Abstract The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.

We prove that the norm of X⊗ˆπY is SSD if either X=ℓp(I) for p>2 and Y is a finite-dimensional Banach space such that the modulus of convexity is of power type q<p (e.g. if Y∗ is a subspace of Lq), or if X=c0(I) and Y∗ is any uniformly convex finite-dimensional Banach space. We also provide a characterisation of SSD elements of a projective tensor product which attain its projective norm in terms of a strengthening of the local Bollobás property for bilinear mappings.

In this article, we study the essential pseudospectra by measure of polynomially strict singular operators, which is a generalization of the class of strict singular operators. We present some new results in essential pseudospectra for closed linear operators in Banach space with polynomially strict singular operators.Furthermore, we apply the obtained results to analyze the incidence of some perturbation results on left(resp. right) Weyl essential pseudospectra and left(resp. right) Fredholm essential pseudospectra. In addition, we will describe the essential pseudospectra of a sum of two bounded linear operators. A final application of the obtained results is to characterize the pseudo-left (right)-Fredholm spectra of 2 x 2 block operator matrices. TRANSLATE with x English Arabic Hebrew Polish Bulgarian Hindi Portuguese Catalan Hmong Daw Romanian Chinese Simplified Hungarian Russian Chinese Traditional Indonesian Slovak Czech Italian Slovenian Danish Japanese Spanish Dutch Klingon Swedish English Korean Thai Estonian Latvian Turkish Finnish Lithuanian Ukrainian French Malay Urdu German Maltese Vietnamese Greek Norwegian Welsh Haitian Creole Persian TRANSLATE with COPY THE URL BELOW Back EMBED THE SNIPPET BELOW IN YOUR SITE Enable collaborative features and customize widget: Bing Webmaster Portal Back

Let B be a Banach space and X a quasi-Banach function lattice. In this paper, we introduce the B -valued weak martingale Hardy spaces associated with X . We then establish the \infty -atomic characterizations of these B -valued weak martingale Hardy-type spaces. As an application, we obtain the boundedness of the \sigma -sublinear operators from B -valued weak martingale Hardy type spaces to weak quasi-Banach function spaces. Using this, we further establish the relationships between different Banach-valued weak martingale Hardy spaces.

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of the spectrum of its infinitesimal generator. More precisely, given an arbitrary positive function r r vanishing at ∞ \infty , we construct a Banach space X X and a bounded semigroup ( T ( t ) ) t ≥ 0 (T(t))_{t \geq 0} of operators on it whose infinitesimal generator A A has empty spectrum σ ( A ) = ∅ \sigma (A)=\varnothing , but for which, for some x ∈ X x \in X , lim sup t → ∞ ‖ T ( t ) A − 1 x ‖ X r ( t ) = ∞ . \begin{equation*} \limsup _{t\to \infty } \frac {\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty . \end{equation*}

In this paper, we suggest a novel approach to linear dynamics based on the idea that supercyclicity can be localized. A nonzero vector x in a Banach space X is called an SJ-class vector for an operator T ∈ L ( X ) provided for every open neighborhood U x of x and every nonempty open subset V of X and also for every N ∈ N , there exists a complex number λ and an integer N $ ]]> n > N such that λ T n ( U x ) ∩ V ≠ ∅ . When an operator T has at least one SJ-class vector, it is called an operator of SJ-class. As a consequence, if T is a supercyclic operator, then it falls into the SJ-class. Aside from investigating the SJ-class operators, we provide some examples demonstrating that the SJ-class is a new class in L ( X ) . We would like to emphasize that despite the fact that the non-separable Banach space ℓ ∞ ( N ) does not admit supercyclic operators, it does admit operators of the SJ-class. Also, we provide a characterization of the SJ-class operators on the Banach space X ⊕ C in terms of the J-class operators on the Banach space X.

Discrete (weighted) pseudo $$\mathcal {S}$$-asymptotic affine periodicity on Banach space and applications

ABSTRACT In this paper, we solve for the initial‐boundary value problem involving the viscous Burgers equation with a homogeneous Dirichlet boundary and the given initial condition . We investigate the existence of mild solutions in the Banach space , for a sufficiently small time horizon . The analysis relies on semigroup theory and the formulation of the problem as a fixed‐point equation involving an integral operator. We prove that the associated nonlinear map is compact and continuous, and satisfies suitable a priori estimates. The Leray–Schauder topological degree is then applied to establish the existence of a fixed point, which corresponds to a mild solution of the original problem. Our result confirms local‐in‐time solvability under minimal regularity assumptions on the initial data.

