Existence of positive solutions to n-point nonhomogeneous boundary value problem

In this paper, we are concerned with the existence of positive solutions to a n-point nonhomogeneous boundary value problem. By using the Krasnoselskii's fixed point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of positive solution is established for the n-point nonhomogeneous boundary value problem.     

On strong solutions of a Robin problem modelling heat conduction in materials with corroded boundary

Motivated by a practical problem on a corrosion process, we shall study a third kind of BVP for a large class of elliptic equations in vector-valued L_p spaces. Particularly we will determine optimal spaces for boundary data and get maximal regularity for inhomogeneous equations. Then based on these results we shall treat some nonlinear problems. Our approach will be based on the semigroup theory, the interpolation theory of Banach spaces, fractional powers of positive operators, operator-valued Fourier multiplier theorems and the Banach fixed point theorem.     

Real and complex interpolation methods for finite and infinite families of Banach spaces

A comparative study is made of the various interpolation spaces generated with respect to n-tuples or infinite families of compatible Banach spaces by real and complex interpolation methods due to Sparr, Favini-Lions, Coifman-Cwikel-Rochberg-Sagher-Weiss, and Fernandez. Certain inclusions are established between these spaces and examples are given showing that in general they do not coincide. It is also shown that, in contrast to the case of couples of spaces, the spaces generated by the above methods may depend on the structure of the containing space in which the Banach spaces of the n-tuple (nϵ 3) or infinite family are embedded. Finally a construction is given which enables the spaces of Sparr and Favini-Lions, hitherto defined only with respect to n-tuples, to also be defined with respect to infinite families of Banach spaces.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Minimal ideals of n-homogeneous polynomials on Banach spaces

The minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.     

On the stability of an n-dimensional cubic functional equation

Let n?2 be an integer number. In this paper, we investigate the generalized Hyers-Ulam-Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C^∗-algebra, and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces:

     

A two-dimensional inequality and uniformly continuous retractions

We prove a new inequality valid in any two-dimensional normed space. As an application, it is shown that the identity mapping on the unit ball of an infinite-dimensional uniformly convex Banach space is the mean of n uniformly continuous retractions from the unit ball onto the unit sphere, for every n?3. This last result allows us to study the extremal structure of uniformly continuous function spaces valued in an infinite-dimensional uniformly convex Banach space.     

Some Remarks on Operator-Norm Convergence of the Trotter Product Formula on Banach Spaces

We extend the Trotter-Kato-Chernoff theory of strong approximation of C₀ semigroups on Banach spaces to operator-norm approximation of analytic semigroups with error estimate. As application we obtain a criterion for the operator-norm convergence of the Trotter product formula on Banach spaces with error estimate n⁻¹ log n, provided one of the generators has a bounded H^∞ functional calculus. For both results, we present versions for approximation in operator-ideal-norms such as the trace norm or the Hilbert-Schmidt norm. Finally, we give some remarks on the operator-norm convergence of the Trotter product formula for semigroups acting on a scale of L_p-spaces.     

Multiple positive solutions of n-point boundary value problems on the half-line in Banach spaces

In this paper, the fixed point index theory is used to investigate the existence of at least two positive solutions of n-point boundary value problem on the half-line in a Banach space. As an application, we give an example to demonstrate our results.     

On a generalized Mazur-Ulam question: Extension of isometries between unit spheres of Banach spaces

We call a Banach space X admitting the Mazur-Ulam property (MUP) provided that for any Banach space Y, if f is an onto isometry between the two unit spheres of X and Y, then it is the restriction of a linear isometry between the two spaces. A generalized Mazur-Ulam question is whether every Banach space admits the MUP. In this paper, we show first that the question has an affirmative answer for a general class of Banach spaces, namely, somewhere-flat spaces. As their immediate consequences, we obtain on the one hand that the question has an approximately positive answer: Given ε>0, every Banach space X admits a (1+ε)-equivalent norm such that X has the MUP; on the other hand, polyhedral spaces, CL-spaces admitting a smooth point (in particular, separable CL-spaces) have the MUP.     

Polynomial numerical indices of Banach spaces with absolute norm

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n⁽²⁾(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n⁽²⁾(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n⁽²⁾(X)=1 is c₀(Λ).     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Existence of solutions of <Emphasis Type="Italic">n</Emphasis>-point boundary value problems on the half-line in Banach spaces

In this paper, the Mönch fixed point theorem is used to investigate the existence of solutions of a n-point boundary value problem on the half-line in a Banach space. As an application, we give an example in an infinite dimensional space to demonstrate our results.     

Discrete time analytic semigroups and the geometry of Banach spaces

We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂-discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁). Finally, we give some elements to compare these regularities.     

Positive solutions for an n-point nonhomogeneous boundary value problem

This paper is concerned with the solvability of an n-point nonhomogeneous boundary value problem. By using the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution are established for the n-point nonhomogeneous boundary value problem.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

Iterative approximation of a zero of accretive operator in Banach space

This paper introduces an iteration scheme for viscosity approximation of a zero of accretive operator in a reflexive Banach space with weakly continuous duality mapping. A new iterative sequence is introduced and strong convergence of the algorithm {x_n} is proved. The results improve and extend the results of Su [Xiaolong Qin, Yongfu Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 320 (2007) 415-424] and some others.     

Stability,open embeddings,and classification of bounded weak-1 manifolds

It is known that the following spaces are homeomorphic: (1) a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology, (2) the conjugate of a separable, infinite-dimensional Banach space with its bounded weak-¹ topology, and (3) Q^∞ = dir lim Qⁿ, where Q is the illbert cube. Let F denote one of the above three spaces, and let M and N denote paraimpact, connected F manifolds. Here we prove that M × F is homeomorphic to M. Combined with previous work of the author this result implies that M, embeds as an open subset of F and that if M and N have the same homotopy type then they are homeomorphic.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .