Narrow Results

Let E be a Banach space and let B(R)⊂E denote the open ball with centre at 0 and radius R. The following problem is studied: given 0<r<R, ∊>0 and a function f holomorphic on B(R), does there always exist an entire function g on E such that |f-g|<∊ on B(r)?

L. Lempert has proved that the answer is positive for Banach spaces having an unconditional basis with unconditional constant 1. In this paper a somewhat shorter proof of Lemperts result is given.

In general it is not possible to approximate f by polynomials since f does not need to be bounded on B(r).

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator $A$ defined in a Banach space $E$, in which by choosing $E$ and $A$ we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized $\varphi$-pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature.

In this paper we extend the utilization of the Banach spaces-based formulations, usually employed for solving diverse nonlinear problems in continuum mechanics via primal and mixed finite element methods, to the virtual element method (VEM) framework and its respective applications. More precisely, we propose and analyze an ${\mathrm{\text{L}}}^p$ spaces-based mixed virtual element method for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions. To this end, a dual-mixed approach determined by the introduction of a nonlinear tensor linking the usual pseudostress for the Stokes equations with the convective term is employed. As a consequence, this new tensor, say $\sigma$, and the velocity $u$ of the fluid constitute the unknowns of the formulation, whereas the pressure is computed via a post-processing formula. The simplicity of the resulting VEM scheme is reflected by the absence of augmented terms, on the contrary to previous works on this and related models, and by the incorporation in it of only the projector onto the piecewise polynomial tensors and the usual stabilizer depending on the degrees of freedom of the virtual element subspace approximating $\sigma$. In turn, the non-virtual but explicit subspace given by the piecewise polynomial vectors of degree $\le k$ is employed to approximate $u$. The corresponding solvability analysis is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem, both in subspaces of Banach spaces. A Strang-type lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of $\sigma$, the post-processed pressure, and a second post-processed approximation of $\sigma$. Finally, several numerical results illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory are reported.

Stochastic approximation (SA) was introduced in the early 1950s and has been an active area of research for several decades. While the initial focus was on statistical questions, it was seen to have applications to signal processing, convex optimization. In later years, SA has found application in reinforced learning (RL) and led to revival of interest.

While bulk of the literature is on SA for the case when the observations are from a finite dimensional Euclidian space, there has been interest in extending the same to infinite dimension. Extension to Hilbert spaces is relatively easier to do, but this is not so when we come to a Banach space — since in the case of a Banach space, even law of large numbers is not true in general. We consider some cases where approximation works in a Banach space. Our framework includes case when the Banach space $𝔹$ is $C([0,1],{ℝ}^d)$, as well as ${Ł}^1([0,1],{ℝ}^d)$, the two cases which do not even have the Radon–Nikodym property.

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of $\mathrm{\Delta }$-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc.60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and ${\ell }^p$-spaces, but not in $L^p({ℝ}^N)$, $p\ne 2$. $\mathrm{\Delta }$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of $\mathrm{\Delta }$-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including, in particular, hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on the loss function, the geometry of the feature space, and the modulus of total convexity of the regularizer. In addition, the proposed analysis gives new insight into basic tools such as reproducing Banach spaces, feature maps, and universality. Even when specialized to Hilbert spaces, this framework yields new results that extend the state of the art.

Application of comparison theorem is used to examine the validitiy of the "lumped parameter assumption" in describing the behavior of solutions of the continuous cable equation U_t = DU_xx+f(U) with the discrete cable equation dV_n/dt = d*(V_n+1 - 2V_n + V_n-1) + f(V_n), where f is a nonlinear functional describing the internal diffusion of electrical potential in single neurons. While the discrete cable equation looks like a finite difference approximation of the continuous cable equation, solutions of the two reveal significantly different behavior which imply that the compartmental models (spiking neurons) are poor quantifiers of neurons, contrary to what is commonly accepted in computational neuroscience.

