Nonempty Compact Convex Subset Research Articles

Hyers–Ulam stability of bijective $$\varepsilon $$-isometries between Hausdorff metric spaces of compact convex subsets

Let X (resp. Y) be a real Banach space such that the set of all $$w^*$$ -exposed points of the closed unit ball $$B(X^*)$$ (resp. $$B(Y^*)$$ ) is $$w^*$$ -dense in the unit sphere $$S(X^*)$$ (resp. $$S(Y^*)$$ ), (cc(X), H) (resp. (cc(Y), H)) be the metric space of all nonempty compact convex subsets of X (resp. Y) endowed with the Hausdorff distance H, and $$f:(cc(X),H)\rightarrow (cc(Y),H)$$ be a standard bijective $$\varepsilon $$ -isometry. Then there is a standard surjective isometry $$g:cc(X)\rightarrow cc(Y)$$ satisfying that $$(1)\, g|_{X}$$ (the restriction of g on $$\{\{u\}, u\in X\}$$ ) is a surjective linear isometry from $$\{\{u\},u\in X\}$$ onto $$\{\{v\},v\in Y\}$$ and $$g(A)=\cup _{a\in A}g|_{X}(\{a\})$$ for any $$A\in cc(X)$$ ; $$(2)\, H(f(A),g(A))\le 3\varepsilon $$ for any $$A\in cc(X)$$ .

Mixed-Type Functional Differential Equations: A $C_{0}$-Semigroup Approach

In this paper we study certain systems of mixed-type functional differential equations, from the point of view of the $C_{0}$-semigroup theory. In general, this type of equations are not well-posed as initial value problems. But there are also cases where a unique differentiable solution exists. For these cases and in order to achieve our goal, we first rewrite the system as a classical Cauchy problem in a suitable Banach space. Second, we introduce the associated semigroup and its infinitesimal generator and prove important properties of these operators. As an application, we use the results to characterize the null controllability for those systems, where the control $u$ is constrained to lie in a non-empty compact convex subset $\Om{}$ of $\R^n$.

On the Krein-Milman theorem in CAT(κ) spaces

Let κ > 0 and (X, ρ) be a complete CAT(κ) space whose diameter smaller than ... It is shown that if K is a nonempty compact convex subset of X, then K is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT(κ) spaces.

SEMI-ASYMPTOTIC NON-EXPANSIVE ACTIONS OF SEMI-TOPOLOGICAL SEMIGROUPS

Abstract. In this paper we extend Takahashi’s fixed point theorem ondiscrete semigroups to general semi-topological semigroups. Next we de-fine the semi-asymptotic non-expansive action of semi-topological semi-groups to give a partial affirmative answer to an open problem raised byA.T-M. Lau. 1. IntroductionA (not necessarily linear) self-mapping T : E → E on a Banach space Eis called non-expansive if kT(x) − T(y)k ≤ kx − yk for all x,y ∈ E. In 1963R. DeMarr proved the following common fixed point theorem for commutingfamilies of non-expansive self-mappings [6].Theorem 1.1. For any non-empty compact convex subset K of a Banach spaceE each commuting family of non-expansive self mappings on K has a commonfixed point in K.This generalizes the celebrated Markov-Kakutani fixed point theorem (forthe case of Banach spaces) on commuting families of continuous linear trans-formations on Hausdorff topological vector spaces leaving certain nonemptycompact convex subset invariant. DeMarr’s theorem has been generalized inseveral directions by Belluce and Kirk [2,3], Takahashi [16], Mitchell [14], Lauand Holmes [7,8].DeMarr theorem suggests that the action of certain commutative semigrouphas a fixed point. It is then natural to seek the same type offixed point propertyfor the actions of more general semigroups.Let S be a semi-topological semigroup, that is a semigroup with a Hausdorfftopology with separately continuous multiplication. We say that S is rightreversible if it has finite intersection property for closed left ideals. An actionof S on a topological space E is a mapping (s,x) → s(x) from S × E into

Hyperspaces of convex bodies of constant width

For every n≥1, let cc(Rn) denote the hyperspace of all non-empty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric topology. For every non-empty convex subset D of [0,∞) we denote by cwD(Rn) the subspace of cc(Rn) consisting of all compact convex sets of constant width d∈D and by crwD(Rn) the subspace of the product cc(Rn)×cc(Rn) consisting of all pairs of compact convex sets of constant relative width d∈D. In this paper we prove that cwD(Rn) and crwD(Rn) are homeomorphic to D×Rn×Q, whenever D≠{0} and n≥2, where Q denotes the Hilbert cube. In particular, the hyperspace cw(Rn) of all compact convex bodies of constant width as well as the hyperspace crw(Rn) of all pairs of compact convex sets of constant relative positive width are homeomorphic to Rn+1×Q.

