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1
Content available remote The new investigation of the stability ofmixed type additive-quartic functionalequations in non-Archimedean spaces
Thanyacharoen Anurak, Sintunavarat Wutiphol
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
174--192
EN
In this article, we prove the generalized Hyers-Ulam stability for the following additive-quarticfunctional equation: f(x + 3y) + f(x - 3y) + f(x + 2y) + f(x - 2y) + 22f(x) + 24f(y) = 13[ f(x + y) + f(x - y)] + 12f(2y), where f maps from an additive group to a complete non-Archimedean normed space.
2
Content available The equality case in some recent convexity inequalities
Maksa G., Pales Z.
Opuscula Mathematica
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2011
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Vol. 31, no. 2
269-277
EN
In this paper, we investigate a functional equation related to some recently introduced and investigated convexity type inequalities.
3
Content available remote Solutions of some functional equations related to the determinant of certain matrices
Akkouchi M., Rhali M.
Demonstratio Mathematica
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2010
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Vol. 43, nr 1
97-106
EN
We solve the functional equation : [....] where K is a field which is not of characteristic 2 and f, g, h : K4 ->o K are unknown functions. We study also a class of functional equations of 2n variables generalizing the two equations above. This work is motivated by a paper of J. K. Chung and P. K. Sahoo published in 2002 and a recent paper of the authors published in 2005.
4
Content available Iteration groups, commuting functions and simultaneous systems of linear functional equations
Matkowski J.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
529-539
EN
Let (ƒt)t∈R be a measurable iteration group on an open interval I. Under some conditions, we prove that the inequalies g o ƒa ≤ ƒa o g and g o ƒb ≤ ƒb o g for some a, b ∈ R imply that g must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function g commutes in a neighbourhood of ƒ with two suitably chosen elements ƒa and ƒb of an iteration group of ƒ then, in this neighbourhood, g coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.
5
Content available remote An invariance of the geometric mean with respect to Lagrangian conditionally homogeneous mean-type mappings
Głazowska D.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
289-302
EN
We determine all the Lagrangian conditionally homogeneous mean-type mappings for which the geometric mean is invariant.
6
Content available remote On Some Special Morphisms Between Groups and Algebras
Fidytek I.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2006
|
Vol. 11
17-40
7
Content available remote Stability of the Euler-Lagrange-Rassias functional equation
Pietrzyk A.
Demonstratio Mathematica
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2006
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Vol. 39, nr 3
523-530
EN
Let F be a field, a1, a2 is an element of F, K is an element of {R, C}, s an element of K\{0,1}, X be a linear space over F, S C is contained in X be nonempty, and Y be a Banach space over K. Under some additional assumptions on S we show some stability results for the functional equation Q (a1x + a2y) + Q (a2X - a1y) = s[Q{x) + Q{y)} in the class of function Q : S -> Y.
8
Content available remote Some remarks on subquadratic functions
Kominek Z., Troczka K.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
751-758
EN
Some basic properties of subquadratic functions, i.e. functions fulfilling the inequality phi (x + y) + phi(x - y) is less than or equal to 2 phi(x) + 2 phi(y) are proved. In this note X be always a real linear space and R be denotes the set of all reals. Every function phi : X approaches R satisfying the following inequality (1) phi(x + y) + phi(x - y) is less than or equal 2phi(x) + 2phi(y), x, y is an element of X, is called subquadratic. If the sign "is less than or equal to" is replaced by "is more than or equal to" then phi is called superquadratic and if we have "=" instead of " is less than or equal to" in (1) then we say that phi is quadratic function. There are plenty papers devoted to quadratic functions [1], [2], [3] (and references there). In this note some properties of the solutions of (1) will be proved, particularly we will investigate nonpositive solutions of (1). Also interesting question of finding sucient conditions on subquadratic function to be quadratic one will be considered.
9
Content available remote On subsemigroups of the L1/s
Jabłoński W.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
815-823
EN
In this paper we consider subsemigroups of the group L1/s such that the r–th parameter of xs is the function of the remaining ones. Moreover we generalize some results concerning the existence of certain form subsemigroups of the group L1/s.
10
Content available remote A remark on quasiaffine functions
Lewicki M.
Demonstratio Mathematica
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2006
|
Vol. 39, nr 4
743-750
EN
We consider the functional inequality (*) min{f(x), f(y)} is less than or equal to f (rx + (1 - r)y) is less than or equal to max{f(x), f(y)}, where f is a real valued function on a linear space X and r is an element of (0, 1) is fixed. The purpose of the present paper is to investigate connections between functions satisfying inequality (*) and solutions of (*) with r = 1/2. As a conclusions we get, that under some regularity assumptions, function f is of the form f = g o alpha, where alpha : X approaches R is an additive and g : R approaches R is monotone.
