Jacek Jachymski

Jacek Jachymski
Lodz University of Technology · Institute of Mathematics

About

69
Publications
4,746
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
2,197
Citations
Introduction

Publications

Publications (69)
Article
Full-text available
We obtain characterizations of non-negative functions on \([0,+\infty )\) which preserve some classes of semimetrics. In particular, one of our main results says that for a non-decreasing function \(f:[0,+\infty )\rightarrow [0,+\infty )\) the following statements are equivalent: (i) for any semimetric space (X, d), if d satisfies the relaxed polyg...
Article
Full-text available
Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valu...
Article
We examine semimetric spaces, i.e., spaces fulfilling two of the axioms of a metric space excluding the triangle inequality. We give an overview of many substitutes of the triangle condition which have appeared in the mathematical literature until now and examine the relations between them. We review and compare various ways of introducing a topolo...
Article
We establish a simple extension of Cantor's intersection theorem in which we weaken the assumption that all sets are closed. This result leads to a characterization of a class of mappings (not necessarily continuous) for which the fixed point problem is well posed. We also present an example of a mapping from that class for which the existence of a...
Article
Full-text available
The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings f: ℓ∞ (X) → X, where X is a metric space and ℓ∞ (X) is the space of all bounded sequences of elements from X. Our result generalizes the theorem obtained by Miculescu and Mihail in 2008, who proved a counterpart of the Banach principle for mappings f: X...
Article
Full-text available
We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenović and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone...
Article
We revisit Perov’s fixed point theorem for selfmaps of a set endowed with a vector metric taking values in the Euclidean space ℝm. In particular, we show that this result is subsumed by the classical Banach contraction principle. We also obtain a generalization of Perov’s theorem by considering mappings on K-metric spaces satisfying a nonlinear Lip...
Article
We obtain the following Cantor type intersection theorem which, in fact, is a partial extension of Šmulian's theorem by weakening the convexity assumption. Let (An) be a decreasing sequence of closed subsets of a superreflexive Banach space such that for any n ε , there is k ε such that Ak + Ak ⊆ 2An. Then the intersection of all sets An is nonempt...
Article
Full-text available
Let T be a nonempty family of selfmaps of a metric space (X,d). Recently, Barnsley and Vince showed that if T is a compact (in the compact-open topology) family of affine selfmaps of the Euclidean space (Rm, de), then the following two statements are equivalent: there exists a metric ρ Lipschitz equivalent to de such that each mapping from T is a c...
Article
We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.
Article
We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we s...
Article
Recently, Ćirić [Lj.B. Ćirić, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) 2009–2018] obtained a fixed point theorem with the intention to get a probabilistic version of the Boyd–Wong theorem [D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20...
Article
Full-text available
We generalize the so-called Weighted König Lemma, due to Máté, for a submultiplicative function on a subset of the union Un∈ℕ Σn, where Σ is a set and Σn is the Cartesian product of n copies of Σ. Instead of a combinatorial argument as done by Matè, our proof uses Tychonoff's compactness theorem to show the existence of a König chain for a submulti...
Article
W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, O...
Article
We develop the Hutchinson–Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. In particular, our results subsume a classical theorem of J.E. Hutchinson [J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713–747] on the existence of an invariant set for an iterated function s...
Article
We show that most contractive conditions of integral type given recently by many authors coincide with classical ones. This is done with the help of a geometric lemma on subsets of the quadrant [0,∞)2[0,∞)2. In particular, we extend a recent result of Suzuki who observed that an integral version of the Banach contraction principle given by Branciar...
Article
We revisit a fixed point theorem for contractions established by Felix Browder in 1968. We show that many definitions of contractive mappings which appeared in the literature after 1968 turn out to be equivalent formulations or even particular cases of Browder’s definition. We also discuss the problem of the existence of approximate fixed points of...
Article
Let X be a Banach space and T ε L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T, that is, for the convergence of the sequence (T n)nεN of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ε N, the restriction o...
