The aim of this paper is to obtain fixed point results for interpolative Kannan type contraction mappings. The purpose of this paper is two fold: one relates to the fixed point results for multivalued cyclic interpolative Kannan type contractions, and second deals with proving the Perov fixed point result for interpolative Kannan type contraction mappings in the framework of vector valued metric spaces.
A topological space X is said to be an F-space if the ring C(X), the algebra of all real-valued continuous functions on X, is a Bézout ring (i.e., every finitely generated ideal in C(X) is principal). This is a classical concept in the context of C(X), and various generalizations of this notion have been introduced. In this paper, associated with a subring R of C(X), we introduce and study the notion of R-F-spaces as a new extension of F-spaces. We investigate a number of characterizations of R-F-spaces by extending some of the well-known facts in the literature of C(X) to lattice-ordered subrings, and some other classes of subrings of C(X), namely, SZ -subrings and invertible subrings. It turns out that this new generalization includes previous ones as a special case, thereby establishing strong connections with the existing results in this context.
This article focuses on the study of zero-divisor graph Γ(C(X)P), annihilator graph AG(C(X)P) and weakly zero-divisor graph WΓ(C(X)P) on the ring C(X)P of all real-valued functions on a topological space X that are continuous outside a member of an ideal P of closed subsets of X. We establish that if C(X)P properly contains the ring C(X) of real-valued continuous functions on X, then the radius of Γ(C(X)P) is 2 and it is not triangulated. Moreover, in this situation, both Γ(C(X)P) and AG(C(X)P) are not hypertriangulated and the dominating number of AG(C(X)P) is 2. Furthermore, WΓ(C(X)P) fails to be a complete graph under the hypothesis C(X)P⫌C(X). We establish a connection between the complemented-ness of Γ(C(X)P) and the Von-Neumann regularity of C(X)P under the assumption that C(X)P⊇{χ{p}:p∈X}. We realise that any two of these three graphs coincide if and only if |X|=2 and in this case, the graphs are complete bipartite. We also note that the phenomena of WΓ(C(X)P) being triangulated, hypertriangulated and complemented depend solely on the cardinality of X.
Semitopological groups G and H are said to be HM-equivalent if the Hartman-Mycielski extensions G· and H· of G and H, respectively, are topologically isomorphic. It is shown that G and Gk are HM-equivalent, for every semitopological group G and an integer k ≥ 1 . We also show that if G and H are semitopological groups and G is topologically isomorphic to a subgroup of a finite power of H, then G· admits a topological monomorphism to H· . It is established that the HM-equivalence relation preserves a variety of properties, especially those expressed in terms of cardinal functions. On the other hand, we show under some extra set-theoretic assumptions that the cellularity, Lindelöf property, countable compactness, tightness, Fréchet-Urysohn property, etc., are not preservedby the HM-equivalence in the class of topological groups.
The primary objective is to establish the fixed-point convergence analysis of our newly proposed iteration process, within the framework of hyperbolic spaces, particularly in the presence of contractive-like mapping and further investigate its stability characteristics. To demonstrate its effectiveness, we present a numerical example that highlights the superiority of our iteration process in terms of dominance and faster convergence compared to existing iterative methods. Furthermore, we explore a practical application of our iteration process by employing polynomiographs, which are generated using traditional iteration schemes from the literature and visualized through fractal images. The scheme is implemented computationally using MATLAB, where a variety of visual outputs are produced to illustrate the structure and behavior of the convergence zones. We then compare these results with polynomiographs produced using our novel Picard-New iteration process, demonstrating its advantages in generating fractal images more effectively.
This article presents the construction of a bundle of C*-algebras with a compact base space by means of localization processes resorting to inductive limits. This construction achieved in terms of the Stone-Čech compactification provides a solution to the compactness issue raised by P. Bertozzini, R. Conti and N. Pitiwan in Discrete Non-Commutative Gelfand-Naimark Duality, where the lack of compactness in the base space was an obstacle for the foreseen applications to quantum mechanics.
Let m be any integer ≥ 3 . We consider the polynomial equation Xn+an-1·Xn-1+···+a1·X+a0·I=O, over (m x m)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai ∈ ℝ for each i and n ≥ 1. We discuss its solution set S supplied with the natural Euclidean topology. In particular, we describe the solution set S for m=3 and calculate its dimension.
In this paper, we establish that inverse limits of fuzzy compact spaces remain fuzzy compact, using a direct proof based solely on Lowen’s definitions. This result enables a categorical treatment of compactness analogous to the Tychonoff theorem in the classical setting. Moreover, we prove that the symmetric power functor is normal in the sense adapted to the category of fuzzy compact spaces. It preserves inverse limits of surjective systems, weight, intersections and preimages. It respects embeddings and surjections and it behaves correctly on the empty and one-point spaces. Indeed, we show by these results that the fuzzy symmetric power construction faithfully generalizes its classical counterpart while preserving the essential structural and categorical properties of compactness.
We present crossconvergence, which commonly generalize bornology, preuniform convergence, orderconvergence, b-uniform filtration and grill-determined prenearness. The corresponding construct CCONV forms a strong topological universe, respectively quasitopos, in which quotients are stable under arbitrary products. We also discuss its enlargement to semicrossconvergence and precrossconvergence, respectively.
