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We determine the general solution of the functional equation fxy, \[ f ( x + y 2 ) + f ( x − y 2 ) = 2 f ( x 2 ) + 2 f ( y 2 ) + λ f ( x ) f ( y ) , f\left ( {\frac {{x + y}} {2}} \right ) + f\left ( {\frac {{x - y}} {2}} \right ) =... more
We determine the general solution of the functional equation fxy, \[ f ( x + y 2 ) + f ( x − y 2 ) = 2 f ( x 2 ) + 2 f ( y 2 ) + λ f ( x ) f ( y ) , f\left ( {\frac {{x + y}} {2}} \right ) + f\left ( {\frac {{x - y}} {2}} \right ) = 2f\left ( {\frac {x} {2}} \right ) + 2f\left ( {\frac {y} {2}} \right ) + \lambda f(x)f(y), \] /: [- f : [ − 1 , 1 ] → R f:[ - 1,1] \to {\mathbf {R}} . This equation was used by Lau in order to characterize Rao’s quadratic entropies. The general solution is obtained here as a special case of a more general result for f f mapping a neighborhood of 0 0 in linear topological space into a field.
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ABSTRACT
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Summary. A natural extension of Jensen's functional equation on the real line is the equation $ f(xy)+f(xy^{-1}) = 2f(x) $ where f maps a group G into an abelian group H. When normalized, it defines semi-homomorphisms. It has been... more
Summary. A natural extension of Jensen's functional equation on the real line is the equation $ f(xy)+f(xy^{-1}) = 2f(x) $ where f maps a group G into an abelian group H. When normalized, it defines semi-homomorphisms. It has been solved when G is free with up to two generators, and when $ G=GL_2(\Bbb Z) $. Here, the results are extended to include all free groups and $ GL_n(\Bbb Z),\ n\geq 3 $.
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ABSTRACT Summary. A first extension of Jensen's functional equation on the real line is the equation f(xy) + f(xy-1) = 2f(x) where f maps a group G into an abelian group H. A second extension is f(xy) + f(y-1x) = 2f(x). Results... more
ABSTRACT Summary. A first extension of Jensen's functional equation on the real line is the equation f(xy) + f(xy-1) = 2f(x) where f maps a group G into an abelian group H. A second extension is f(xy) + f(y-1x) = 2f(x). Results were reported on the first equation in two preceding articles. Here, we solve the second on free groups, and illustrate how it leads to solutions on more specific groups including linear groups GLn(\Bbb Z) GL_n({\Bbb Z}) , SL2(\Bbb Z) SL_2({\Bbb Z}) , symmetric groups Sn, alternating groups An, dihedral groups, and finite abelian groups.
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A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a... more
A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a planar mapping admitting quadratic invariant curves y=x +C, except for the case -3/4≥C≤0. This remaining case is now resolved, and we describe the solutions of the functional equation K(x +C)+k(x)=x by iterations of y.
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Let f be a continuous self-map on the real line, f [m] denote its m-th iterate and f n its n-th multiplicative power. In this paper we solve the functional equation f [m] =f n for integers m≥2, n≥2. When m=n, it reveals functions whose... more
Let f be a continuous self-map on the real line, f [m] denote its m-th iterate and f n its n-th multiplicative power. In this paper we solve the functional equation f [m] =f n for integers m≥2, n≥2. When m=n, it reveals functions whose n-th iterate and power agree.
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ABSTRACT The functional equation ƒ(h(x)) − ƒ(h(y)) + ƒ(y) = ƒ(h(x − y) + y) was solved by Aczél, Luce and Marley on the assumption that the functions are different iable. They posed the question of its strictly monotonic continuous... more
ABSTRACT The functional equation ƒ(h(x)) − ƒ(h(y)) + ƒ(y) = ƒ(h(x − y) + y) was solved by Aczél, Luce and Marley on the assumption that the functions are different iable. They posed the question of its strictly monotonic continuous solutions. The problem is solved using a uniqueness theorem. The continuity assumption on the functions is removed and the equation is also solved on a restricted domain.
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ABSTRACT Let (G, .) be a group, (H, +) an abelian group and f : G → H. The first and the second Cauchy differences of f are $$\begin{aligned} & \quad C^{1}f(x,y) = f(xy) - f(x) - f(y), \\ & C^{2}f(x,y,z) = f(xyz) - f(xy) -... more
ABSTRACT Let (G, .) be a group, (H, +) an abelian group and f : G → H. The first and the second Cauchy differences of f are $$\begin{aligned} & \quad C^{1}f(x,y) = f(xy) - f(x) - f(y), \\ & C^{2}f(x,y,z) = f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$ Higher order Cauchy differences \({C^{m}f}\) are defined recursively. The functional equation $$C^{m}f = 0$$ is studied. Some earlier results on the equation \({C^{2}f = 0}\) are extended to higher m. For m = 3 we present its general solution on free groups G. When the free group has just one generator the solution is obtained for all m.
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Uniqueness theorems for the solution f of the functional equation {*20c \textf[ \textG( \textx, \texty ) ] = \textH [ \textf ( \textx ), \textf ( \texty ), \textx, \texty ] \textf | \textA = \textg \left\{{\begin{array}{*{20}c}... more
Uniqueness theorems for the solution f of the functional equation {*20c \textf[ \textG( \textx, \texty ) ] = \textH [ \textf ( \textx ), \textf ( \texty ), \textx, \texty ] \textf | \textA = \textg \left\{{\begin{array}{*{20}c} {{\text{f}}\left[ {{\text{G}}\left( {{\text{x,}}\,{\text{y}}} \right)} \right] = {\text{H}}\,\left[ {{\text{f}}\,\left( {\text{x}} \right),\,{\text{f}}\,\left( {\text{y}} \right),\,{\text{x,}}\,{\text{y}}} \right]} \\ {{\text{f}}\,\mid{\text{A}} = {\text{g}}} \\ \end{array} } \right. (*) where ${*{20}c} {\text{f}} \hfill & : \hfill &
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In order to understand the dynamics of a second order delay differential equation with a piecewise constant argument, we study the derived planar mappings and their invariant curves.
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ABSTRACT Let (G,·) be a group, (H,+) an abelian group, and f, g, h : G → H. The Pexider-Jensen functional equation $$f(xy) + g(xy^{{ - 1}} ) = h(x)$$ is studied. Some results on Jensen’s equation f(xy) + f(xy−1) = 2f(x) are extended. We... more
ABSTRACT Let (G,·) be a group, (H,+) an abelian group, and f, g, h : G → H. The Pexider-Jensen functional equation $$f(xy) + g(xy^{{ - 1}} ) = h(x)$$ is studied. Some results on Jensen’s equation f(xy) + f(xy−1) = 2f(x) are extended. We obtain the solution on free groups and outline a process to find the solution on other groups. Examples include certain linear groups and semidirect products.