共查询到20条相似文献,搜索用时 0 毫秒
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Imre Kocsis 《Aequationes Mathematicae》2007,73(3):280-284
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Margherita Fochi 《Aequationes Mathematicae》1996,52(1):246-253
Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX
1 = {(x, y) X
2/x, y = 0} andX
2 = {(x, y) X
2/x = y}. In this paper we prove that the equation (1) restricted toX
1 is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f2(x) – 1. Using this result, we prove the equivalence between (1) restricted toX
2 and (1) on the whole spaceX.
This research follows similar previous studies concerning the additive, exponential and quadratic functional equations. 相似文献
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Gian Luigi Forti 《Aequationes Mathematicae》1982,24(1):195-206
We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG. 相似文献
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Let S be a real interval with
, and
be a function satisfying
We show that if h is Lebesgue or Baire measurable, then there
exists
such that
That result is motivated by a question of E. Manstaviius.
Received: 11 February 2003 相似文献
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Henrik Stetkaer 《Aequationes Mathematicae》1997,53(1-2):91-107
Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation
n = 0
N – 1
f(x + wny) = Nf(x)g(y), x, y C, given a primitiveN
th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g
(z) = N–1
n = 0
N – 1
E(wnz) for some C2. HereE
(1, 2)(x + iy) = exp(
1x + 2).For fixedg = g
the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z
N
on the continuous functions on C. We prove that ther
th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation
n = 0
N – 1
f(x + wny)wnr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz
n = 0
N – 1
wnr E(wnz), except forg = 1 andr 0 where they are spanned by
andz
N – r. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ
N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z
N
andSO(2). 相似文献
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C. T. Ng 《Aequationes Mathematicae》1990,39(1):85-99
Summary A natural extension of Jensen's functional equation on the real line is the equationf(xy) + f(xy
–1
) = 2f(x), wheref maps a groupG into an abelian groupH. We deduce some basic reduction formulas and relations, and use them to obtain the general solution on special groups. 相似文献
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Henrik Stetkær 《Aequationes Mathematicae》2003,66(1-2):100-118
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Summary Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy
–1) = 2f(x), f(e) = 0, wherex, y G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional equation. The case whenG is non-abelian is also discussed. 相似文献
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Che Tat Ng 《Aequationes Mathematicae》2005,70(1-2):131-153
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Pavlos Sinopoulos 《Aequationes Mathematicae》2004,67(1-2):188-194
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Janusz Brzdek 《Aequationes Mathematicae》1996,52(1):105-111
Summary A new shorter proof is given for the Theorem of P. Volkmann and H. Weigel determining the continuous solutionsf:R R of the Baxter functional equationf(f(x)y + f(y)x – xy) = f(x)f(y). The proof is based on the well known theorem of J. Aczél describing the continuous, associative, and cancellative binary operations on a real interval. 相似文献