A Fixed Point Theorem for Left Amenable Semi-Topological Semi Groups

In this note, we extend and improve the corresponding result of Takahashi. Explanation of DeMarr’s theorem is further generalized for some semi groups of non-expansive self- maps on K by the following considerations which are explained in the paper. The application of Zorn’s lemma and its application are explained. An application of Zorn’s lemma shows that there exists a minimal non-empty compact convex and S-invariant subset.

On the convergence of Ishikawa iterative process for a pair of single-valued and multi-valued nonexpansive mappings

Let K be a nonempty compact convex subset of a complete CAT(0) space X and let t: K → K and T: K → CB(K) be a single-valued nonexpansive mapping and a multi-valued nonexpansive mapping, respectively. Assume, in addition, that F(t) ∩ F(T) ≠ ∅ and Tp = {p} for all p ∈ F(t) ∩ F(T). We prove that the sequence of modified Ishikawa iteration method generated from an arbitrary x0 ∈ K by yn = (1 - βn) xn ⊕ βn zn, xn+1 = (1-αn) xn ⊕ αntyn, where zn ∈ Txn and {αn}, {βn} are sequences of positive numbers satisfying 0< a ≤ αn, βn ≤ b < 1 converges strongly to a common fixed point of t and T; that is, there exists x ∈ K such that x = tx ∈ Tx. Our results extends and generalizes some theorems of T. Puttasontiphot [24].

Averaging of Multivalued Integral Equations

Integral equations are extensively used in various fields of knowledge, such as the theory of elasticity, heat and mass transfer, the theory of oscillations, fluid dynamics, the theory of filtration, electrostatics, electrodynamics, biomechanics, game theory, control theory, voting theory, electrical engineering, economics, and medicine. The investigations of actual processes based on idealized mathematical models often lead to equations with small parameters. For their investigation, various asymptotic methods are extensively used. The choice of a specific asymptotic method depends on the structure of the equation used to describe the dynamics of an object. In recent years, averaging methods are rapidly developed in the nonlinear mechanics and in the theory of oscillations. The mathematical substantiation of the averaging method for ordinary differential equations was originated in 1937 by Krylov–Bogolyubov in their fundamental work [1]. The works by Grebenikov, Mitropol’skii, Moiseev, Perestyuk, Plotnikov, Samoilenko, Filatov, and other researchers [2–4, 7–11, 14] played a great role in the development of the averaging method for different classes of differential equations. Later, the ideas of the averaging method were generalized for differential equations with multivalued and fuzzy right-hand sides (see [5, 6, 9, 12, 13] and the references therein). In the present paper, we substantiate the averaging method for a multivalued integral equation. Let conv .R/ be a metric space of nonempty compact convex subsets R with the Hausdorff metric h.F;G/: Consider a multivalued integral equation

Orlicz metrics for convex bodies

The \(L_p\) metrics, extensions of the Hausdorff metric for convex bodies, were investigated by Vitale in 1985. In this paper, we extend \(L_p\) metrics to Orlicz metrics, and show that these Orlicz metrics generate the same topology as the Hausdorff one in the space of all convex bodies (i.e., non-empty compact convex subsets), consequently, the space of all convex bodies with the Orlicz metric is a complete, separable metric space. Furthermore, we also show that the space of all convex bodies with the Orlicz metric is a metric segment space.

Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach space

In this paper, we approximate a fixed point of the semigroup of Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition and with respect to a finite family of sequences of left strong regular invariant means defined on an appropriate invariant subspace of . Our result extends the main results announced by several others. MSC:47H09, 47H10, 47J25.

