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We study the Bishop–Phelps–Bollobas property for operators from l ∞ 4 to a Banach space. For this reason we introduce an appropriate geometric property, namely the AHSp- l ∞ 4 . We prove that spaces Y satisfying AHSp- l ∞ 4 are precisely those spaces Y such that ( l ∞ 4 , Y ) has the Bishop–Phelps–Bollobas property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, C 0 ( L ) and L 1 ( μ ) satisfy AHSp- l ∞ 4 .

We give an extension of the classical Bartle–Dunford–Schwartz theorem for weakly compact operators on a C ( K ) space, to weakly compact operators on a separable L 1 -predual space. Using this we show that for operators on these spaces, the set of weakly compact operators that attain their norm is dense in the space of weakly compact operators. For operators from the space of affine continuous functions on a metrizable Choquet simplex with values in an uniformly convex space, we show that the operator theoretic version of the Bishop–Phelps–Bollobas property is valid. This gives an extension of some recent work of Kim and Lee.

The polynomial cluster value problem replaces the role of the continuous linear functionals in the original cluster value problem for the continuous polynomials to describe the corresponding cluster sets and fibers. We prove several polynomial cluster value theorems for uniform algebras $H(B)$ between $A_u(B)$ and $H^{\infty}(B)$, where $B$ is the open unit ball of a complex Banach space $X$. We also obtain new results about the original cluster value problem, especially for $A_{\infty}(B)$. Examples of spaces $X$ considered here are spaces of continuous functions, $\ell_1$ and locally uniformly convex spaces., Comment: 14 pages

In this work we show that the family of closed convex sets with the almost fixed point property is not stable under renormings for non-reflexive Banach spaces. This, together with a result by Reich, shows that a Banach space is reflexive if and only if it has the same family of closed convex sets with the almost fixed point property for every equivalent norm.

We introduce weakly compactly locally uniformly rotund Banach spaces and present their properties.

In this paper, we study two properties ( P 1 ) and ( P 2 ) introduced in the literature for studying the existence and stability of relative Chebyshev centers. We reformulate these properties in a way that leads naturally to our results. We mainly relate these properties with some geometric properties of Banach spaces. In particular, reflexive spaces having the Kadec–Klee property are characterized in terms of property ( P 1 ) and uniform convexity is characterized in terms of property ( P 2 ) . Some continuity results for the center map are also presented.

In this article, fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings and equilibrium problems are investigated based on a monotone projection algorithm. Strong convergence theorems are established without the aid of compactness in the framework of reflexive Banach spaces.

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}nɛN be a family of monotone and Lipschitz continuos mappings of C into E*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for {An} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

We exhibit a new class of Banach spaces Y such that the pair ( c 0 , Y ) has the Bishop–Phelps–Bollobas property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

We characterize real Banach spaces Y such that the pair ( l ∞ n , Y ) has the Bishop-Phelps-Bollobas property for operators. To this purpose it is essential using an appropriate basis of the domain space R n . As a consequence of the mentioned characterization, we provide examples of spaces Y satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and L 1 ( μ ) (μ a positive measure) satisfy the previous property.

A Banach space X admits an equivalent strongly uniformly Gâteaux smooth norm if and only if it contains the dense range of a super weakly compact operator, which is equivalent to say that X is generated by a convex super weakly compact set. Moreover, if X is strongly generated by a convex super weakly compact set, then there is an equivalent norm on X such that its restriction to any reflexive subspace of X is both uniformly convex and uniformly Frechet smooth.

In this paper, we establish the demiclosedness principle for the class of mean nonexpansive mappings, introduced in 2007 by Goebel and Japon 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 Japon Pineda.

In the present paper we investigate close connections between the combinatorial geometry of convex bodies and Banach space theory. Inspired by the still unsettled covering problem of Hadwiger (asking for the least number of smaller homothets of a convex body K sufficient to cover K), we derive new results on covering functionals of convex bodies which are closely related to this famous problem. In addition, we show that for the subcase that K is centrally symmetric (and thus can be interpreted as the unit ball of a normed space), these investigations yield new results involving moduli of convexity, James and Schaffer constants and other notions from Banach space theory.