11
Content available remote Study of some functional equations related to some algebras of matrices
Akkouchi M., Rhali M.
Demonstratio Mathematica
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2005
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Vol. 38, nr 3
611--622
EN
Let K be the real or complex field. Let 1 < n < p be two positive integers. We denote Mp(K) the usual algebra of p x p-matrices. Let An be an n- dimensional subalgebra of Mp(K). Then there exists an injective linear mapping A : Kn- Mp(K) such that A^") = An' Therefore K" may be equipped with a product denoted * such that (Kn, +,*) is an associative algebra. The aim of this paper is to investigate the general solutions of the functional equation: f(x*y) = f(x) f(y), and its Pexider form f(x*y) = g(x) h(y), for all x, y in Kn, where f,g, h: Kn - K are unknown functions. This work is inspired by the paper [2].
12
Content available remote Stability of difference equations generated by parabolic differential functional equations
Ciarski R.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
101--117
EN
The aim of this paper is to present a numerical approximation for the initial boundary value problem for quasilinear parabolic differential functional equations. The convergence result is proved for the difference scheme with the property that the difference operators approximating mixed derivatives depend on local properties of coefficients of the differential equation. A numerical example is given.
13
Content available remote On the equation associated with bounded paraconcave entropies
Dudek Z.
Demonstratio Mathematica
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2005
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Vol. 38, nr 2
401--411
EN
A paraconcave entropy function [2] is represented by a pair of two real functions of a real variable satisfying certain natural conditions. The subject of this paper is the functional equation, L(sum j f(pj)) = sum j g(pj)i, that describes equivalence between two representations of a paraconcave entropy function with concave functions f and g satisfying the condition for a bounded entropy. With the use of E-transforms of the functions f and g we reduce the problem of solvability of the equation to the problem ofinjectivity of a certain nonlinear operator denned on the set of concave homeomorphisms of the interval [0,1] onto itself. Additionally, we prove some facts about concavity of the E-transform f.
14
Content available remote On a functional equation related to an automorphism of a unit circle
Kir'ytzkii E.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
119--134
EN
In this article the complete description of the decisions of a functional equation f(w(z)) = f(w(0))f(z) is given, where w(z)-automorphism of a unit circle E and the decisions are searched among analytical in E functions. It is established, that research of a given functional equation is closely connected to property of stationary points of automorphism w(z).
15
Content available remote On probability distribution solutions of a functional equation
Morawiec J., Reich L.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
Vol. 53, no 4
389-399
EN
Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation [WZÓR] and its solutions in two classes of functions, namely Z ={φ: R→ R∣ φ is increasing, φ|(−∞,0] = 0, φ|[1,∞) = 1}, C = {φ: R → R∣ φ is continuous, φ|(−∞,0] = 0, φ|[1,∞) = 1}. We prove that the above equation has at most one solution in C and that for some parameters α, β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in Z and we show the exact connection between solutions in both classes.
16
Content available remote Euler's Beta function diagonalized and a related functional equation
Choczewski B., Wach-Michalik A.
Opuscula Mathematica
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2004
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Vol. 24, no. 1
35-41
EN
Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function beta: R+ → R+, beta(x) := B(x, x), where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if varphi: R+ → R+ satisfies equation (5) and the function h o varphi is continuous and ultimately convex, then varphi = beta.
17
Content available remote Linear approximation and asymptotic expansion associated with the system of functional equations
Long N.
Demonstratio Mathematica
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2004
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Vol. 37, nr 2
349--362
18
Content available remote Abstract variable domain hyperbolic differential equations
Guezane-Lakoud A.
Demonstratio Mathematica
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2004
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Vol. 37, nr 4
883--892
EN
An abstract problem is studied for a class of linear hyperbolic differential equations with variable domain and non-local boundary conditions. Existence and uniqueness of the strong solution are proved.
19
Content available remote 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.
20
Content available remote Semicontinuous solutions of systems of functional equations
Herzog G.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2003
|
[Z] 43(2)
207-219
EN
We use the method of upper and lower solutions to prove the existence of upper and lower semicontinuous solutions of functional equations of the form F(w,u(w),u(g_1(w)),...,u(g_m(w)) = ) in R^n under monotonicity and quasimonotonicity assumptions on F, and for w from a metrizable topological spaces.
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