Explore the vast range of titles available.

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\\mathcal {K}(E, F) = \\mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\\mathcal {L}(E,F), \\tau _c)^* = (\\mathcal {L}(E, F), \\left \\Vert \\cdot \\right \\Vert )^*$ , where $\\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\\sigma (T)$ of an operator $T:X\\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

Based on a characterization of upper semi-Fredholm operators due to A. Lebow and M. Schechter, we introduce and investigate a new quantity characterizing upper semi-Fredholm operators. This quantity and several well-known quantities are used to characterize bounded compact approximation property. Similarly, a new quantity characterizing lower semi-Fredholm operators is introduced, investigated and used to characterize the bounded compact approximation property for dual spaces.

We establish inter alia a compactness criterion in metric spaces involving a sequence of completely continuous mappings, which is continuously convergent, in the sense of H. Hahn, to the identity mapping. For Banach spaces, the linear version of that result coincides with the compactness theorem due to Mazur. We also present a new proof of the Ascoli–Arzelà theorem, in which we use the above compactness criterion applied to the sequence of Bernstein operators.

In this paper, we consider stochastic mixed vector variational inequality problems. Firstly, we present an equivalent form for the stochastic mixed vector variational inequality problems. Secondly, we present a deterministic bi-criteria model for giving the reasonable resolution of the stochastic mixed vector variational inequality problems and further propose the approximation problem for solving the given deterministic model by employing the smoothing technique and the sample average approximation method. Thirdly, we obtain the convergence analysis for the proposed approximation problem while the sample space is compact. Finally, we propose a compact approximation method when the sample space is not a compact set and provide the corresponding convergence results.

In this paper, based on the second-order compact approximation of first-order derivative, the numerical algorithm with second-order temporal accuracy and fourth-order spatial accuracy is developed to solve the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative; the solvability, convergence, and stability of the numerical algorithm are analyzed in detail by algebraic theory and Fourier analysis, respectively; the numerical experiment support our theoretical analysis results.

In this paper, a compact difference operator, termed CWSGD, is designed to establish the quasi-compact finite difference schemes for approximating the space fractional diffusion equations in one and two dimensions. The method improves the spatial accuracy order of the weighted and shifted Grünwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949 ) from 2 to 3. The numerical stability and convergence with respect to the discrete L 2 norm are theoretically analyzed. Numerical examples illustrate the effectiveness of the quasi-compact schemes and confirm the theoretical estimations.

In this paper we give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property.

This paper discusses about a new compact 2-level implicit numerical method in the form of exponential approximation for finding the approximate solution of nonlinear fourth order Kuramoto–Sivashinsky and Fisher–Kolmogorov equations, which have applications in chemical engineering. The described method has an accuracy of temporal order two and a spatial order three (or four) on a variable (or constant) mesh. The approach has been demonstrated to be applicable to both non-singular and singular issues. This article has established the stability of the current technique. The suggested approach is used to solve several benchmark nonlinear parabolic problems associated in chemistry and chemical engineering, and the computed results are compared with the existing results to demonstrate the proposed method's superiority.