A normed space is said to have the weak fixed point property (WFPP) if every nonexpansive self map on a weakly compact convex set has a fixed point. Kirk proved that if a normed space has normal structure, then it has WFPP. It is known that if a normed space is uniformly convex in every direction (UCED), then it has normal structure. Also known is that every normed space that is uniformly convex in all but countably many directions has normal structure. We show that a normed space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a countable union of n-dimensional subspaces of X for some positive integer n. We also show that in such a space, the Chebyshev center C ( K ) of a weakly compact convex set K is a common invariant set for the collection of all isometries that map K into K and also that there is a common fixed point in C ( K ) for this collection of maps. This was previously known to be true only in the case of a normed space that is UCED. Another observation made in this paper is that a Banach space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a linear subspace with a countable Hamel basis.

In this paper we prove some norm inequalities in the classical James sequence space J , and use them to prove that orthogonal convexity is a stable property in J .

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E, A be a single valued monotone and Lipschitz continuous mapping of C into E* and T be a single valued relatively nonexpansive mapping of C into itself. In this paper, we consider the composition and the convex combination of T and the gradient projection method for A which Goldstein (1964) proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for A and fixed points of T by the hybrid method in mathematical programming (Haugazeau, 1968). And we get several results which improve the well-known results in a 2-uniformly convex and uniformly smooth Banach space and a Hilbert space.

In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centred at that point to convex sets, even if it is perturbed by adding C 1 , 1 smooth mappings with controlled Lipschitzian behaviour. This result, which is valid for mappings defined on a subclass of uniformly convex Banach spaces, can be regarded as a set-valued generalization of the Polyak convexity principle. The latter, indeed, can be derived as a special case of the former. Such an extension of that principle enables one to build large classes of nonconvex multifunctions preserving the convexity of small balls. Some applications of this phenomenon to the theory of set-valued optimization are then proposed and discussed.

Several Banach–Stone-like generalizations of Shirota's result for metrizable uniform spaces are proved. Namely, if complete uniform spaces X , Y have isomorphic lattices U ( X ) , U ( Y ) of their real-valued uniformly continuous functions, and both X , Y are either some products of spaces having monotone bases (metrizable or uniformly zero-dimensional), or are locally fine and of non-measurable cardinality, then X and Y are uniformly homeomorphic.

A Urysohn-type theorem is introduced for a subalgebra of the algebra C b ( Ω ) of all bounded complex-valued continuous functions on a Hausdorff topological space Ω. With use of this theorem, it is shown that a type of the Bishop-Phelps-Bollobas theorem holds for certain classes of holomorphic functions on the unit ball of a complex Banach space X if X is either a locally uniformly convex space or a locally c-uniformly convex, order-continuous sequence space.

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gâteaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly Gâteaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.

Let C be a closed, bounded and convex subset of a uniformly convex and uniformly smooth Banach space. Let { T t } t ≥ 0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations x k + 1 = c k T t k + 1 ( x k + 1 ) + ( 1 − c k ) x k . We prove that, under certain conditions, the sequence { x k } converges weakly to a common fixed point of the semigroup { T t } t ≥ 0 . There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property. However, many important spaces like L p for 1 ≤ p ≠ 2 do not possess the Opial property. In this paper, we do not assume the Opial property. We do assume instead that X is uniformly convex and uniformly smooth. L p for p > 1 are prime examples of such spaces.

We show that the pair ( C ( K ) , X ) has the Bishop–Phelps–Bollobas property for operators if K is a compact Hausdorff space and X is a uniformly convex space.

We show that nests of finite subsets of a separable real Banach space are reflexive, and obtain bounds for the reflexivity index.

