Applied General Topology

A new modified mixed-type Ishikawa iteration scheme with error for common fixed points of enriched strictly pseudocontractive self mappings and ΦΓ-enriched Lipschitzian self mappings in uniformly convex Banach spaces

2024-05-13T06:54:36+02:00

Let E be a uniformly convex Banach space and C a nonempty closed bounded convex subset of E. Let Γ : C ⟶ C and G : C ⟶ C be enriched strictly pseudocontractive mapping and Φ_Γ -enriched Lipschitzian mapping respectively. We introduce the above two mappings in uniformly convex Banach space and thereafter prove that a new modified mixed-type lshikawa iteration scheme converges strongly to the common fixed points of Γ and G. In addition, we incorporate error terms to enhance the convergence of
the method and also to improve the stability and robustness of the method. Our results extend and generalize the results obtained in [5] and so many other recent results currently existing in literature.

Solution of boundary value problem via C-algebra valued bipolar metric spaces

2023-12-08T18:15:13+01:00

In this manuscript, we prove common fixed point theorems using commuting mapping on C*-algebra valued bipolar metric space. Some of the well-known outcomes in the literature are expanded and generalized by the results shown. An example and applications to support our result is presented.

Some generalizations of Darbo's fixed point theorem under weak topology features with application to a Volterra-type integral equation

2025-01-05T20:39:26+01:00

In this paper, we provide some generalizations of Darbo's fixed point theorem for larger classes of contraction. Our results are investigated under the weak topology of a Banach space using the measure of weak noncompactness. The results presented in the paper generalize and extend several well-known comparable results in the literature. Further, We illustrate the applicability of our theoretical findings by studying the existence of solutions for a coupled of nonlinear Volterra-type integral equations.

Infinity valued partial metrics

2024-09-17T16:24:37+02:00

A partial metric is a generalised metric incorporating non-zero self-distance. It was introduced by Matthews in Computer Science, and connected well with Kopperman's long standing interest in generalised metrics and non-Hausdorff topology. In this article we further generalise partial metrics by incoporating ∞ as a distance. We prove that a so-called infinite partial metric induces a partial metrizable topology, but not in general vice verse.

Basic constructions of topological n-ary polygroups

2024-10-21T12:01:02+02:00

In this paper, we begin the study of topological n-ary polygroups. We begin by stating some properties of topological n-ary polygroups and introducing quotient n-ary polygroups. We consider topological isomorphism and prove some topological isomorphism theorems. Moreover, We define the concept of cp-resolvable topological polygroups and investigate a few properties of them.

Strong chain transitivity in hyperspaces of uniform spaces

2024-02-05T09:39:09+01:00

We apply topological definitions of strong chain transitivity and characterizetopological strong chain transitivity for the induced maps on hyperspaces ofuniform spaces. We prove that, if the induced maps on such hyperspaces exhibittopological strong chain transitivity then the base map satisfies the propertyof topological strong chain transitivity.

Proximal motion planning algorithms

2024-05-20T16:45:10+02:00

In this paper, we transfer the problem of measuring navigational complexity in topological spaces to the nearness theory. We investigate the most important component of this problem, the topological complexity number (denoted by TC), with its different versions including relative and higher TC, on the proximal Schwarz genus as well as the proximal (higher) homotopic distance. We outline the fundamental properties of some concepts related to the proximal (or descriptive proximal) TC numbers. In addition, we provide some instances of (descriptive) proximity spaces, specifically on basic robot vacuum cleaners, to illustrate the results given on proximal and descriptive proximal TC.

Totally bounded ultrametric spaces generated by labeled rays

2024-09-02T12:18:46+02:00

We will say that a tree T is almost a ray if T is the union of a ray and a finite tree. Let l be a non-degenerate labeling of the vertex set V of almost a ray T and let d_I be the corresponding ultrametric on V. It is shown that the ultrametric space (V, d_I ) is totally bounded iff this space contains an infinite totally bounded subspace. We also prove that the last property characterizes the almost rays.

