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Abstract  

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | xy}. We give the solution of the equation

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$F(G_1 (x,y),G_2 (u,v)) = G(F(x,u),F(y,v)),$$ \end{document}
which holds for all (x, y) ∈ Δ and (u, υ) ∈ Δ, where the functions F: XX, G 1: Δ → X, G 2: Δ → X, and G: F(X, X) × F(X, X) → X are continuous and strictly monotonic in each variable.

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remark on “two-sided” monotonicity condition: an application to L p convergence, Acta Math. Hungar. 113 (2006), no. 1–2, 159–169. MR 2007f :42006 Le R. J. A remark on

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Federer, H. , Geometric Measure Theory , Springer-Verlag, New York Inc., 1969. MR 41 //1976 Fischer, P. and Slodkowski, Z. , Monotonicity of the

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Wituła, R., Słota, D., and Seweryn, R. , On Erdős’ theorem for monotonic subsequences, Demonstratio Math. , 40 (2007), 239–259. Seweryn R. On Erdős’ theorem for monotonic

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Ismail, M. E. H. et Laforgia, A. , Monotonicity properties of determinants of special functions, Constr. Approx. , 26 (2007), 1–9. Laforgia A. Monotonicity properties of

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5 103 165 Elbert, Á. and Laforgia, A. , Some monotonicity properties for the zeros of ultraspherical polynomials, Acta

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47 53 Heath, R. W., Lutzer, D. J. and Zenor, P. L. , Monotonically normal spaces, Trans. Amer. Math. Soc. , 178 (1973), 481–493. MR 0372826 ( 51 #9030

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approximation 1993 Doğru, O. and Gupta, V. , Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn

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Monotonicity , Lecture Notes in Economics and Mathematical Systems, 502 , pages 21–47. Springer, 2001. MR 2002d :90055 Prékopa, A. and Alexe, G. , Dual Methods for the Numerical Solution of the Univariate Power Moment

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Cecchi M., Marini, M. and Villari, G. , On the monotonicity property for a certain class of second order differential equations, J. Differential Equations , 82 (1) (1989), 15–27. G V

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