This paper provides a quantitative version of the classical image recovery problem to find an ϵ -approximate solution of the problem. The rate of asymptotic regularity of the iteration schemas, connected with the problem of image recovery, coincides with the existing optimal and quadratic bound for Krasnoselskii–Mann iterations. We then provide explicit effective and uniform bounds on the approximate fixed points of the mappings under consideration to be an approximate solution of the image recovery problem up to a uniform change from ϵ to δ ( ϵ ) . When combined, these results provide algorithms with explicit rates of convergence for the recovery of an ϵ -perturbation of the original image in different settings.

In this paper, we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space. Our results generalize simultaneously the approximation results of Rhoades (1994) [18], Suantai (2005) [20] and Xu and Noor (2002) [26] on a nonlinear domain. Our results are refinements and generalizations of the corresponding ones in uniformly convex Banach spaces and CAT ( 0 ) spaces.

In this paper, we prove that the existence of an e -isometry from a separable Banach space X into Y (the James space or a reflexive space) implies the existence of a linear isometry from X into Y . Then we present a set valued mapping version lemma on non-surjective e -isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by e -isometries.

In this article, we extend the definition of uniformly starlike functions and uniformly convex functions on the unit disk to the unit ball in n , give the discriminant criterions for them, and get some inequalities for them.

This article presents a survey of some aspects of the theory of non-commutative Banach function spaces which draw their inspiration from the contributions of A.C. Zaanen to the theory of normed Kothe spaces.

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ϕ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.

We show that the space of bounded linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop–Phelps–Bollobas property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an L 1 -space.

We study sharp square function inequalities for martingales taking values in a 2-convex Banach space B . We show that an appropriate weak-type bound holds true if and only if B is isometric to a Hilbert space.

According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is $C^{1,1}$ around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present paper considers the extension of such result to mappings defined on a certain subclass of uniformly convex Banach spaces. This enables one to extend to such setting a variational principle for constrained optimization problems, already observed in finite dimension, that establishes a convex behaviour for proper localizations of them. Further variational consequences are explored., Comment: 13 pages

The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound 1 / N ≤ λ n ( 1 − λ n ) , the error ϵ > 0 and a modulus η of uniform convexity.

Let X be a uniformly convex and uniformly smooth Banach space. Let T : C → C be an asymptotic pointwise nonexpansive mapping, where C is a bounded, closed and convex subset of X . In this paper we investigate conditions sufficient for the weak convergence of the generalized Mann and Ishikawa processes to a fixed point of T .

We study different facets of property ( β ) of Rolewicz. We remark that the notions of compact uniform convexity and property ( β ) are isometrically equivalent and present new examples of spaces with that property. An observation is made that the property ( β ) can be formulated in terms of graphs and an estimate of the ( β ) -modulus is also given.

Let μ be a probability measure on a separable Banach space X . A subset U ⊂ X is μ -continuous if μ ( ∂ U ) = 0 . In the paper the μ -continuity and uniform μ -continuity of convex bodies in X , especially of balls and half-spaces, is considered. The μ -continuity is interesting for study of the Glivenko–Cantelli theorem in Banach spaces. Answer to a question of F. Topsoe is given.

In this paper, we introduce and analyze the new generalized mixed equilibrium problems (NGMEP) and the bilevel new generalized mixed equilibrium problems (BNGMEP) in Banach spaces. First, by using a minimax inequality, some new existence theorems of the solution and the behavior of solution set for the NGMEP and the BNGMEP are obtained in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the NGMEP and the BNGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are also proved in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. These results are new and generalize some recent results in this field.

Let B be a Banach space, Φ 1 , Φ 2 be two generalized convex Φ -functions and Ψ 1 ,Ψ 2 the Young complementary functions of Φ 1 , Φ 1 respectively with ∫ t 0 t ψ 2 ( s ) s d s ≤ c 0 ψ 1 ( c 0 t ) ( t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ 1 1 and ψ 2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p -uniformly smooth space (or q -uniformly convex space, respectively), then there exists a constant c > 0 such that for any B -valued martingale f =( f n ) n ≥ 0, ‖ f * ‖ Φ 1 ≤ c ‖ S ( p ) ( f ) ‖ Φ 2 ( or ‖ S ( q ) ( f ) ‖ Φ 1 ≤ c ‖ f * ‖ Φ 2 , respectively ) , where f * and S (p) (f) are the maximal function and the p -variation function of f respectively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = ( v n ) n ≥ 0 ( v * ≤ 1 ) , then ‖ ( T v f ) * ‖ Φ 1 ≤ c ‖ f * ‖ Φ 2 .