Article
We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some lin...
Article
Full-text available
We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {Fi : i } is a family of Matkowski's contractions on a complete metric space (X, d) such that (Fix0)iN is bounded for some x0 X, then there exists a non-empty bounded and separable set K which is inva...
Article
We establish a Banach-Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of L 1(μ) × L 1(μ) and c 0 × c 0. As another consequence, we get a Banach-Mazurkiewicz type theorem on some residual subset of C involving Kharaz...
Article
Full-text available
In the paper of Lee et al. an equivalent condition for a function $f$ to be of the first Baire class has been established. This condition is of an $\eps-\de$ type, similarly as in Cauchy's definition of continuity of a function. In the first part of this paper we examine a problem whether it is possible to obtain other classes of functions by furth...
Article
Recently Kirk introduced the notion of asymptotic contractions on a metric space and using ultrapower techniques he obtained an asymptotic version of the Boyd–Wong fixed point theorem. In this paper we extend this result and moreover, we give a constructive proof of it. Furthermore, we obtain a complete characterization of asymptotic contractions o...
Article
Full-text available
We show that a discrete fixed point theorem of Eilenberg is equivalent to the restriction of the contraction principle to the class of non-Archimedean bounded metric spaces. We also give a simple extension of Eilenberg's theorem which yields the contraction principle.
Article
We show how some results of the theory of iterated function systems can be derived from the Tarski-Kantorovitch fixed-point principle for maps on partially ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson-Barnsley theorem with the only restriction that a metric space considered has the Heine-Bor...
Article
Let X be an abstract nonempty set and T be a self-map of X. Let and denote the sets of all periodic points and all fixed points of T, respectively. Our main theorem says that if , then there exists a partial ordering ≼ such that every chain in (X,≼) has a supremum and for all x∈X, x≼Tx. This result is a converse to Zermelo's fixed point theorem. We...
Article
Using the Zermelo Principle, we establish a common fixed point theorem for two progressive mappings on a partially ordered set. This result yields the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings.
Article
This chapter is intended to present connections between two branches of fixed point theory: The first, using metric methods which is the main subject of this Handbook, and the second, involving partial ordering techniques. We shall concentrate here on the following problem: Given a space with a metric structure (e.g., uniform space, metric space or...
Article
We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Bore...
Article
We simplify a proof of Bessaga's theorem given in the mono- graph of Deimling. Moreover, our argument let us also obtain the following result.
Article
The classic Banach Contraction Principle states that any contraction on a complete metric space has a unique fixed point. Rather than requiring that a single operator be a contraction, we consider a minimum involving a set of powers of that operator and derive fixed-point results. Ordinary analytical techniques would be extremely unwieldy, and so w...
Article
The classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weake...
Article
The classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weake...
Article
Full-text available
We introduce the notion of iterative equivalence of two classes of map­ pings on metric spaces and we demonstrate its utility in metric fixed-point theory. In particular, we show that the fixed-point theorem for Matkowski's contractions can be derived from the corresponding theorem for Browder's contractions, though the first class of mappings is e...
Article
We show (without the axiom of choice) that the Zermelo theorem implies directly a restriction of the Caristi fixed point theorem to continuous functions. Under the axiom of choice, this restriction is proved to be equivalent to Caristi's theorem. We also discuss Kirk's problem on an extension of the Caristi theorem and we establish two selection th...
Article
We show that the Tarski-Kantorovitch Principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.
Article
Full-text available
We establish an equivalence between eight contractive denitions. Next, we formulate a separation theorem for right upper semicontinuous func- tions. As its application, we give a complete characterization of relations between xed point theorems of Boyd and Wong (1969), and Browder (1968).
Article
A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results...
Article
Full-text available
We demonstrate a usefulness of the notion of a connected graph for obtaining some common fixed point theorems. In particular, we establish two theorems of this type involving one, two and four sequences of maps. This generalizes among others the recent results of S. Chang [1], J. Jachymski [2], S. Sessa, R. N. Mukherjee and T. Som [3], and T. Tanig...
