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Second order Sobolev spaces are important in applications to partial differential equations and geometric analysis, in particular to equations such as the bi-Laplacian. The main purpose of this paper is to establish some new characterizations of the second order Sobolev spaces W 2 , p ( R N ) in Euclidean spaces. We will present here several types of characterizations: by second order differences, by the Taylor remainder of first order and by the differences of the first order gradient. Such characterizations are inspired by the works of Bourgain et al. (2001) and Nguyen (2006, 2008) on characterizations of first order Sobolev spaces in the Euclidean space.

International audience; We give short simple proofs of Uspenskii's results characterizing Besov spaces as trace spaces of weighted Sobolev spaces. We generalize Uspenskii's results and prove the optimality of these generalizations. We next show how classical results on the functional calculus in the Besov spaces can be obtained as straightforward consequences of the theory of weighted Sobolev spaces.

We study the generalized fractional integral transforms associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between L p -spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces.

In this paper we are concerned with Trudinger’s inequality and continuity for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces on metric measure spaces.

We prove complex interpolation theorems for analytic families of multilinear operators defined on quasi-Banach spaces, with explicit constants on the intermediate spaces. We obtain analogous results for analytic families of operators defined on spaces generated by the Calderon method applied to couples of quasi-Banach lattices with nontrivial lattice convexity. As an application we derive a multilinear version of Stein’s classical interpolation theorem for analytic families of operators taking values in Lebesgue, Lorentz, and Hardy spaces. We use this theorem to prove that the bilinear Bochner–Riesz operator is bounded from L p ( R n ) × L p ( R n ) into L p / 2 ( R n ) for 1 p 2 .

In this article we define variable exponent Bergman spaces and show that polynomials are dense in the spaces. We also show that the Bergman projection and the Berezin transform are bounded in these spaces.

In this article we study variable exponent Sobolev spaces on Riemannian manifolds. The spaces are examined in the case of compact manifolds. Continuous and compact embeddings are discussed. The paper contains an example of the application of the theory to elliptic equations on compact manifolds.

In this paper, we develop the boundedness of Riesz transform in Morrey spaces, which generalize the corresponding result in the usual L p spaces. As an application of this boundedness, we establish the well-posedness and blow-up criterion for the surface quasi-geostrophic equations in the framework of Besov–Morrey spaces.

We show that for a wide class of non-locally convex quasi-Banach spaces X that includes the spaces l p for 0 p 1 , there exists a continuous function f : [ 0 , 1 ] → X failing to have a primitive, thus solving a problem raised by M.M. Popov in 1994. We also construct the first known examples of functions in C ( 1 ) ( [ a , b ] , X ) that fail to be Lipschitz.

In this paper, we consider multipliers from Sobolev spaces to Lebesgue spaces. We establish some wavelet characterization of multiplier spaces without using capacity. Further, we give a sharp logarithmic Morrey space condition for multipliers which lessens Fefferman’s Morrey space condition to the logarithm level and generalizes Lemarie’s counter-example to non-integer cases and expresses his results in a more precise way.

In this paper we present a simple and unified approach to the fixed point results on cone symmetric spaces and metric type spaces based on symmetric spaces fixed point theory. We also give a new characterization of semi-metric spaces with open balls.

We study the stability properties of the class of weak*-extensible spaces introduced by Wang, Zhao, and Qiang showing, among other things, that weak*-extensibility is equivalent to having a weak*-sequentially continuous dual ball (in short, w*SC) for duals of separable spaces or twisted sums of w*SC spaces. This shows that weak*-extensibility is not a 3 -space property, solving a question posed by Wang, Zhao, and Qiang. We also introduce a restricted form of weak*-extensibility, called separable weak*-extensibility, and show that separably weak*-extensible Banach spaces have the Gelfand–Phillips property, although they are not necessarily w*SC spaces.

We study extrapolation spaces generated by the Banach scales of L p and L p , ∞ -spaces defined on [ 0 , 1 ] . We give new extrapolatory descriptions for the Marcinkiewicz and the Orlicz–Lorentz spaces. Using interpolation of sublinear positive operators, we also prove interpolation embeddings for extrapolation spaces.

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.

