General Banach Spaces Research Articles

On the convergence of sequence of maximal monotone operators of type (D) in Banach spaces

Within the setting of general real Banach spaces, we prove that the sequence of maximal monotone operators of type (D) graphically converges provided, their corresponding class of representative functions converge epigraphically. Moreover, we provide a condition to guarantee that the lower limit of a sequence of maximal monotone operators of type (D) is a maximal monotone operator of type (D) in real Banach spaces.

Random coupled Caputo-Hadamard fractional differential systems with four- point boundary conditions in generalized banach spaces

This paper deals with some existence and uniqueness of random solutions for a coupled system of Caputo–Hadamard fractional differential equations with four-point boundary conditions and random effects in generalized Banach spaces. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces. An illustrative example is presented in the last section.

Criteria for Convexity of Closed Sets in Banach Spaces

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.

"Existence and topological structure of solution sets for φ-Laplacian impulsive stochastic differential systems "

In this article, we present results on the existence and the topological struc- ture of the solution set for initial-value problems for the rst-order impulsive dierential equation with innite Brownian motions are proved.The approach is based nonlinear alternative Leary-Schauder type theorem in generalized Banach spaces

WCG spaces and their subspaces grasped by projectional skeletons

Weakly compactly generated Banach spaces and their subspaces are characterized by the presence of projectional skeletons with some additional properties. We work with real spaces. However the presented statements can be extended, without much extra effort, to complex spaces.

Effective Reifenberg theorems in Hilbert and Banach spaces

A famous theorem by Reifenberg states that closed subsets of $$\mathbb {R}^n$$ that look sufficiently close to k-dimensional at all scales are actually $$C^{0,\gamma }$$ equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure $$\mu $$ in $$\mathbb {R}^n$$ , one may introduce the k-dimensional Jones’ $$\beta $$ -numbers of the measure, where $$\beta ^k_\mu (x,r)$$ quantifies on a given ball $$B_r(x)$$ how closely in an integral sense the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these $$\beta $$ -numbers satisfy the uniform summability estimate $$\int _0^2 \beta ^k_\mu (x,r)^2 \frac{dr}{r}<M$$ , then $$\mu $$ must be rectifiable with uniform measure bounds. Note that one only needs the square of the $$\beta $$ -numbers to satisfy the summability estimate, this power gain has played an important role in the applications, for instance in the study of singular sets of geometric equations. One may also weaken these pointwise summability bounds to bounds which are more integral in nature. The aim of this article is to study these effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $$\mathbb {R}^n$$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a weak version of the effective Reifenberg theorem holds in that if one assumes a summability estimate $$\int _0^2 \beta ^k_\mu (x,r)^1 \frac{dr}{r}<M$$ without power gain, then $$\mu $$ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain turns out to be a subtle issue. For $$k=1$$ we will see for a uniformly smooth Banach space that if $$\int _0^2 \beta ^1_\mu (x,r)^\alpha \frac{dr}{r}<M^{\alpha /2}$$ , where $$\alpha $$ is the smoothness power of the Banach space, then $$\mu $$ is again rectifiable with uniform measure estimates.

Equivalence between almost-greedy and semi-greedy bases

In [3] it was proved that almost-greedy and semi-greedy bases are equivalent in the context of Banach spaces with finite cotype. In this paper we show this equivalence for general Banach spaces.

On functionals of the Wiener process in a Banach space

In development of stochastic analysis in a Banach space one of the main problem is to establish the existence of the stochastic integral from predictable Banach space valued (operator valued) random process. In the problem of representation of the Wiener functional as a stochastic integral we are faced with an inverse problem: we have the stochastic integral as a Banach space valued random element and we are looking for a suitable predictable integrand process. There are positive results only for a narrow class of Banach spaces with special geometry (UMD Banach spaces). We consider this problem in a general Banach space for a Gaussian functional.

Sharp growth rates for semigroups using resolvent bounds

We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an L^{p}-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article, we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

Acceleration of sequential subspace optimization in Banach spaces by orthogonal search directions

Conjugate Gradient (CG) methods are one of the most effective iterative methods to solve linear equations in Hilbert spaces. So far, they have been inherently bound to these spaces since they make use of the inner product structure. In more general Banach spaces one of the most prominent iterative solvers are Landweber-type methods that essentially resemble the Steepest Descent method applied to the normal equation. More advanced are subspace methods that take up the idea of a Krylov-type search space, wherein an optimal solution is sought. However, they do not share the conjugacy property with CG methods. In this article we propose that the Sequential Subspace Optimization (SESOP) method can be considered as an extension of CG methods to Banach spaces. We employ metric projections to orthogonalize the current search direction with respect to the search space from the last iteration. For the l2-space our method then exactly coincides with the Polak-Ribi\`ere type of the CG method when applied to the normal equation. We show that such an orthogonalized search space still leads to weak convergence of the subspace method. Moreover, numerical experiments on a random matrix toy problem and 2D computerized tomography on lp-spaces show superior convergence properties over all p compared to non-orthogonalized search spaces. This especially holds for lp-spaces with small p. We see that the closer we are to an l2-space, the more we recover of the conjugacy property that holds in these spaces, i. e., as expected, the more the convergence behaves independently of the size of the truncated search space.

Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space

In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in its classical Tikhonov version as well as two further—Ivanov type and Morozov type—versions. In these two alternative versions, regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit, respectively. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an L^infty source. For the latter, we also provide numerical results with the Ivanov type IRGNM.

Topological transitivity and mixing of composition operators

Let X=(X,B,μ) be a σ-finite measure space and f:X→X be a measurable transformation such that the composition operator Tf:φ↦φ∘f is a bounded linear operator acting on Lp(X,B,μ), 1≤p<∞. We provide a necessary and sufficient condition on f for Tf to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions.

An Approach to Stochastic Integration in General Separable Banach Spaces

We suggest a new approach to stochastic integration in infinite-dimensional spaces that is based on representing random variables on Banach spaces as real-valued processes on an interval. We prove stochastic integrability of operator-valued processes on general separable Banach spaces under the conditions that do not depend on the norm of the space and show how our methods can be applied to studying infinite-dimensional stochastic differential equations. In particular, our results provide a natural construction of the stochastic integral in abstract Wiener spaces.

On a resolvent approach for perturbed semigroups and application to $$L^1$$ L 1 -neutron transport theory

We give new sufficient and practical conditions in terms of the generators ensuring the stability of the critical or the essential type of a perturbed $$C_0$$ -semigroup in general Banach spaces. We apply our theoretical results in order to investigate the control and in particular the time asymptotic behavior of solutions to a broad class of transport equations in $$L^1$$ -spaces and higher dimension. Our results improve, complete and enrich several earlier works.

Iterates of Markov operators and their limits

It is well known that iterates of quasi-compact operators converge towards a spectral projection, whereas the explicit construction of the limiting operator is in general hard to obtain. Here, we show a simple method to explicitly construct this projection operator, provided that the fixed points of the operator and its adjoint are known which is often the case for operators used in approximation theory. We use an approach related to Riesz–Schauder and Fredholm theory to analyze the iterates of operators on general Banach spaces, while our main result remains applicable without specific knowledge on the underlying framework. Applications for Markov operators on the space of continuous functions C(X) are provided, where X is a compact Hausdorff space.

Gradient Dynamics: Motion Near a Manifold of Quasi-Equilibria

In a general Banach space we consider gradient-like dynamical systems with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction...

Some Compactness Criteria for Weak Solutions of Time Fractional PDEs

The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $\gamma\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces. The compactness criteria are then established using the time regularity estimates. The existence of weak solutions for a special case of time fractional compressible Navier-Stokes equations with constant density and time fractional Keller-Segel equations in $\mathbb{R}^2$ are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs.

Well-posedness and flow invariance for semilinear functional differential equations governed by non-densely defined operators

The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild solutions is introduced through a new type of variation of constants formula and the well-posedness is established under a semilinear stability condition with respect to a metric-like functional and a subtangential condition. The abstract result is applied to a size-structured model with birth delay.

Effective Rates of Convergence for the Resolvents of Accretive Operators

ABSTRACTWe extract explicit, computable, and highly uniform rates for the strong convergence of the resolvents of set-valued, m-accretive, and uniformly accretive at zero/ϕ-expansive operators on general real Banach spaces to the zero of each operator. This is achieved through proof mining on the proof of a theorem by García–Falset the motivation of which originates from a classical work by Reich. For the bound extraction we make use of a modulus of accretivity at zero, a notion introduced recently by Kohlenbach and the author, as well as a modulus of ϕ-expansivity, a notion introduced analogously here.

About Subtransversality of Collections of Sets

We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.