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F(x)=0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity condition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails.

This paper presents a type of variational principles for real valued w* lower semicontinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.

Let F be a non archimedean local field and let G be an algebraic connected almost F-simple group over F, whose Lie algebra contains sl₃(F). We prove that G(F) has strong Banach property (T) in a stronger sense than in the article "Un renforcement de la propriété (T)", published in Duke Math. J. As a consequence, families of expanders built from a lattice in G(F) do not embed uniformly in Banach spaces of type > 1. Also any affine isometric action of G(F) on a Banach space of type > 1 has a fixed point.

Based on the so-called degree of nondensifiability, DND, we provide a generalization of the recently introduced $(b,𝜃)$-enriched contractions, as well as a fixed point existence result for this new class of mappings. From our main result, and under suitable conditions, we derive a result to state the existence of fixed points for the sum of two mappings, one of them being compact. Also, our main result is more general than the other fixed point theorem based on the DND.

In this paper, we introduce a hybrid iterative method for approximating the common solution of a systems of generalized mixed equilibrium problem and the class of $(\alpha ,\beta ,\delta ,\gamma )$-generalized Bregman nonspreading mappings in reflexive Banach space. We prove the asymptotic fixed point property of the class of $(\alpha ,\beta ,\delta ,\gamma )$-generalized Bregman nonspreading mappings in reflexive Banach spaces and provide a strong convergence theorem for solving the problem under some mild conditions. Furthermore, some numerical results were presented to show the important and computational performance of the proposed method. This result extends and generalized many interesting results in the contemporary literature.

In this paper, based on the so-called degree of nondensifiability (DND), we introduce the concept of DND-convex-power condensing mapping which generalizes that of the DND-condensing one. A fixed point result for this new class of mappings is proved, and with some examples we evidence the differences between it and other fixed point theorems of the same type. As application of our results, we prove under suitable conditions the existence of fixed point for the superposition operator, defined in the Banach spaces of the continuous functions from $[0,1]$ and with values in a Banach space.

The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of experiment as a set of transformations. The main ingredient of the mathematical construction is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the adjoint of a transformation.

Due to properties of the topological tensor product of ultrametric Banach spaces, the algebraic notion of coalgebra has a natural ultrametric counterpart. An important class of ultrametric Banach coalgebras is provided by the spaces of continuous functions from a totally discontinuous compact group G with values in a complete ultrametric valued field K. The coproduct on is induced by the law of multiplication of the group G.

Related to ombral calculus is the characterization of the endomorphisms of a coalgebra. For a wide class of ultrametric Banach colagebras the monoid of the continuous coalgebra endomorphisms is anti-isomorphic to the monoid of continuous and weakly continuous algebra endomorphims of dual algebra. Hence, we recover earlier results obtained on the coalgebra , where ℤ_p, is the additive group of the ring of p-adic integers and K is a complete valued field of the field extension of p-adic numbers ℚ_p. In addition for such ground fields K and more generally for K of residue characteristic p, we consider the coalgebra , where V_q is the infinite compact monothetic subgroup of the group of units of K generated by q which is not a root of unit such that |q^h - 1) < 1, for h an integer ≥ 1. One obtains, as for formal power series, a procedure of substitution on the algebra of p-adic bounded measures M(V_q, K) which gives the continuous and weak*-continuous algebra endomorphisms of M(V_q, K) and which in turn gives the continuous coalgebra endomorphisms of .

This paper is devoted to the analysis of abstract hyperbolic differential equations in C([0, T];E^θ), C^α([0,T];E) and L^p([0,T];E^θ) spaces. The presentation is based on C₀-cosine operators theory and a functional analysis approach. For the solutions of second order differential equations, the weak maxima regularity estimates are established.

We survey some connections of the bounded approximation property with Banach operator ideals, both historical and recent ones. The exposition is self-contained and includes an introduction to Banach operator ideals.