Article
We present some results of metric fixed-point theory, which can be derived from the following fixed point theorems involving a partial ordering: Zermelo’s Theorem, the Knaster-Tarski Theorem and the Tarski-Kantorovich Theorem. Using ideas of B. Fuchssteiner [Pac. J. Math. 68, 73-79 (1977; Zbl 0339.26007)] and R. Mańka [Jahrbuch der Kurt-Gödel-Gesel...
Article
We establish a common fixed point principle for a commutative family of self-maps on an abstract set. This principle easily yields the Markoff-Kakutani theorem for affine maps, Kirk's theorem for nonexpansive maps and Cano's theorem for maps on the unit interval. As another application we obtain a new common fixed point theorem for a commutative fa...
Article
Full-text available
Let g be a continuous self-map of the unit interval I. Equivalent conditions are given to ensure that g has a common xed point with every continuous map f : I 7! I that commutes with g on a suitable subset of I. This extends a recent result of Gerald Jungck.
Article
Full-text available
Let C be a non-empty subset of a linear topological space X, and T be a selfmap of C such that the range of I-T is convex, where I denotes the identity map on X. We give conditions under which a map T has a fixed point or a V-fixed point (i.e. a point x 0 ∈C such that Tx 0 ∈x 0 +V, where V is a neighborhood of the origin). Our theorems generalize t...
Article
Full-text available
The Hahn-Banach extension theorem is generalized to the case of continuous linear operators mapping a subspace Y of a normed space X into a normed space V. In contrast with known results of this kind, we do not equip V with a partial ordering neither impose any restrictions on V. The extension property is fully characterized by the sign of the one...
Article
Fixed point theorems for expansive mappings are given generalizing the recent results of T. L. Hicks and L. M. Saliga [Math. Jap. 38, No. 5, 953-956 (1993; Zbl 0803.47053)], and T. Taniguchi [ibid. 34, No. 1, 139-142 (1989; Zbl 0677.54038)]. A common fixed point theorem for four mappings that are not necessarily compatible is proved, which extends...
Article
Let (X,d) be a Hausdorff semimetric (d need not satisfy the triangle inequality) and d-Cauchy complete space. Let f be a selfmap on X, for which d(fx,fy)≤ϕ(d(x,y)), (x,y∈X), where ϕ is a non-decreasing function from ℝ + , the nonnegative reals, into ℝ + such that ϕ n (t)→0, for all t∈ℝ + . We prove that f has a unique fixed point if there exists an...
Article
Fixed point theorems are obtained for some contractive type maps on a complete metric space. Examples are given to show that our main theorem does improve some recent results of J. Kincses and V. Totok [ibid. 4, 69–90 (1990; Zbl 0723.54039)]. A case of maps on an arbitrary metric space is also discussed.
Article
Letf be a self-map on a metric space (X, d). We give necessary and sufficient conditions for the sequences {f n x} (x ∈ X) to be equivalent Cauchy. As a typical application we get the following result. Letf be continuous and (X, d) be complete. If, for anyx, y ∈ X d(f n x, f n y) → 0 and for somec > 0, this convergence is uniform for allx, y inX wi...
Article
Let f be a continuous self-map of the unit interval I. It is proved among others that the family of iterates of f is equicontinuous on I if and only if the set of periodic points of f is a closed interval and it coincides with the set of fixed points of f 2 . This improves upon recent results of Z. Liu [Math. Jap. 38, No. 3, 509-511 (1993; Zbl 0807...
Article
Let f be a continuous self-map on a complete metric space X and p X. Let c be a positive real. Equivalent conditions are given for the singleton {p} to be an attractor of a set of c−fixed points of f. We also establish equivalent conditions for the existence of a contractive fixed point of f. These results subsume a body of fixed point theorems.(Re...
Article
In the monograph [A. Alexiewicz, Functional Analysis (Polish) (1969; Zbl 0181.130)] a certain class of linear normed spaces was introduced; the so called weakly sequentially complete spaces. The idea of the author of this paper was to find an analogon of Cantor’s theorem on completeness for weakly sequentially complete spaces. Besides the fact that...
Article
Several authors extended fixed point theorems for metric spaces to quasi- metric spaces indicating that in quasi-metric setting it is necessary to make some additional assumptions like Hausdorffness of the space and continuity of the mapping [I. L. Reilly, P. V. Subrahmanyam and M. K. Vamanamurthy, Monatsh. Math. 93, 127-140 (1982; Zbl 0472.54018);...
Article
The author solves two problems posed by T. Świa̧tkowski at the end of his paper [ibid. 602(23), 59-65 (1993; Zbl 0816.54013)]. In particular, he extends the Dini theorem to monotonic sequences of functions belonging to the class C 1 (X), the closure of the set of all continuous functions on X having compact supports, with respect to the topology of...

Network

Cited By