The paper encompasses a complete study of the weak-star extensibility of subspaces of Banach spaces introduced by B.S. Mordukhovich and B. Wang when studying the restrictive metric regularity in variational analysis. Many useful properties are presented; in particular, a new point of view and formulation of the property in the framework of Banach space theory is established. We also propose the concept of weak-star extensible Banach spaces that has interesting applications, and is fairly broad in that it contains many important classes of Banach spaces. Some further applications of this extensibility to the theory of linear operators between Banach spaces are also explored.

In this paper we derive the existence and comparison results for second order initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and the theory of locally HL integrable vector-valued functions.

In this paper we obtain gradient estimates in Sobolev spaces and Orlicz spaces for weak solutions of parabolic A -harmonic equations under the assumptions that A satisfies some proper conditions and the given function satisfies some moderate growth condition.

In this paper we study the regularity of geodesic Ptolemy spaces and apply our findings to metric fixed point theory. It is an open question whether such spaces with a continuous midpoint map are CAT(0) spaces. We prove that if a certain uniform continuity is imposed on such a midpoint map then these spaces, if complete, are reflexive (that is, the intersection of decreasing families of bounded closed and convex subsets is nonempty) and that bounded sequences have unique asymptotic centers. These properties will then be applied to yield a series of fixed point results specific to CAT(0) spaces.

We investigate Hilbertian Hardy type spaces of complex analytic functions of infinite many variables, associated with compact unitary groups and the corresponding invariant Haar’s measures. For such analytic functions we establish a Cauchy type integral formula and describe natural domains. Also we show some relations between constructed spaces of analytic functions and the symmetric Fock space.

The unit sphere in an infinite dimensional Banach space is a lipschitzian retract of the unit ball. The aim of this paper is to present a new upper bound for the optimal retraction constant in some classical Banach spaces. In particular, an improved estimate from above is obtained for the space C [ 0 , 1 ] .

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in p -uniformly convex and q -uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.

We study solid hulls of abstract Hardy spaces of analytic functions on the unit disk. Interpolation of sublinear operators is used to description of solid hulls of quasi-Banach spaces generated by interpolation methods associated to classical Hardy spaces H p for 0 p 1 . We specialize our results to solid hulls of Hardy–Orlicz and Hardy–Lorentz spaces and we are able to characterize multipliers from Hardy–Lorentz spaces to any solid sequence space. We give, for example, a complete description of multipliers with the domain in Hardy–Lorentz–Zygmund spaces.

In this paper we derive existence and comparison results for initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and theory of HL integrable vector-valued functions. Concrete examples are presented and solved.

The aim of the paper is to collect some results concerning l -stability of mappings with values in possibly infinite-dimensional Banach spaces. We show that any l -stable function from finite-dimensional space to an arbitrary Banach space is Lipschitz near the reference point. Further, we show that any l -stable function from finite-dimensional space to a Banach space having Radon–Nikodým property is strictly differentiable at the reference point. As an application we present the second-order sufficient condition for the unconstrained optimization problem previously obtained for finite-dimensional range space.

Several fixed point theorems for symmetric spaces are proved. Then these theorems are used in symmetric PPM-space to prove and generalize Theorem 6 of [T.L. Hicks, B.E. Rhoades, Fixed point theory in symmetric spaces with applications to probabilistic spaces, Nonlinear Anal. 36 (1999) 331–344].

In the present paper we consider weighted modulation spaces. Weights of exponential growth will be admissible. With the molecular decomposition we shall obtain an unconditional basis. As applications, we obtain the greedy basis of coorbit spaces.

We give, in this work, a new basis definition for Banach spaces and investigate some structural properties of certain vector-valued function spaces by using it. By novelty of the new definition, we prove that l ∞ has a basis in this sense, and so we deduce as a result that it has approximation property. In fact, we obtain a more general result that the linear subspace P ( B , X ) of l ∞ ( B , X ) of all those functions with a precompact range has an X -Schauder basis. Hence P ( A , X ) has approximation property if and only if the Banach space X has. Note that P ( B , X ) = l ∞ ( B , X ) for some finite-dimensional X . Further, we give a representation theorem to operators on certain vector-valued function spaces.

We analyze a nonlinear equation in Banach spaces, with the nonlinearity composed of multiple terms of different degrees. We prove a theorem regarding the existence of solutions for such equations. Moreover, we show how this result may be applied to obtain the well-posedness of various parabolic initial value problems.