Some examples of affine isometries of Banach spaces arising from 1-D dynamics

In this paper, we defined new norms in 2-normed spaces derived from the 2-norm with respect to its quotient spaces. Moreover, these norms were used to observe some aspects of 2-normed spaces namely, a Convergent sequence, a Cauchy sequence, completeness, a closed set, and a bounded set. Furthermore, we used these aspects to prove the Fixed-Point Theorem in a 2-Banach Space.

This paper explores the concepts of nilpotent and quasinilpotent linear relations in the context of Banach spaces. We establish sufficient conditions under which a quasinilpotent linear relation can be classified as nilpotent, focusing on the finiteness of ascent and descent and the closure of iterated images. We then introduce and analyze the class of meromorphic linear relations, emphasizing conditions that lead to their classification as quasinilpotent. Furthermore, we expand on the notion of quasinilpotent linear relations by introducing the concept of regularity and defining R-quasinilpotent relations. As an application, we investigate generalized Drazin invertible relations related to a given regularity.

$$\lambda (t,s)$$-Dichotomy and its Robustness on Banach Spaces

The Banach spaces is the most popular and well-known spaces in the subject of pure mathematics. Many mathematicians have worked on different concepts from different angles in the field of functional analysis due to wide range of utilizations in quantum mechanics, optimization, and numerical analysis. In this paper, we explore the new iterative approach, namely HK-iteration method in Banach spaces. In addition, we investigate the some new convergence results in Banach Spaces via newly introduced concept with an application to a first-order delay differential equation. For the paper quality and reader’sinterest, numerical example and performance comparison via newly introduced concept have been added.

New convergence conditions for Secant-type higher-order methods in Banach spaces

This study addresses the existence and approximate controllability of a type of higher-order Hilfer fractional evolution differential (HOHFED) system with time delays in Banach spaces. Using the properties of the Mittag–Leffler function, cosine families, and Hilfer-type fractional differential operators, we first demonstrate the existence and uniqueness of mild solutions using a fixed-point method. Furthermore, a sequential technique is proposed to establish adequate conditions for approximate controllability. A detailed example is provided to illustrate the applicability and effectiveness of the theoretical results.

This paper addresses the $L^p([0,\nu], U)$ exact controllability of the abstract semilinear differential inclusion with nonlocal conditions within the context of a uniformly convex Banach space ($U$). By presuming exact controllability for the linear system, we apply an approximate solvability technique to reduce the problem to finite-dimensional subspaces. Consequently, the solutions for the primary problem are the limiting functions within these finite dimensional subspaces. The paper offers a unique solution to a challenge introduced by assuming $U$ as a uniformly convex Banach space, which presents issues of convexity during the construction of the necessary control. Such issues are not present when $U$ is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for $L^p([0,\nu], U)$ controllability of the semilinear differential control system in which $U$ is a uniformly convex Banach space.

In this research artricle, we present modifications to the results reported in the paper of Chung and Park (2012). The modifieded theorems strengthen the theoretical framework of functional equations in 2-Banach spaces and ensure consistency in their applications to stability theory.

Abstract We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $$p>1$$ p > 1 . The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems.

This paper investigates a coupled system of nonlinear implicit fractional differential equations of order α∈(1,2] subject to anti-periodic boundary conditions. The analysis is conducted using the ψ-Caputo fractional derivative, a generalized operator that incorporates several well-known fractional derivatives. The system features implicit coupling, where each equation depends on both unknown functions and their first derivatives, as well as an implicit dependence on the fractional derivatives themselves. The boundary value problem is transformed into an equivalent system of integral equations. Sufficient conditions for the existence and uniqueness of solutions are established using Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen Banach space. Furthermore, the Ulam–Hyers stability of the system is analyzed. The applicability of the theoretical results is demonstrated through a detailed example of a coupled system where all hypotheses are verified.

In this paper, we investigate stochastic versions of the Hopf-Lax formula which are based on compositions of the Hopf-Lax operator with the transition kernel of a Lévy process taking values in a separable Banach space. We show that, depending on the order of the composition, one obtains upper and lower bounds for the value function of a stochastic optimal control problem associated to the drift controlled Lévy dynamics. Dynamic consistency is restored by iterating the resulting operators. Moreover, the value function of the control problem is approximated both from above and below as the number of iterations tends to infinity, and we provide explicit convergence rates and guarantees for the approximation procedure.