In this paper, we study two classes of topological spaces: cellular-almost Lindelöf spaces and cellular-weakly Lindelöf spaces. We prove that the classes of cellular-Lindelöf, cellular weakly Lindelöf and cellular-almost Lindelöf are distinct. In addition, we present a comparative study of these classes. We also establish some cardinality results. In particular, we prove that, under the assumption of 2
In this paper a new contractive type mapping known as mapping contracting transverse axis of hyperbola is introduced. This mapping can reduce the length of transverse axis of a hyperbola. This is a geometric technique that is connected to the study of geometric characteristics of certain curves defined over metric spaces. The paper is decorated by some suitable examples that support our proven results and showing the distinctness of our mapping from the usual contractive type mappings. Our proposed mapping also admits discontinuity at fixed point, thus gives a new solution to an open problem posed by B.E. Rhoades. Finally a geometric figure is illustrated to describe the speciality of our mapping.
In this paper, we study selectively highly divergent (SHD) spaces on hyperspaces with the Vietoris topology and the Pixley-Roy topology. Moreover, we show that they are preserved under quasi-perfect countable-to-one mappings and preserved inversely under quasi-open mappings.
The degree of nondensifiability (DND) quantifies, in a specific sense, the distance between a bounded subset of a metric space and the Peano Continua it contains. In this paper, we use the DND to establish a quantitative version of the concepts of collective compactness and equicompactness for families of bounded linear operators between Banach spaces. Specifically, we prove inequalities that relate the DND to these notions and through some examples, we demonstrate that these inequalities are the best possible
We present an algebraic approach to the Collatz conjecture by studying the topology τf on ℕ induced by the Collatz function f, where the open sets θ ⊂ ℕ satisfy f-1 ( θ ) ⊂ θ . This topology, known as \emph{primal topology}, turns τf into a commutative semiring. We prove that the Collatz conjecture holds if and only if τf is local. More generally, we show that any compact primal topology corresponds to a semiring that decomposes as a finite direct sum of certain local semirings and that primal compactness connectedness characterises locality. In addition, we establish that a topological space is not w-R0 if and only if its associated semiring of open sets has a unique maximal ideal such that it is an avoidance ideal of a closed set.
Given a metric continuum X, we consider C(X) as the collection of all subcontinua of X. S. López introduced the concepts of pseudo-linearity and pseudo-circularity in order to characterize continua having a positive Whitney level that is an arc or a simple closed curve. In this paper, we introduce the concepts of w-unicoherence and top-irreducibility. We study the relations between these and the well-known and more naturally-related properties defined in continuum theory, and with the concepts of pseudo-linearity and pseudo-circularity. Moreover, by using these new concepts, we obtain a new characterization of continua which have a positive Whitney level that is an arc or a simple closed curve. Also, we provide necessary conditions for a continuum to have a positive Whitney level which is a simple n-od.
In this paper, we introduce and study two new classes of subrings of C(X): norm-closed subrings and norm-reflecting subrings. A subring ℝ ⊆ S ⊆ C ( X ) is said to be norm-closed if for every f ∈ S , the function |f| ∈ S; it is norm-reflecting if |f| ∈ S implies f ∈ S. These concepts are inspired by the lattice structure of C(X), particularly the operation of taking absolute values. We provide characterizations of norm-closed (norm-reflecting) subrings. Several examples and counterexamples are presented to illustrate the distinctions and connections between these classes.
We introduce an iterative technique with an inertial term that converges strongly to a fixed point of mappings satisfying Condition (E). Our results extend existing work by providing a robust numerical method for solving fixed point problems in Banach spaces. To demonstrate the effectiveness of our approach, we present numerical examples of a mapping that is not nonexpansive but satisfies Condition (E). Furthermore, we illustrate the convergence behaviour of our algorithm for different choices of initial guesses and coefficients, using MATLAB to validate the theoretical results. This work contributes to the broader framework of fixed point theory and offers practical insights for solving nonlinear problems in applied mathematics.
We consider various strengthenings of the notion of topological transitivity in non-autonomous discrete dynamical systems. We give many equivalent conditions for each of these notions and present the implications among them. We also consider rearrangements of non-autonomous dynamical systems.
We show that the polynomial entropy of homeomorphisms on regular curves is bounded above by one. Moreover, the polynomial entropy equals one under the fairly mild condition that the homeomorphism possesses a wandering point. We obtain a rigidity result for homeomorphisms on local dendrites (and therefore on graphs and dendrites): their polynomial entropy is either zero or one.
In the context of the category Cap of convergence approach spaces and contractions, we introduce and study approach analogs of the upper and lower Kuratowski convergences, upper-Fell and Fell topologies on the set of closed subsets of the coreflection on the category Conv of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the Conv-coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the Conv-coreflection of the base space, while the Conv-reflection is the lower Kuratowski convergence associated with the Conv-reflection. The Conv-coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the Conv-reflection of the base space, while the Conv-reflection is the upper Kuratowski convergence associated with the Conv-coreflection of the base space. We show that, over an approach space, the lower Kuratowski convergence approach structure is in fact an approach structure that coincides with the ∨ -Vietoris approach structure introduced by Lowen and his collaborators, though it may be strictly finer over a general convergence approach space. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach, but finer than the upper Fell approach structure introduced by the first and third author. We also obtain a Cap abstraction of the classical result that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.
