Aequationes Mathematicae最新文献

The following two alternative functional equations are investigated

and

where (f,g:mathbb {R} rightarrow mathbb {R}); assuming that in the first equation f and g are in (C^2(mathbb {R})) and in the second one that f is in (C^1(mathbb {R})) and g in (C^2(mathbb {R})), it is proved that both equations have only trivial solutions, that is for the first equation either f is quadratic on (mathbb {R}) or g is quadratic on (mathbb {R}); for the second equation either f is additive on (mathbb {R}) or g is quadratic on (mathbb {R}).

We say that a map (f:S_X rightarrow S_Y) between the unit spheres of two Banach spaces X and Y is a max-phase-isometry (min-phase-isometry, respectively) if it satisfies

Let (Gamma ,Delta ) be two arbitrary index sets, and let (p>0) and (pne 1). Here, all (ell ^p(Gamma ))-type spaces are over the real numbers. We show that for every surjective max-phase-isometry or min-phase-isometry (f:S_{ell ^p(Gamma )}rightarrow S_{ell ^p(Delta )}), there exists a phase function (varepsilon : S_{ell ^p(Gamma )} rightarrow {-1, 1}) such that (varepsilon cdot f) is an isometry. This isometry is the restriction of a linear isometry from (ell ^p(Gamma )) onto (ell ^p(Delta )). Furthermore, for (p=1), this result is valid for min-phase-isometries but fails, in general, for max-phase-isometries.

In this paper we present coincidence point (and fixed point) results for compact maps with a selection property. We also present homotopy results for a subclass of these maps.

The functional equation

occurs in connection with some characterization problems in probability theory. We find the form of its solutions defined in a vicinity of zero and fulfilling a natural asymptotic condition at zero: real-valued ones for a wide class of functions (p_1, ldots ,p_n) and complex-valued solutions when (p_1= cdots =p_n=1). The obtained results generalize some theorems proved by Vincze (Magyar Tud. Akad. Mat. Kutató Int., Közl 7:357–361, 1962), Laha and Lukacs (Aequationes Math. 16:259–274, 1977), Kuczma, Choczewski and Ger (Iterative Functional Equations, Encyclopedia of Mathematics and its Applications 32, Cambridge University Press, Cambridge, 1990), and Baker (Proc. Amer. Math. Soc. 121:767–773, 1994). As a consequence we obtain an extension of a result by Zdun (Aequationes Math. 8:229–232, 1972). It provides a new characterization of the complex exponential functions. We record also the form of complex-valued solutions of the equation

with some asymptotics at zero.

The pioneer work of János Aczél on functional equations led to a comprehensive theory on this field. His basic book “Lectures on functional equations and their applications” served as a cornerstone of this theory. A great number of classical functional equations are of or can be reduced to convolution-type systems of functional equations. It turned out that this class can effectively be studied using the methods of harmonic analysis and spectral synthesis. In this paper we give a short summary starting from the basics of the theory to the present state and main results.

Under certain simple conditions for real functions f, g, h, defined on a real interval, the bivariable functions (A_{f}), (G_{g}) and (H_{h}) given, respectively, by

are natural generalizations of the classical weighted arithmetic, geometric and harmonic means. The article concerns the following invariance equations involving these means

where f, g and h are unknown functions. The first two of these equations are investigated under the assumption that f is twice differentiable, and g, h are differentiable. If (A_{f}) is translative and (G_{f}) and (H_{f}) are homogeneous, we determine the solutions without any regularity conditions.

A quasisum is a function ( F: I_1 times dots times I_n longrightarrow mathbb {R}) of the form

where ( n ge 2 ) is an integer and ( f_k: I_k longrightarrow mathbb {R}) is a continuous, strictly monotone function defined on a nonempty open interval of ( mathbb {R}) (for ( k = 1 , dots , n )), moreover ( g: f_1(I_1) + dots + f_n(I_n) longrightarrow mathbb {R}) is also continuous, strictly monotone. In this paper we will show that if ( p in mathbb {N}) and the quasisum F is p-times continuously differentiable then each of the generator functions ( g , f_1 , dots , f_n ) are p-times continuously differentiable as well. We present applications of our results for p-times continuously differentiable semigroup operations and additively separable utility functions as well.

The Sincov equation (f(x,z)=f(x,y)+f(y,z)) has a long history. An excellent source is Gronau(2014). Under usual circumstances the general solution is given by (f(x,y)=g(y)-g(x)) with arbitrary g. This is also true, when the equation is satisfied for all (xle yle z) in a linearly ordered domain and for abelian groups as co-domain. In Pia̧tek(2005) a result in this context is presented in the case that the domain is (only) partially ordered. We present a counter example and suggest positive results under mild additional hypotheses. In Bögel-Tasche(1974) and much better in chap. 7, The Lebesgue-Stieltes Integral of McShane(1944) the notion of additive interval functions is introduced. It seems that it went unnoticed till now that there is an intimate connection to the Sincov equation. This will be discussed in detail here.

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the x-axis and return to the x-axis. Versions where the return to the x-axis isn’t required are also considered. A path is peakless (valleyless) if UD (if DU) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of UD resp. DU. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.

We consider additive operators that approximately preserve the Birkhoff-James orthogonality relation. In particular, we show that such operators must be linear.