Summary
LetE be a real Hausdorff topological vector space. We consider the following binary law * on ℝ ·E:(α, β) * (α′, β′) = (λαα′, α′ k β + α′β′) for(α, β), (α′, β′) ∈ ℝ × E where λ is a nonnegative real number,k andl are integers.
In order to find all subgroupoids of (ℝ ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y)k x +f(x)l y) =λf(x)f(y) (x, y ∈ E). (1)
In this paper, all solutionsf: ℝ → ℝ of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E →ℝ of (1). The subgroupoids of (ℝ* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL 1 n of the form {(F(x 2,x 3, ⋯,x n ),x 2,x 3, ⋯,x n ); (x 2, ⋯,x n ) ∈ ℝn − 1}, where the mappingF: ℝn − 1 → ℝ* has some regularity property, is {1} × ℝn − 1.
We may noitice that the Gołąb-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, λ = 1). So we can say that (1) is of Gołąb—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gołąb—Schinzel type if it is of the form:f(f(y)k x +f(x)l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y)k x +f(x)l y) =f(xy) (x, y ∈ ℝf: ℝ → ℝ). (4)
This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).
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References
Aczél, J. andDhombres, J.,Functional equations in several variables. Cambridge University Press, Cambridge, 1989.
Benz, W.,The cardinality of the set of discontinuous solutions of a class of functional equations. Aequationes Math.32 (1987), 58–62.
Brillouët, N. andDhombres, J.,Equations fonctionnelles et recherche de sous-groupes. Aequationes Math.31 (1986), 253–293.
Bruckner, A. M. andCeder, J. G.,Darboux continuity. Jahresber. Deutsch. Math.-Verein.67 (1965), 93–117.
Dhombres, J.,Some aspects of functional equations. (Lecture Notes). Department of Mathematics, Chulalongkorn University, Bangkok, 1979.
Javor, P.,Continuous solutions of the functional equation f(x + yf(x)) = f(x)f(y). InProc. Internat. Sympos. on Topology and its Applications (Herceg—Novi, 1968), Savez Društava Mat. Fiz. i Astronom., Belgrade, 1969, pp. 206–209.
Midura, S.,Sur la détermination de certains sous-groupes du groupe L 1s à l'aide d'équations fonctionnelles. [Dissertationes Math., No. 105]. PWN, Warsaw, 1973.
Midura, S.,Solutions des équations fonctionnelles qui déterminent les sous-demigroupes du groupe L 14 . Wyż.Szkoła Ped. Rzeszów Kocznik Nauk.-Dydakt. Mat.7 (1985), 51–56.
Midura, S.,Sur les homomorphismes du groupe L 1 s pour s < 5. InReport of Meeting. Aequationes Math.39 (1990), 288.
Midura, S. andUrban, P.,O Rozwiaząniach Równania Funkcyjnego ϕ(αϕ k (β) + βϕ l(α)) = ϕ(α)ϕ(β). Wyż.Szkola Ped. Rzeszów. Rocznik Nauk.-Dydakt. Mat.6 (1982), 93–99.
Urban, P.,Rownania funkcyjne typu Gołąba-Schinzla. Uniwersytet Śląski, Instytut Matematyki, Katowice, 1987.
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Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth
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Brillouët-Belluot, N. On some functional equations of Goląb-Schinzel type. Aeq. Math. 42, 239–270 (1991). https://doi.org/10.1007/BF01818494
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DOI: https://doi.org/10.1007/BF01818494