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1

On two functional equations connected with distributivity of fuzzy implications

Ger R., Kuczma M. E., Niemyska W.

Commentationes Mathematicae

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2015

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Vol 55, No. 2

163--169

EN

The distributivity law for a fuzzy implication I:[0,1]2→[0,1] with respect to a fuzzy disjunction S:[0,1]2→[0,1] states that the functional equation I(x,S(y,z))=S(I(x,y),I(x,z)) is satisfied for all pairs (x,y) from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: h(min(xg(y),1))=min(h(x)+h(xy),1), x∈(0,1), y∈(0,1], and h(xg(y))=h(x)+h(xy), x,y∈(0,∞), in the class of increasing bijections h:[0,1]→[0,1] with an increasing function g:(0,1]→[1,∞) and in the class of monotonic bijections h:(0,∞)→(0,∞) with a function g:(0,∞)→(0,∞), respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.

2

Mean values for vector valued functions and corresponding functional equations

Ger R., Sablik M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

255--269

EN

Although, in general, a straightforward generalization of the Lagrange mean value theorem for vector valued mappings fails to hold we will look for what can be salvaged in that situation. In particular, we deal with Sanderson's and McLeod's type results of that kind (see [9] and [7], respectively). Moreover, we examine mappings with a prescribed intermediate point in the spirit of the celebrated Aczél's theorem characterizing polynomials of degree at most 2 (cf. [1]).

3

Riemann integrability and quasi-uniform convergence

Ger R., Jędrzejewski J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2010

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Vol. 15

31--34

EN

We consider quasi-uniform convergence of sequences of functions in a context of Riemann integrability of its limit. Some generalizations are discussed as well.

4

Stability of the equation of ring homomorphisms

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2007

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Vol. 12

29-35

EN

Let R be a unitary ring and (A,║‧║) stand for a Banach algebra with a unit. In connection with some stability results of R. Badora [1] and D.G. Bourgin [2] concerning the system of two Cauchy functional equations [formula] for mappings f : R→ A, we deal with Hyers-Ulam stability problem for a single equation f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y). The basic question whether or not equation (**) is equivalent to the system (*) has widely been examined by J. Dhombres [3] and the present author in [4] and [5].

5

Sublinear Functionals and Weak - compatness

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2005

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Vol. 10

67--76

EN

Let X be a real locally convex linear topological space. A functional f : X → lR is called sublinear provided that f is subadditive and f(nx) = nf(x),x ∈ X,n ∈ IN. We establish a one-to-one correspondence between the collectron of all sublinear functional satisfying some mild regularity conditions and the family of all nonempty convex and weakly"- compact subsets of the dual space X*.

6

Fischer-Muszély additivity on Abelian groups

Ger R.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

83-96

EN

A description of a general solution f : X -> Y mapping a commutative group (X, +) into a real normed linear space (Y, || o ||) of the functional equation [formula] is given in terms of isometrics and additive mappings. Several results describing the solutions of this equation that were obtained earlier under some alternative assumptions regarding the domains, ranges and//or by imposing some regularity upon the map f become special cases of our main result. To gain a proper proof tool we have also established an improvement of E. Berz's [4] representation theorem for sublinear functionals on Abelian groups.

7

Abstract Boundedness and the Stability of the Pexider Equation

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2003

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Vol. 9

41--49

EN

Let (S, +) be a semigroup (not necessarily Abelian) and let (X,+) be a commutative group. We deal with an axiomatically given family B ⸦ 2x of "bounded sets" and with mappings f, g, h : S → X such that the transformation S x S ∋ (x, y) → f (x + y) - g (x) - h (y) ∈ X remains B-bounded. Stability results existing in the literature in connection with the Pexider functional equation become special cases of our theorems up to the magnitude of approximating constants.

8

On a characterization of the logarithm by a mean value property

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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1998

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Vol. 4

1-7

EN

Any real polynomial f(x) = ax2 + bx + c, x ∈ IR, has the property that f (x)-f (y) x-y for every (x, y) ∈ IR, x ꞊ y. It turns out that that particular form of the Lagrange mean value theorem characterizes polynomials of at most second degree. Much more can be proved: J. Aczél [1] has shown that, with no regularity assumptions, a triple (/, g, h) of functions mapping IR into itself satisfies the equation f(x)-g(y) x-y= h(x + y) for all (x, y) ∈ IR, x ≠ y, if and only if there exist real constants a, 6, c such that f (x) = g(x) = ax2 + b, x + c, x ∈ IR, and h(x) = ax + b, x ∈ IR. Generalizations involving weighted arithmetic means were also considered (see e.g. M. Falkowitz [3] and the references therein) and characterizations of polynomials of higher degrees (in the same spirit) were obtained (see [4] and [5], for instance). In what follows we are going to characterize the logarithm in a similar way. To this end, denote by D the open first quadrant of the real plane IR2 with the diagonal removed, i.e. D := (O, ∞)2 \ {(x, x) e IR2 : x ∈ (0, ∞) }.Applying the classical Lagrange mean value theorem to the logaritmic function we derive the existence of a function D 3 (x, y) -> £(x,y) € intcony {x, y} such that the equality log a:-log y x-y £(z,y)