Digital quotient spaces and digital quotient maps

2024-10-13T13:22:12+02:00

In this paper we give the definitions of digital quotient space as well as digital quotient map. We also show some propositions about digital quotient maps and show that the subdivision projection is also a digital quotient map.

Some innovative results for interpolative Kannan type and Reich-Rus-Ćirić type cyclic contractions

2024-08-26T13:19:55+02:00

In this manuscript, we present innovative findings on the existence and uniqueness of fixed points for cyclic mappings. Our discussion covers results for interpolative Kannan-type cyclic contractions in two cases: when the sum of the interpolative exponents is less than one and when it is greater than one. Additionally, we provide results for interpolative Reich-Rus-Ćirić cyclic contractions specifically for cases where the sum of the interpolative exponents is greater than one. Furthermore, we verify our results with suitable examples. Our results are new and complement some results in the literature.

Dynamical behavior of a general non-autonomous dynamical system

2024-11-19T18:44:40+01:00

In this work, we investigate dynamical notions such as minimality, almost periodicity, equicontinuity, transitivity, proximality and periodicity for a general non-autonomous dynamical system. We provide necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that for an equicontinuous system (X,F), if x is almost periodic then OH(x) is uniformly almost periodic. We also establish preservance of almost periodicity under uniform convergence. We also establish some results capturing the qualitative behavior of a general non-autonomous dynamical system.

Non-contractive mappings and application to a drug diffusion problem

2024-05-15T10:40:16+02:00

For mappings that admit multiple fixed points, we find conditions that imply a unique fixed point. Our conclusions unify and extend numerous existing fixed point theorems. We also establish common fixed point conclusions that may not satisfy a contractive condition. Since general techniques for determining common fixed points of non-contractive mappings are not available, our results introduce new techniques for such studies. Further, we find out the geometric properties of multiple fixed points whenever there is no possibility of the persistence of a unique fixed point. An application of fixed point results has also been given to model a problem emerging while a diffusing drug is confined within an absorbing agent bounded by parallel walls with fixed concentrations.

On the topology of topological fundamental groupoids

2024-11-13T12:31:44+01:00

‎This paper is devoted to the Lasso and the Whisker topology of the fundamental groupoid‎. ‎We prove that topological fundamental groupoid of a given space X is locally path connected ‎in‎ ‎general‎ and is path connected if X is simply connected‎. ‎We show that for locally path connected space X‎, ‎the unit map 1 : X → π X is an embedding if and only if X is a semilocally simply connected space‎. ‎Also‎, ‎we give conditions that guarantee Hausdorffness of the topological fundamental groupoid‎.

L-Fuzzy rough point-wise proximity spaces

2024-10-24T21:47:16+02:00

In this paper, we study the concept of nearness between L-fuzzy rough sets and L-fuzzy points. The nearness between two L-fuzzy rough sets are viewed in terms of this new concept. Also, we investigate the relationship of L-fuzzy rough point-wise proximity space with L-fuzzy rough grills, L-fuzzy rough δ_ℛ -clan and L-fuzzy rough δ_ℛ-clusters. Moreover, we introduce the L-fuzzy rough point-wise LO-proximity spaces and study a few of its fundamental properties. The research is supported by a sufficient number of examples.

Fixed point property and degree of coincidence

2024-05-26T21:14:17+02:00

We introduce the concept of degree of coincidence to measure the divergence from the fixed point property in case a space does not have the FPP.

