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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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LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR
+ = [0, ). A one parameter family of operators {S(t);t R
+},S:R
+ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S
2(s))S(t –s), for alls, t R
+,s t. Let
,fD(A), and
,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A
2) implies
0
t
(t –u(S(u)fduD(B) and B
0
t
(t –u)S(u)fdu =S(t)f –f +tAf – 2A
0
t
S(u)fdu + 2A
2
0
t
(t –u)S(u)fdu.D(B) is seen to be dense inD(A
2). Some regularity properties ofS(t) have also been obtained. 相似文献
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Janusz Brzdęk 《Aequationes Mathematicae》1993,46(1-2):56-75
Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}.
Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A
1,A
2,B
1,B
2, F\ {0}with L(ax, y) = A
1
L(x, y), L(x, ay) = A
2
L(x, y), L(bx, y) = B
1
L(x, y), and L(x, by) = B
2
L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A
1 =A
2,B
1 =B
2,A = A
1
2
,and B = B
1
2
. (3)
Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y
0
Y and an additive function h: X Y such that if A + B 1, then y
0 = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y
0 + 1/2A
1
-1
B
1
-1
L(x, x)for x P. 相似文献
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John A. Baker 《Aequationes Mathematicae》1996,52(1):302-312
Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology. 相似文献
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C. T. Ng 《Aequationes Mathematicae》1985,28(1):161-169
The functional equationg(u, x)+g(v, y)=g(u, y)+g(v, x) for allu, v, x, y>0 withu+v=x+y is initiated by F. A. Cowell and A. F. Shorrocks in their research on the aggregation of inequality indices. We solve the equation by extension theorems.Dedicated to Professor Janos Aczél on his 60th birthday 相似文献
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《Quaestiones Mathematicae》2013,36(5):651-663
AbstractLet G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formulaQk f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k2 ? 1)f (y)for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Qk f (x, y) = 0, after it, we show that if Qk f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7]. 相似文献
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Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth. 相似文献