On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c ₀. In particular, a Schauder frame of a Banach space with no copy of c ₀ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c ₀. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.     

Existence of solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions

In this paper, we prove the existence of mild solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces. The results are obtained by using the theory of resolvent operators and Leray–Schauder nonlinear alternative.     

Existence Results on Semi-Infinite Intervals for Nonlocal Evolutions Equations and Inclusions via Semigroup Theory

In this paper, we prove existence and controllability results on semi-infinite intervals for first- and second-order evolutions equations and inclusions in Banach spaces with nonlocal conditions. Our theory makes use of semigroups, cosine families, nonlinear alternative of Leray–Schauder, and a diagonalization argument.     

Banach空间半线性积微分方程的非局部Cauchy问题

运用预解算子理论和Schauder不动点定理,在Banach空间证明了一类非局部半线性积微分方程积分解的存在性.     

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.     

Spaces of diagonal operators

Properties of space of diagonal operators on semi-normalized Schauder bases are studied. A necessary and sufficient condition is given for the containment of in the space of compact diagonal operators on a semi-normalized Schauder basis. Using this condition it is shown that there exist semi-normalized Schauder bases of reflexive Banach spaces such that does not embed in the space of compact diagonal operators on these Schauder bases. Received January 21, 1999; in final form April 10, 2000 / Published online February 5, 2001     

Schauder Bases under Uniform Renormings

Let X be a separable superreflexive Banach space with a Schauder basis. We prove the existence of an equivalent uniformly smooth (resp. uniformly rotund) renorming under which the given basis is monotone. First author supported by the grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain) and AP2003-4453 of MECD (Spain), Second author supported by AV0Z10190503 and A100190502.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

On Schauder equivalence relations

We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that and are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality .     

A quasistatic viscoplastic contact problem with normal compliance and friction

We consider a quasistatic contact problem between a viscoplasticbody and an obstacle, the so-called foundation. The contactis modelled with normal compliance and the associated versionof Coulomb's law of dry friction. We derive a variational formulationof the problem and, under a smallness assumption on the normalcompliance functions, we establish the existence of a weak solutionto the model. The proof is carried out in several steps. Itis based on a time-discretization method, arguments of monotonicityand compactness, Banach fixed point theorem and Schauder fixedpoint theorem.     

Constructive utility functions on Banach spaces

In this paper we prove the existence of continuous order-preserving functions on subsets of ordered Banach spaces using a constructive approach.     

Existence theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007     

Nonlinear functional differential equations of monotone-type in Hilbert spaces

In this paper, we study a class of semilinear functional evolution equations with finite delay in which the nonlinearity satisfies the weak condition of demicontinuity with respect to functional variable and also a semimonotone condition. We will prove the existence, uniqueness and measurability of the mild solutions based on an extension of the analogous results for non-delay initial value problems on Banach spaces and a version of recently developed random Schauder’s fixed point theorem. This will be done first for the generalized solutions and then we follow an approximating procedure to obtain the same results for the mild solutions.     

New Characterizations of Operator-Valued Bases on Hilbert Spaces

In this paper we study operator valued bases on Hilbert spaces and similar to Schauder bases theory we introduce characterizations of this generalized bases in Hilbert spaces.We redefine the dual basis associated with a generalized basis and prove that the operators of a dual g-basis are continuous.Finally we consider the stability of g-bases under small perturbations.We generalize two results of KreinMilman-Rutman and Paley-Wiener [7] to the situation of g-basis.     

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodym property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Split equilibrium problems for related games and applications to economic theory

ABSTRACT

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.     

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.     

ON SOLUTIONS TO SEMILINEAR INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper, using the theory of resolvent operators, Banach,s contraction prin-ciple and Schauder,s fixed point theorem, we study the existence of integral solutions to semilinear integrodifferential equations under nonlocal conditions in Banach space. An example is provided to illustrate the results obtained.     

Isomorphic Schauder decompositions in certain Banach spaces

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ?_ψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.