A generalization of m-topology and U-topology on rings of measurable functions

2024-05-11T18:39:03+02:00

In this paper, we generalize m_μ and U_μ topologies on M ( X , A ) via an ideal I in the ring M ( X , A ) of all real-valued measurable functions. The collection L_I^∞ ( μ ) of all essentially I-bounded functions over the measure space (X,A,μ)$ and the ideal I_μ (X,A) ={ f ∈ M (X,A) : for every g ∈ M(X,A), fg is essentially I-bounded } are the components of 0 in the space m_μIand U_μI respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide. In particular, it is proved that they coincide if and only if the three sets M(X,A), L_I^∞ ( μ ) , and I_μ (X,A) coincide and this holds when and only when M(X,A), with either topology, is connected, which is further equivalent to each of these two topological spaces being a topological ring as well as a topological vector space. Furthermore, L_I^∞ ( μ ) is a complete pseudonormed linear space regardless of the choice of I. Finally, we examine when these later spaces are separable.

Non-metrizable manifolds and contractibility

2024-07-16T16:30:55+02:00

We investigate whether non-metrizable manifolds in various classes can be homotopy equivalent to a CW-complex (in short: heCWc),
and in particular contractible. We show that a non-metrizable manifold cannot be heCWc if it has one of the following properties: it contains a countably compact non-compact subspace; it contains a copy of an ω₁-compact subspace of an ω₁-tree; it contains a non-Lindelöf closed subspace functionally narrow in it. (These results hold for more general spaces than just manifolds.) We also show that the positive part of the tangent bundle of the longray is never heCWc for any smoothing. These theorems follow from stabilization properties of real valued maps. On a more geometric side, we also show that the Prüfer surface, which has been shown to be contractible long ago, has an open submanifold with trivial homotopy groups which is not heCWc. On the other end of the spectrum, we show that there is a non-metrizable contractible Type I surface.

C1(X) on the edge of C2(X)

2024-07-22T11:08:36+02:00

Given a continuum X and a positive integer n, let C_n(X) be the hyperspace consisting of all nonempty closed subsets of X having at most n components. For a subcontinuum A of X having empty interior, consider the following properties: A is a subcontinuum of colocal connectedness, X\A is continuumwise connected, A is a nonblock subcontinuum, A is a shore subcontinuum, A is not a strong centre. In this paper, we prove that C₁(X) has all of these properties in C_n(X) if n ≥ 3, and we study when C₁(X) has one of these properties in C2(X) .

Countably regular good spaces and almost Cκ-spaces

2024-07-29T12:56:36+02:00

In this paper, we build upon our previous studies by redefining fundamental concepts and transferring them from rings of continuous functions to rings of continuous functions with countable images. This article will focus on regular good spaces (``"RG-spaces", for short) and related concepts. Specifically, we introduce and examine two generalizations of RG-spaces, calling countably regular good spaces and strongly countably regular good spaces.

Enriched rational type contractions in quasi-Banach spaces and generalized convex b-metric spaces

2024-08-14T13:50:46+02:00

In this paper, we define enriched rational type contraction in the setting of quasi-Banach spaces and prove the existence and uniqueness of fixed point of the enriched contraction by using Krasnselskii iteration procedure. Morever, we introduce weak enriched rational type contraction and give some fixed point theorems related to Kirk's iteration method. We also give the notion of generalized convex b-metric space and extend the results for quasi-Banach spaces to the concept of generalized convex b-metric spaces. Moreover some examples to support our results are also given.

Fixed points of Oslike contraction mapping: stability and convergence of a new iteration method

2024-09-18T20:21:42+02:00

This work introduces a novel three-type iteration method for estimating the fixed point approximation for a contraction mapping in the context of Banach space. Furthermore, we present a stability finding for an iteration scheme, demonstrating the strong convergence of our new iteration scheme. In conclusion, we proved that our iteration runs faster than some other iterations previously documented in the literature. We also provide a numerical example using the computer application Matlab to support our claim.

Subdivision-based homotopy equivalence of digital circles

2024-10-30T14:42:27+01:00

A direct translation of the notion of homotopy equivalence from algebraic topology to digital images leads to a much more rigid definition in the context of digital topology. This results in two digital circles of different radii being not homotopic. A more suitable idea of equivalences, called "subdivision-based homotopy equivalence" which is introduced by Lupton et al., involves the concept of subdivision of digital images. We prove that, in this sense, any digital circle is subdivision-based homotopic to the Diamond, the prototypical digital circle of radius 1, which implies, as a consequence, that the fundamental group of any digital circle is isomorphic to Z.

