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1
Content available A generalization of Matthews partial metric space and fixed point theorems
Antal A., Gairola U. C., Khantwal D., Matkowski J., Negi S.
Fasciculi Mathematici
|
2021
|
nr 65
5--15
EN
In the present paper, we introduce the notion of a generalized partial metric space which is an extension of the partial metric space due to S. G. Matthews (Partial metric topology, Papers on general topology and applications, Ann. New York Acad. Sci.,728 (1994), 183-197). We investigate some basic properties of the generalized partial metric spaces and establish some new fixed point theorems for linear and non-linear contraction on such spaces.
2
Content available Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
Guerrero J. A., Matkowski J., Merentes N., Wróbel M.
Journal of Applied Mathematics and Computational Mechanics
|
2015
|
Vol. 14, nr 4
41--51
EN
We show that the one-sided regularizations of the generator of any uniformly bounded set-valued Nemytskij composition operator mapping the space of bounded variation functions in the sense of Wiener into the space of bounded variation functions with closed bounded convex values (in the sense of Wiener) are affine functions.
3
Content available Conjugate functions, lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity
Matkowski J.
Opuscula Mathematica
|
2014
|
Vol. 34, no. 3
523-560
EN
For h : (0,∞) → R, the function h* (t) := th( 1/t ) is called (*)-conjugate to h. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of (*)-conjugacy are proved. If φ and φ* are bijections of (0,∞) then [formula]. Under some natural rate of growth conditions at 0 and ∞, if φ is increasing, convex, geometrically convex, then [formula] has the same properties. We show that the Young conjugate functions do not have this property. For a measure space (Ω,Σ,μ) denote by S = (Ω,Σ,μ) the space of all μ-integrable simple functions x : Ω → R, Given a bijection φ : (0,∞) → (0,∞) define [formula] by [formula] where Ω(x) is the support of x. Applying some properties of the (*) operation, we prove that if ƒ xy ≤ Pφ(x)Pψ (y) where [formula] and [formula] are conjugate, then φ and ψ are conjugate power functions. The existence of nonpower bijections φ and ψ with conjugate inverse functions [formula] such that Pφ and Pψ are subadditive and subhomogeneous is considered.
4
Content available Note on some infinite products for π
Kahlig P., Matkowski J.
Journal of Applied Mathematics and Computational Mechanics
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2014
|
Vol. 13, nr 2
43--50
EN
After a brief review of (slowly converging) Wallis-type infinite products for π , (faster converging), Dido-type infinite products for π are treated. The notion of “alternating products” facilitates error checking.
5
Content available remote A counterpart of the Taylor theorem and means
Matkowski J.
Fasciculi Mathematici
|
2014
|
Nr 53
85--93
EN
For an n-times differentiable real function ƒ defined in an a real interval I, some properties of the Taylor remainder means Tn[ƒ] are considered. It is proved that Tn[ƒ] is symmetric iff n – 1, and a conjecture concerning the equality Tn[g]- Tn[ƒ] is formulated. The main result says that if ƒ (n) is one-to-one, there exists a unique mean Mn[ƒ] : ƒ(n) (I) x ƒ(n) (I) → ƒ(n) (I) such that, for all x, y ϵ I, …[wzór]. The connection between Tnƒ ana Mnƒ is given. A functional equation related to M2 ƒ is derived and an open problem is posed.
6
Content available remote Mean-value theorems and some symmetric means
Matkowski J.
Demonstratio Mathematica
|
2013
|
Vol. 46, nr 3
461--474
EN
Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.
7
Results of Tadeusz Świątkowski on algebraic sums of sets and their applications in the theory of subadditive functions
Matkowski J.
Journal of Applied Analysis
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2012
|
Vol. 18, nr 1
97-116
EN
A refinement of Steinhaus' theorem on the algebraic sum of subsets of R due to Raikov (1939) was not known to the mathematical community and still is not popular. In 1994, Tadeusz Świątkowski, being not aware of the existence of Raikov's theorem, proved another result of this type. Unfortunately, a few days later he passed away. In this paper we present the theorems of Świątkowski and Raikov and we apply them in the theory of subadditive type inequalities. An improvement of a converse of Minkowski's inequality theorem is presented.
8
Content available Subadditive periodic functions
Matkowski J.
