Point Existence Theorem Research Articles

Hereditarily decomposable continua have non-block points

In this note we expand upon our results from [1] to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e. points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

Fixed points of set-valued mappings satisfying a Banach orbital condition

In this note, we prove a fixed point existence theorem for set-valued functions by extending the usual Banach orbital condition concept for single valued mappings. As we show, this result applies to various types of set-valued contractions existing in the literature.

A Hybrid WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems

In this paper, we develop a simplified hybrid weighted essentially non-oscillatory (WENO) method combined with the modified ghost fluid method (MGFM) [28] to simulate the compressible two-medium flow problems. The MGFM can turn the two-medium flow problems into two single-medium cases by defining the ghost fluids status in terms of the predicted the interface status, which makes the material interface invisible. For the single medium flow case, we adapt between the linear upwind scheme and the WENO scheme automatically by identifying the regions of the extreme points for the reconstruction polynomial as same as the hybrid WENO scheme [50]. Instead of calculating their exact locations, we only need to know the regions of the extreme points based on the zero point existence theorem, which is simpler for implementation and saves computation time. Meanwhile, it still keeps the robustness and has high efficiency. Extensive numerical results for both one and two dimensional two-medium flow problems are performed to demonstrate the good performances of the proposed method.

Multiplicative Normed Linear Space and its Topological Properties

The aim of this article is to propose a new space called multiplicative normed linear space and discuss different topological properties of this space. We establish equivalence between multiplicative compactness and multiplicative sequentially compactness. Some examples are given which demonstrate the validity of our results. Further we apply our results to prove fixed point existence theorems.

Hölder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

Suppose that $S$ is a left amenable semitopological semigroup. We prove that if $\mathcal{S}=\{ T_{t}:t\in S\} $ is a uniformly $k$-Lipschitzian semigroup on a bounded closed and convex subset $C$ of a Hilbert space and $k< \sqrt{2}$, then the set of fixed points of $\mathcal{S}$ is a Hölder continuous retract of $C$. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.

Topological Methods in Analysis

Topological methods have played a seminal role in functional analysis since its birth in the early twentieth century. The Baire category theorem, for example, is the bedrock on which rest such basic principles of functional analysis as the open mapping theorem and the principle of uniform boundedness. Initial (weak) topologies, compactness, and the Tikhonov theorem drive such classical results of duality theory as the Banach-Alaoglu and Krein-Milman theorems. Topological methods also play a crucial role in Banach algebra theory (Gelfand topology), harmonic analysis (locally compact groups and function spaces), differential equations (fixed point theorems and Ascoli-Arzela theorem), and nonlinear analysis (fixed point existence theorems and topological degree theory) to mention just a few. The lofty goal of this special issue is to present some new outstanding contributions in this research area. They include results on fixed point theorems, metric spaces, C-algebras, modular spaces, algebraic hypersurfaces, and topological groups. Our hope is that this wide spectrum of topics will be of interest to many researchers in the area.

Noncompatible Mappings in Intuitionistic Fuzzy Metric Space

In this paper, the new concepts normal product of two multi-valued mappings, \(s\)-weakly compatible, \(s\)-common fixed point and the (EAs) property for two pairs of multi-valued mappings are introduced, and the \(s\)-common fixed point existence theorems for two pairs of multi-valued noncompatible mappings under strict contractive condition are proved. without appeal to continuity of any map involved there in and completeness of underlying space. The results presented in this paper generalize, improve, and unify some recent results in this field.

On the structure of fixed-point sets of asymptotically regular semigroups

We show that the set of fixed points of an asymptotically regular mapping acting on a convex and weakly compact subset of a Banach space is, in some cases, a Hölder continuous retract of its domain. Our results qualitatively complement the corresponding fixed point existence theorems and extend a few recent results of Górnicki [15–17]. We also characterize Bynum’s coefficients and the Opial modulus in terms of nets.

A Brézis–Browder principle on partially ordered spaces and related ordering theorems

Through a simple extension of Brézis–Browder principle to partially ordered spaces, a very general strong minimal point existence theorem on quasi ordered spaces, is proved. This theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems, and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then, they are applied to prove strong minimal point existence results, generalizations of Bishop–Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion.

S-COINCIDENCE AND S-COMMON FIXED POINT THEOREMS FOR TWO PAIRS OF SET-VALUED NONCOMPATIBLE MAPPINGS IN METRIC SPACE

In this work, the new concepts, normal product of two set-valued mappings, s-weakly compatible, s-common fixed point and the (\(EA_s\)) property for two pairs of set- valued mappings are introduced, and the s-common fixed point existence theorems for two pairs of set-valued noncompatible mappings under strict contractive condition are proved, without appeal to continuity of any map involved therein and completeness of underlying space. The results presented in this paper generalize, improve, and unify some recent results in this field.

Formula omitted] connected topological spaces and cut points

A non-cut point existence theorem is proved for non-degenerate H ( i ) connected topological spaces. In an H ( i ) connected space, each component of the complement of a cut point contains a non-cut point. It is also shown that an H ( i ) connected topological space having exactly two non-cut points is a connected ordered topological space.

A General Existence Theorem of Zero Points

Let X be a non-empty, compact, convex set in R and o an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of R;. It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under o has a non-empty intersection with the normal cone of X at the point. In case for every point in X it holds that the intersection of the image under o with the normal cone of X at the point is either empty or contains the origin O; then o must have a zero point on X, i.e. there exists a point in X satisfying that O lies in the image of the point. Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point. In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases. We also discuss what kind of solutions may exist when no further conditions are stated on the mapping o. Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.

Common fixed point theorems for commuting k‐uniformly Lipschitzian mappings

We give a common fixed point existence theorem for any sequence of commuting k‐uniformly Lipschitzian mappings (eventually, for k = 1 for any sequence of commuting nonexpansive mappings) defined on a bounded and complete metric space (X, d) with uniform normal structure. After that we deduce, by using the Kulesza and Lim (1996), that this result can be generalized to any family of commuting k‐uniformly Lipschitzian mappings.

Quasi-weak convergence with applications in ordered Banach space

In the paper quasi-weak convergence is introduced in ordered Banach space and it is weaker than weak convergence. Besed on it, the fixed point existence theorem of increasing operator is proved without the suppose of continuity and compactness in the sense of norm and weak compactness and is applied to the Hammerstein nonlinear intergal equation.

Cut-point spaces

The notion of a cut-point space is introduced as a connected topological space without any non-cut point. It is shown that a cut-point space is infinite. The non-cut point existence theorem is proved for general (not necessarily T 1 T_1 ) topological spaces to show that a cut-point space is non-compact. Also, the class of irreducible cut-point spaces is studied and it is shown that this class (up to homeomorphism) has exactly one member: the Khalimsky line.

Pure strategy Nash equilibrium points and the Lefschetz fixed point theorem

A pure strategy Nash equilibrium point existence theorem is established for a class ofn-person games with possibly nonacyclic (e.g. disconnected) strategy sets. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps, due to Eilenberg and Montgomery, which extends their better known. Eilenberg-Montgomery fixed point theorem (EMT) [Eilenberg/Montgomery, Theorem 1, p. 215] to nonacyclic spaces. Special cases of the existence theorem are also discussed.

AN UNSOLVED PROBLEM CONCERNING CONTRACTIVE TYPE MAPPINGS

In this paper, comparing to (25) contractive type mappings defined by B.E. Rhoades we get a class of more general contractive type mappings. In virtue of this, we discuss two new fixed point theorems. And further, we derive two fixed point existence theorems for (25) contractive type mappings.