszewska--Orlicz.

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References

[1] J. Alexopoulos, `De la Vallée Poussin's Theorem and Weakly Compact Sets in Orlicz Spaces´, QM 17, (1994), 231-248.

[2] T. Andô, `On Some Properties of Convex Functions´, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys 8, (1960), 413-418.

[3] C. E. Finol and M. Wójtowicz, `Multiplicative Properties of Real Functions with Applications to Classical Functions´, Aequationes Math 59, (2000), 134-149.

[4] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer-Math. Soc, Providence, United States, 1957.

[5] M. A. Krasnosielskii and Y. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff Ltd, Groningen, Holland, 1961.

[6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy's Equation and Jensen's Inequality, P.W.N, Warszawa-Kraków, Polska, 1985.

[7] K. Lindberg, `On Subspaces of Orlicz Sequences Spaces´, Studia Math 45, (1973), 119-146.

[8] J. Lindenstrauss and L. Tzafiri, `On Orlicz Sequence Spaces II´, Israel J Math 10, 3 (1971), 379-390.

[9] J. Lindenstrauss and L. Tzafiri, `On Orlicz Sequences Spaces III´, Israel J Math 10, 4 (1973), 368-389.

[10] J. Lindenstrauss and L. Tzafiri, Classical Banach Spaces I, Ergebnisse der Math. und ihrer Grenzgebiete, Springer-Verlag, Berlin, Germany, 1977.

[11] W. Matuszewska and W. Orlicz, `On Certain Properties of φ-Functions´, Bull Acad. Polon. Sci. Sér. Math. Astronom Phys 8, (1960), 439-443.

[12] W. Matuszewska and W. Orlicz, `On Some Classes of Funtions with Regard to the their Order of Growth´,Studia Math 26, (1965), 11-24.

[13] W. Orlicz, On spaces of φ-integrable functions, `Proc. of the International Symp. on Linear Spaces held at the Hebrew University of Jerusalem´, (1960), Academic Press, Jerusalem, Israel, p. 357-365.

(Recibido en marzo de 2011. Aceptado en agosto de 2011)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv45n2a06, 
    AUTHOR  = {Bárcenas, Diomedes and Finol, Carlos}, 
    TITLE   = {{On a Problem of Krasnosel'skii and Rutickii}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2011}, 
    volume  = {45}, 
    number  = {2}, 
    pages   = {179--186} 
}