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Any abstract convex cone S with a uniformity satisfying the law of cancellation can be embedded in a topological vector space \(\widetilde{S}\) (Urbański, Bull Acad Pol Sci, Sér Sci Math Astron Phys 24:709–715, 1976). We introduce a notion of a cone symmetry and decompose in Theorem 2.12 a quotient vector space \(\widetilde{S}\) into a topological direct sum of its symmetric subspace \(\widetilde{S}_s\) and asymmetric subspace \(\widetilde{S}_a\). In Theorem 2.19 we prove a similar decomposition for a normed space \(\widetilde{S}\). In section 3 we apply decomposition to Minkowski–Rådström–Hörmander (MRH) space with three best known norms and four symmetries. In section 4 we obtain a continuous selection from a MRH space over ℝ2 to the family of pairs of nonempty compact convex subsets of ℝ2.

Existence Solutions of Vector Equilibrium Problems and Fixed Point of Multivalued Mappings

LetKbe a nonempty compact convex subset of a topological vector space. In this paper-sufficient conditions are given for the existence ofx∈Ksuch thatF(T)∩VEP(F)≠∅, whereF(T)is the set of all fixed points of the multivalued mappingTandVEP(F)is the set of all solutions for vector equilibrium problem of the vector-valued mappingF. This leads us to generalize and improve some existence results in the recent references.

Affine group acting on hyperspaces of compact convex subsets of Rn

For every n ≥ 2, let cc(R) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space R endowed with the Hausdorff metric topology. Let cb(R) be the subset of cc(R) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(R). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(R). This is applied to show that cb(R) is homeomorphic to the product Q×R, where Q stands for the Hilbert cube. Furthermore, we investigate the action of the orthogonal group O(n) on cc(R). In particular, we show that if K ⊂ O(n) is a closed subgroup that acts non-transitively on the unit sphere Sn−1, then the orbit space cc(R)/K is homeomorphic to the Hilbert cube with a removed point, while cb(R)/K is a contractible Q-manifold homeomorphic to the product (E(n)/K) × Q. The orbit space cb(R)/Aff(n) is homeomorphic to the Banach-Mazur compactum BM(n), while cc(R)/O(n) is homeomorphic to the open cone over BM(n).

On lower semi-continuous metric projections onto finite dimensional subspaces of spaces of continuous functions

The context of the paper is: a locally compact Hausdorff space T; the space C0(T), equipped with the uniform norm, of real continuous functions on T which vanish at infinity; a linear subspace G, of finite dimension n, of the space C0(T); and the set-valued metric projection PG of C0(T) into the family of non-empty compact convex subsets of G, defined by PG(f)= the set of best uniform approximations to f from G. Those G which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those G for which the metric projection PG is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for PG to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case T is a compact space in which no net of components is convergent to a single point set, of those subspaces G of C(T) for which PG is lower semi-continuous. Simple examples show that if the space T is compact and totally disconnected, but not finite, the situation is essentially different.

Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces

In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subset C of a complete CAT(0) space X and prove that the proposed sequence converges to a common fixed point for these semigroups of nonexpansive mappings. Note that our results generalize Cho et al. result (Nonlinear Anal. 74:6050-6059, 2011) and related results.

Kakutani-type fixed point theorems: A survey

A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or semigroup S of continuous affine transformations s : Q → Q, where Q is a nonempty compact convex subset of a Hausdorff locally convex linear topological space, then under suitable conditions S has a common fixed point in Q, i.e., a point $${a \in Q}$$ such that s(a) = a for each $${s \in S}$$ . In 1938, Kakutani gave two conditions under each of which a common fixed point of S in Q exists. They are (1) the condition that S be a commutative semigroup, and (2) the condition that S be an equicontinuous group. The present survey discusses subsequent generalizations of Kakutani’s two theorems above.

Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces

Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping. For the implicit iterates , , , . We proved that converges strongly to a fixed point of under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007).

Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

AbstractLet "Equation missing" be a nonempty compact convex subset of a uniformly convex Banach space "Equation missing", and let "Equation missing" and "Equation missing" be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that "Equation missing" and "Equation missing" for all "Equation missing". We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary "Equation missing" by "Equation missing", "Equation missing", where "Equation missing" and "Equation missing", "Equation missing" are sequences of positive numbers satisfying "Equation missing", "Equation missing", converges strongly to a common fixed point of "Equation missing" and "Equation missing"; that is, there exists "Equation missing" such that "Equation missing".

Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and 𝒯 and 𝒮 are two families of self set-valued mappings of X such that 𝒮 is generalized KKM w.r.t. 𝒯, under some natural conditions, the set-valued mappings S ∈ 𝒮 have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.