Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

Cascade search: Stability of reachable limit points

This paper is a continuation of the series of author’s previous works focused on the cascade search method. The stability of the subset of limit points of a given searching multi-cascade reachable from x is considered with respect to a small variation of the initial point x and to a small perturbation of the searching functional φ generating the multi-cascade. The results are applied to the cascade search problem for common preimages of a closed subspace under n (n ≥ 1) multi-valued mappings.     

Cascade search for preimages and coincidences: Global and local versions

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.     

Stability of coincidence points and properties of covering mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given α-covering mapping and a mapping satisfying the Lipschitz condition with constant β < α have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an α-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of α-covering set-valued mappings is an (α–?)-covering for an arbitrary ? > 0.     

Rectifiable sets and coarea formula for metric-valued mappings

We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the coarea formula. As a consequence, we prove the coarea formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.     

Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

In this paper, we establish coincidence and common fixed point theorems for contractive mappings on a metric space endowed with an amorphous binary relation. The presented theorems extend the results of Samet and Turinici in [Commun. Math. Anal. 12 (2012), 82– 97] and generalize many existing results on metric and ordered metric spaces. We apply also our main results to derive coincidence and common fixed point theorems for cyclic contractive mappings.     

An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds

LetX be a projective manifold of dimension n ≥ 2 andY →X an infinite covering space. EmbedX into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection ofX by a linear subspace of codimension <n extends toY. The proof uses a Hausdorff duality theorem for L₂ cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein.     

On w-compatible mappings and common coupled coincidence point in cone metric spaces

Recently, Abbas et al. [M. Abbas, M.A. Khan, S. Radenovi?, Common coupled fixed point theorems in cone metric spaces for $w$-compatible mapping, Applied Mathematics and Computation (2010) doi:10.1016/j.amc.2010.05.042] introduced the concept of $w$-compatible mappings and obtained results on coupled coincidence point for nonlinear contractive mappings in a cone metric space. In the present paper, we introduce the concept of a common coupled coincidence point of the mappings $F,G:X\times X\to X$ and $f:X\to X$ and we prove some theorems for nonlinear contractive mappings in a cone metric space with a cone having nonempty interior. Our results generalize several well known comparable results in the literature.     

Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces

In this paper we introduce the concept of a tripled coincidence point for a pair of nonlinear contractive mappings F : X³ → X and g : X → X. The obtained results extend recent coincidence theorems due to ?iri? and Lakshmikantham [V. Lakshmikantham, L. ?iri?, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341-4349].     

Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums ∑_nμ(A_n)F(t_n). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.     

Common coupled fixed point theorems in cone metric spaces for w-compatible mappings

In this paper we introduce the concept of a w-compatible mappings to obtain coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in cone metric space with a cone having non-empty interior. Coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend and unify several well known comparable results in the literature. Results are supported by three examples.     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

KKM-type theorems for best proximity points

Let us consider two nonempty subsets A,B of a normed linear space X, and let us denote by 2^B the set of all subsets of B. We introduce a new class of multivalued mappings {T:A→2^B}, called R-KKM mappings, which extends the notion of KKM mappings. First, we discuss some sufficient conditions for which the set ∩{T(x):x∈A} is nonempty. Using this nonempty intersection theorem, we attempt to prove a extended version of the Fan-Browder multivalued fixed point theorem, in a normed linear space setting, by providing an existence of a best proximity point.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions

In this paper, we obtain some common fixed point theorems for occasionally weakly compatible mappings on a set X together with the function d:X×X→[0,∞) without using the triangle inequality and assuming symmetry only on the set of points of coincidence.     

Functional characterizations of p-spaces

We show that a completely regular space Y is a p-space (a ?ech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G _δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.     

Lineability criteria,with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.     

Interpolation and vector-valued functions

Let A be a subspace of C(X), and let K ? X be an interpolation set for A. Let F be a Banach space. We study the following question: When is K a set of interpolation for A ? F, a space of vector-valued functions naturally associated with A ?