Convex Subsets Of Banach Spaces Research Articles

Fixed point theorems of generalized contractions and their applications

The paper presents some theorems about fixed points of nonexpansive mappings of certain convex subsets of Banach spaces, aswell as examples of applications of these theorems, among others, to justify the correctness of the square roots approximation process in Heron’s algorithm (from the 1st century AD).

On the asymptotics of (c)-mapping iterations

In this paper, we give an investigation of the asymptotics for the iterations associated with (c)-mappings acting on unbounded closed convex subsets of Banach spaces. In particular, the almost fixed point property (in short; AFPP) and conditions ensuring an ergodic type theorem for this class of mappings are established.

Stochastic order on metric spaces and the ordered Kantorovich monad

In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. Our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction.The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances.We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively.In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.

Split equilibrium problems for related games and applications to economic theory

ABSTRACTIn this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

On the Ornstein–Uhlenbeck operator in convex subsets of Banach spaces

On the Ornstein–Uhlenbeck operator in convex subsets of Banach spaces

Split Equilibrium Problems for Related Games and Applications to Economic Theory

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan–KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

Best Proximity Points of Contractive-type and Nonexpansive-type Mappings

The purpose of this paper is to obtain best proximity point theorems for multivalued nonexpansive-type and contractive-type mappings on complete metric spaces and on certain closed convex subsets of Banach spaces. We obtain a convergence result under some assumptions and we prove the existence of common best proximity points for a sequence of multivalued contractive-type mappings.

The demiclosedness principle for mean nonexpansive mappings

In this paper, we establish the demiclosedness principle for the class of mean nonexpansive mappings, introduced in 2007 by Goebel and Japón Pineda, defined on closed, convex subsets of Banach spaces satisfying Opial's condition. We also establish the demiclosedness principle for a subclass of the mean nonexpansive maps whose domains are closed, bounded, convex sets in uniformly convex spaces. These results extend known demiclosedness results for nonexpansive maps and lead to some fixed point results for mean nonexpansive maps which partially answer an open question posed by Goebel and Japón Pineda.

On super fixed point property and super weak compactness of convex subsets in Banach spaces

For a nonempty convex set C of a Banach space X, a self-mapping on C is said to a linear (respectively, affine) isometry if it is the restriction of a linear (respectively, affine) isometry defined on the whole space X. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a Banach space has super fixed point property for affine (or, equivalently, linear) isometries if and only if it is super weakly compact; and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a Banach space under equivalent renorming sense. With the application of Fabian–Montesinos–Zizler's renorming theorem, we finally show that every strongly super weakly compact generated Banach space can be renormed so that every weakly compact convex set has super fixed point property.

On the axiomatization of convex subsets of Banach spaces

We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown’s algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces.

Fixed point theorems and spproximating fixed points of generalized (C)-condition mappings

In this paper, we present fixed point theorems of generalized (C)condition mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Moreover, we prove weak and strong convergence theorems for Ishikawa iteration of generalized (C)condition mappings in uniformly convex Banach spaces.

DOMAIN PERTURBATION FOR PARABOLIC EQUATIONS

We study the effect of domain perturbation on the behaviour of parabolic equations. The first aspect considered in this thesis is the behaviour of solutions under changes of the domain. We show how solutions of linear and semilinear parabolic equations behave as a sequence of domains Ωn converges to an open set Ω in a certain sense. In particular, we are interested in singular domain perturbations so that a change of variables is not possible on these domains. For autonomous linear equations, it is known that convergence of solutions under domain perturbation is closely related to the corresponding elliptic equations via a standard semigroup theory. We show that there is also a relation between domain perturbation for non-autonomous linear parabolic equations and domain perturbation for elliptic equations. The key result for this is the equivalence of Mosco convergences between various closed and convex subsets of Banach spaces. An important consequence is that the same conditions for a sequence of domains imply convergence of solutions under domain perturbation for both parabolic and elliptic equations. By applying variational methods, we obtain the convergence of solutions of initial value problems under Dirichlet or Neumann boundary conditions. A similar technique can be applied to obtain the convergence of weak solutions of parabolic variational inequalities when the underlying convex set is perturbed. Using the linear theory, we then study domain perturbation for initial boundary value problems of semilinear type. We are also interested in the behaviour of bounded entire solutions of parabolic equations defined on the whole real line. We establish a convergence result for bounded entire solutions of linear parabolic equations under L2 and Lp-norms. For the Lp-theory, we also prove Holder regularity of bounded entire solutions with respect to time. In addition, the persistence of some classes of bounded entire solutions is given for semilinear equations using the Leray-Schauder degree theory. The second aspect is to study the dynamics of parabolic equations under domain perturbation. In this part, we consider parabolic equation as a dynamical system in an

Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

A new condition for mappings, called condition (C), which is more general than nonexpansiveness, was recently introduced by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095]. Following the idea of Kirk and Massa Theorem in [W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 364–375], we prove a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. This covers a result obtained by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095]. We also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Consequently, we extend the results in [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095] to many other Banach spaces.

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let \(X\subset W\) and \(Y\subset Z\) be closed subspaces. Let \(\mathcal{L}\) be a subspace of \(\mathcal{L}(W^*, Z^{**})\), the Banach space of bounded linear operators from W* to Z**, containing \(X\otimes Y\). We describe, for \(x^{*}\in X^{*}\) and \(y^*\in Y^*\), all norm-preserving extensions of \(x^{*}\otimes y^{*}\in (X\otimes Y)^\ast\) to the space \(\mathcal{L}\) in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented.

Variational conclusions of set-valued bifunctions on convex subsets of Banach spaces with applications

In this paper, variational conclusions of set-valued bifunctions on convex subsets of Banach spaces are investigated and several new results are obtained. The results are applied to the fixed point theory and variational inequalities. We obtain two fixed point theorems and existence theorems of solutions of variational inequalities.

A Note about Differentiability of Maps Defined on Convex Subsets of Banach Spaces That May Be Nowhere Dense

In connection with continuum mechanics there are physically meaningful choices of infinite-dimensional Banach spaces such that the domain of constitutive maps is nowhere dense in them, as4pointed out. Thus the usual differential calculus on open sets cannot be applied there. Here we give a differentiability notion for mapsfdefined on any convex subset of a Banach space that may be nowhere dense. When the domain offis open, this notion coincides with the usual one. We give the definitions and prove the theorems related to first and higher order derivatives off.

Nonlinear variational inequalities on convex subsets of Banach spaces

The solvability of a class of nonlinear variational inequality problems involving p-monotone and p-Lipschitz types of operators is presented.

Equilibria of set-valued maps on nonconvex domains

We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if K K is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space E E , and Φ : K ⟶ 2 E \Phi :K\longrightarrow 2^E is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition Φ ( x ) ∩ T K r ( x ) ≠ ∅ for all x ∈ K , \begin{equation*} \Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*} then there exists x 0 ∈ K x_0\in K such that 0 ∈ Φ ( x 0 ) . 0\in \Phi (x_0). Here, T K r ( x ) T_K^r(x) denotes a new concept of retraction tangent cone to K K at x x suited for compact neighborhood retracts. When K K is locally convex at x , T K r ( x ) x,T_K^r(x) coincides with the usual tangent cone of convex analysis. Special attention is given to neighborhood retracts having “lipschitzian behavior”, called L − L- retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.

Best approximation theorems for composites of upper semicontinuous maps

Let (E, τ) be a Hausdorff topological vector space and (X, ω) a weakly compact convex subset of E with the relative weak topology ω. Recently, there have appeared best approximation and fixed point theorems for convex-valued upper semicontinuous maps F: (X, ω) → 2(E, τ) whenever (E, τ) is locally convex. In this paper, these results are extended to a very broad class of multifunctions containing composites of acyclic maps in a topological vector space having sufficiently many linear functionals. Moreover, we also obtain best approximation theorems for classes of multifunctions defined on approximatively compact convex subsets of locally convex Hausdorff topological vector spaces or closed convex subsets of Banach spaces with the Oshman property.

Fixed points of multivalued nonexpansive maps

Fixed point theorems for multivalued contractive‐type and nonexpansive‐type maps on complete metric spaces and on certain closed bounded convex subsets of Banach spaces have been proved. They extend some known results due to Browder, Husain and Tarafdar, Karlovitz and Kirk.