Non-symmetric convergence and regularity

2024-09-20T23:44:00+02:00

We study regularity in quasi-convergence spaces and biconvergence spaces. We show that a notion weaker than the usually considered pairwise regularity is sufficient in important applications. This regularity can be defined in terms of closures of pair filters or by a diagonal condition. We show its appropriateness by characterizing it in terms of continuous convergence and in terms of extensions of continuous mappings.

New fixed point theorems in metric space with w-distance

2024-09-27T18:38:27+02:00

This paper presents some new fixed point results for a class of weakly contractive maps in a metric space. We use the concept of w-distance in the framework of metric spaces. As far as we know, the concept of w-distance is being utilized for the first time to study Suzuki-type antecedent conditions with weakly contractive maps. An illustrative example demonstrates the generality of the main result.

Some topological aspects of ideals in quantales

2024-11-14T10:30:43+01:00

As a natural extension of the ongoing development of a theory of ideals in commutative quantales with an identity element, this article aims to study into the analysis of certain topological properties exhibited by distinguished classes of ideals. These ideals are equipped with quantale topologies. The primary objectives encompass characterizing quantale spaces exhibiting sobriety, examining several conditions pertaining to quasi-compactness, and demonstrating that quantale spaces comprised of proper ideals adhere to the spectral properties as defined by Hochster. We introduce the notion of a strongly disconnected spaces and show that for a quantale with zero Jacobson radical, strongly disconnected quantale spaces containing all maximal ideals of the quantale imply existence of non-trivial idempotent elements in the quantale. Additionally, a sufficient criterion for establishing the connectedness of a quantale space is presented. Finally, we discuss on continuous maps between quantale spaces.

Remarks on fixed point assertions in digital topology, 9

2024-10-07T09:58:19+02:00

We continue a discussion of published assertions that are incorrect, incorrectly proven, or trivial, in the theory of fixed points in digital topology.

Pre-symmetric c-distances and characterization of complete cone metric spaces

2024-10-04T15:00:34+02:00

The main goal of this research is to express the definition of pre-symmetric c-distances in cone metric spaces inspired by pre-symmetric w-distances defined by Romaguera and Tirado [Mathematics. 12 (2024), 2:305] and give the properties of this type of c-distances along with some examples. Then we introduce Suzuki's theorem regarding a pre-symmetric c-distance and obtain
relationship between several important contractions with respect to cone metric spaces. Finally, we will state a characterization of the completeness of cone metrics using fixed point theorems, which is different from previous descriptions of complete cone metric spaces.

β-Normality in locales

2024-12-12T10:25:31+01:00

In this paper, we establish the theory of β-normal locales which are the point- free counterparts of β-normal spaces which were introduced by Arkhangel’skii and Ludwig. We give characterizations of β-normal locales using some types of open sublocales. Certain circumstances are exhibited in which normality coincides with β-normality. For instance, we use the localic Kateˇtov-Tong insertion theorem to prove that every β-normal locale which is also a coframe is normal. This result argues that there does not exist a finite β-normal locale which is not normal. Included here is also an answer to Murtinova ́’s question about the existence of a regular β-normal space which is not Tychonoff. The study of β-normal locales leads to some variants of β-normal locales, namely α-normal locales and almost weakly β-normal locales. This paper also examines preservation and reflection of β-normal locales by localic maps.

Cauchy G-spaces: Local convexity

2024-12-12T10:29:10+01:00

Cauchy continuous actions of a Cauchy monoid acting on a locally convex Cauchy space X are investigated. The local convexity is determined by a partial order on X. Sufficient conditions are given to ensure the existence of a Hausdorff Cauchy space completion T for which the Cauchy continuous action can be appropriately extended to T.