Opuscula Mathematica
|
2011
|
Vol. 31, no. 1
75-96
EN
Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function ƒ0 : [0, 1) ? R is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.
9
Content available remote Mean-value type equalities with interchanged function and derivative
Matkowski J.
Fasciculi Mathematici
|
2011
|
Nr 47
19--27
EN
According to a new mean value-theorem, if a func¬tion f satisfies the classical conditions ensuring the existence and uniqueness of Lagrange’s mean, then there also exists a unique mean M such that ...[wzór]. The main result gives necessary and sufficient conditions for the equality ...[wzór] The relevant equality for the Lagrange mean-value theorem is also considered.
10
Content available The bounded local operations in the Banach space of Hölder functions
Matkowski J., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2010
|
Vol. 15
91--98
EN
It is known that every locally defined operator acting between two Hölder spaces is a Nemytskii superposition operator. We show that if such an operator is bounded in the sense of the norm, then its generator is continuous.
11
Content available On some generalizations of Gołąb-Schinzel functional equation
Matkowski J., Okrzesik J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2010
|
Vol. 15
81--90
EN
Composite functional equations in several variables generalizing the Gołąb-Schinzel equation are considerd and some simple methods allowing us to determine their one-to-one solutions, bijective solutions or the solutions having exactly one zero are presented. For an arbitrarily fixed real p, the functional equation Φ([pφ(y) + (1−p)]x +[(1−p)φ(x)+p]y) = φ(x)φ(y), x,y ∈ R, being a special generalization of the Gołąb-Schinzel equation, is considered.
12
Content available Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
Azocar A., Guerrero J. A., Matkowski J., Merentes N.
Opuscula Mathematica
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2010
|
Vol. 30, no. 1
53-60
EN
We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.
13
A composite functional equation and invariant curves
Matkowski J., Okrzesik J.
Journal of Applied Analysis
|
2010
|
Vol. 16, nr 1
121-133
EN
The continuous solutions of a composite functional equation are characterized. An applications to the problem of invariant curves is presented.
14
Content available remote Uniformly continuous composition operator in the space of φ-variation functions in the sense of Riesz
Acosta A., Aziz W., Matkowski J., Merentes N.
Fasciculi Mathematici
|
2010
|
Nr 43
5-11
EN
Assuming that a Nemytskii operator maps a subset of the space of bounded variation functions in the sense of Riesz into another space of the same type, and is uniformly continuous, we prove that the generator of the operator is an affine function.
15
Content available remote Generalizations of Lagrange and Cauchy Mean-Value theorems
Matkowski J.
Demonstratio Mathematica
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2010
|
Vol. 43, nr 4
765-774
EN
Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented.
16
Content available remote Uniformly continuous composition operators in the space of functions of two variables of bounded φ-variation in the sense of Wiener
Guerrero J.A., Matkowski J., Merentes N.
Commentationes Mathematicae
|
2010
|
Vol. 50, [Z] 1
41-48
EN
Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener φ-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable.
17
Content available On extension of solutions of a simultaneous system of iterative functional equations
Matkowski J.
Opuscula Mathematica
|
2009
|
Vol. 29, no. 4
415-421
EN
Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form φ(x)=h(x, φ[ƒ1(x)],…, φ[ƒm(x)]) φ(x)=H(x,φ[F1(x),…, [Fn(x)]) to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [1-3]).
18
Content available Extensions of solutions of a functional equation in two variables
Matkowski J.
Opuscula Mathematica
|
2009
|
Vol. 29, no. 4
423-425
EN
An extension theorem for the functional equation of several variables ƒ (M(x,y))=N(ƒ(x), ƒ (y)), where the given functions M and N are left-side autodistributive, is presented.
19
Content available Iteration groups, commuting functions and simultaneous systems of linear functional equations
Matkowski J.
Opuscula Mathematica
|
2008
|
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.
20
Content available remote On a composite functional equation
Matkowski J., Okrzesik J.
Demonstratio Mathematica
|
2003
|
Vol. 36, nr 3
653--658
EN
We determine all continuous functions f : (0, oo) ->(0,oo) satisfying the functional equation f(xG(f(x)))=f(x)G(f(x)) where G is continuous and strictly increasing function such that 1 e G((0, oo)).
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