This paper is devoted to multiplicative inequalities in some generalized Sobolev spaces associated with Lie algebras. These Lie algebras are generated by the differential operator of variable coefficients or by pseudo-differential operators having non-regular symbols. Under geometrical assumptions we show that the norms of two suitable classes of generalized Sobolev spaces are equivalent. This leads to the proof that the composition operator u → | u | p acts on such spaces.

In this paper some properties of the gap function for multiple-objective optimization problems in Banach spaces are established.

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.

A CAT(0) space is a geodesic space for which each geodesic triangle is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently several years ago by Lim and Kuczumow is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular many Banach space results involving weak convergence have precise analogues in this setting. At the same time, many questions remain open.

In this paper, a new maximal element theorem is proved in H -space and, as applications, the Browder–Hartman–Stampacchia variational inequalities and the complementarity problem are discussed in Banach spaces and locally convex space, respectively.

In this paper, the existence of unbounded nonnegative solutions of a boundary value problem for n th-order impulsive integro-differential equations on an infinite interval in Banach spaces is obtained by means of the Monch fixed point theory in a Banach space.

We give new sharper valuations for the optimal retraction constant in some Banach spaces.

In this paper, we introduce a new class of Banach spaces, locally k nearly uniformly smooth spaces, which is the dual notion to locally k nearly uniformly convex spaces introduced by Denka Kutzarova. The relationships between locally k nearly uniformly smooth spaces, locally k uniformly smooth Banach spaces, and locally fully k smooth Banach spaces are discussed. We also introduce the two new smooth properties, ( S k - I ) and ( S k - II ), which are the extensions of smooth properties (S-I) and (S-II) that were introduced by Suyalatu, and discuss the relations between ( S k - I ), ( S k - II ) and other smooth properties. At last, the locally fully k -smoothness, compact locally fully k -smoothness, smooth properties ( S k - I ) and ( S k - II ) in product space l p ( X i ) are discussed.

We re-prove various uniqueness theorems for the Navier–Stokes equations, stating the assumptions in terms of multipliers between Sobolev spaces instead of Lebesgue or Lorentz spaces.

One purpose of this paper is to extend the conjugate gradient method to minimize functions on Banach spaces that are norm reflexive, strictly convex and non-Hilbert. The algorithms in this paper are based upon the notion of the metric gradient defined by Golomb and Tapia (Numer. Math. 20 (1972) 115–124). Generalizations are made for the algorithms of Daniel (The Approximate Minimization of Functionals, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971; The conjugate gradient method for linear and nonlinear operator equations, Ph.D. Thesis, Stanford University, Palo Alto, CA, 1965), Polak (Computational Methods in Optimization—a Unified Approach, Academic Press, New York and London, 1971), Polak–Ribiere (Rev. Francaise Informat. Recherche Operationnelle 3 (16) (1969) 35–43), and Fletcher–Reeves (Comput. J. 7 (1964) 149–154). A local convergence theorem is given for a class of descent conjugate gradient methods for functionals defined on Banach spaces. Applications include problems in differential equations and the calculus of variations where the Banach spaces are Sobolev spaces and the second Frechet differential has finite signature and nullity as defined by Hestenes (Pacific J. Math. 1 (1951) 525–581).

We study relationships between two normal compactness properties of sets in Banach spaces that play an essential role in many aspects of variational analysis and its applications, particularly in calculus rules for generalized differentiation, necessary optimality and suboptimality conditions for optimization problems, etc. Both properties automatically hold in finite-dimensional spaces and reveal principal features of the infinite-dimensional variational theory. Similar formulations of these properties involve the weak ∗ convergence of sequences and nets, respectively, containing generalized normal cones in duals to Banach spaces. We prove that these properties agree for a large class of Banach spaces that include weakly compactly generated spaces. We also show that they are always different in Banach spaces whose unit dual ball is not weak ∗ sequentially compact. Moreover, the sequential and topological normal compactness properties may not coincide even in non-separable Asplund spaces that admit an equivalent C∞-smooth norm.

We consider a family of spaces wider than UNC spaces introduced by Prus, and we give some fixed point results in the setting of these spaces.