We exhibit examples of Fréchet Montel spaces E which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite-dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2. They are of independent interest and show for example that the canonical inclusion between James spaces Jp¿Jq, 1 pJq has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces. © 2011 Elsevier Inc., This research was partially supported by MEC and FEDER Project MTM2010-15200 and by GV Project Prometeo/2008/101.

In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.

Criteria for nearly strict convexity of Musielak–Orlicz–Bochner function spaces equipped with the Luxemburg norm are given. We also prove that, in Musielak–Orlicz–Bochner function spaces generated by strictly convex Banach space, nearly strict convexity and strict convexity are equivalent.

In this paper, we prove that if a Banach space X contains some uniformly convex subspace in certain geometric position, then the C ( K , X ) spaces of all X-valued continuous functions defined on the compact metric spaces K have exactly the same isomorphism classes that the C ( K ) spaces. This provides a vector-valued extension of classical results of Bessaga and Pelczynski (1960) [2] and Milutin (1966) [13] on the isomorphic classification of the separable C ( K ) spaces. As a consequence, we show that if 1 p q ∞ then for every infinite countable compact metric spaces K 1 , K 2 , K 3 and K 4 are equivalent: (a) C ( K 1 , l p ) ⊕ C ( K 2 , l q ) is isomorphic to C ( K 3 , l p ) ⊕ C ( K 4 , l q ) . (b) C ( K 1 ) is isomorphic to C ( K 3 ) and C ( K 2 ) is isomorphic to C ( K 4 ) .

We point out that uniform noncreasiness is a super-property in Banach spaces. Moreover, we prove that Orlicz–Bochner function spaces are noncreasy if and only if they are rotund or smooth. Finally, we obtain that Orlicz–Bochner function spaces are uniformly noncreasy if and only if they are uniformly rotund or uniformly smooth.

In this paper, we introduce a new implicit iterative algorithm for finding a common element of a countable family of continuous pseudocontractions in a uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results extend the recent results announced by many others.

Estimates for the James constant for various norms in real interpolation spaces for finite families of Banach spaces are given. As a corollary it is shown that if a family contains at least one space which is uniformly nonsquare, then the interpolation space is uniformly nonsquare.

We introduce the class of α -nonexpansive mappings in Banach spaces. This class contains the class of nonexpansive mappings and is related to the class of firmly nonexpansive mappings in Banach spaces. In addition, we obtain a fixed point theorem for α -nonexpansive mappings in uniformly convex Banach spaces.

In this paper, we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi- ϕ -asymptotically nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec–Klee property. Our results improve and extend the corresponding results announced by Qin et al. (2010) and many authors.

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k -strongly convex space and k -very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.

In this paper, we prove weak and strong convergence theorems for Ishikawa iteration of Suzuki-generalized nonexpansive mappings in uniformly convex Banach spaces. Furthermore, we extend the results to CAT(0) spaces. Our work extends the results of Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mapping, J. Math. Anal. Appl. 340 (2008) 1088–1095] and Takahashi and Kim [W. Takahashi, G.E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Jpn. 48 (1998) 1–9].

We introduce a new iterative algorithm for finding a common fixed point of a countable family of strict pseudocontractions in q-uniformly smooth and uniformly convex Banach spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common fixed point of an infinite family of strict pseudocontractions. Our results mainly improve and extend the results announced by Yao et al. [Y. Yao, Y.-C. Liou, G. Marino, Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl. 2009 (2009) 7 pages. doi:10.1155/2009/279058. Art. ID 279058].