Weak Orlicz space and its convergence theorems

1 Geometric properties. -1.1 Introduction. -1.2 Uniformly convex spaces. -1.3 Strictly convex Banach spaces. -1.4 The modulus of convexity. -1.5 Uniform convexity, strict convexity and reflexivity. -1.6 Historical remarks. -2 Smooth Spaces. -2.1 Introduction. -2.2 The modulus of smoothness. -2.3 Duality between spaces. -2.4 Historical remarks. -3 Duality Maps in Banach Spaces. -3.1 Motivation. -3.2 Duality maps of some concrete spaces. -3.3 Historical remarks. -4 Inequalities in Uniformly Convex Spaces. -4.1 Introduction. -4.2 Basic notions of convex analysis. -4.3 p-uniformly convex spaces. -4.4 Uniformly convex spaces. -4.5 Historical remarks. -5 Inequalities in Uniformly Smooth Spaces. -5.1 Definitions and basic theorems. -5.2 q-uniformly smooth spaces. -5.3 Uniformly smooth spaces. -5.4 Characterization of some real Banach spaces by the duality map. -5.4.1 Duality maps on uniformly smooth spaces. -5.4.2 Duality maps on spaces with uniformly Gateaux differentiable norms. -6 Iterative Method for Fixed Points of Nonexpansive Mappings. -6.1 Introduction. -6.2 Asymptotic regularity. -6.3 Uniform asymptotic regularity. -6.4 Strong convergence. -6.5 Weak convergence. -6.6 Some examples. -6.7 Halpern-type iteration method. -6.7.1 Convergence theorems. -6.7.2 The case of non-self mappings. -6.8 Historical remarks. -7 Hybrid Steepest Descent Method for Variational Inequalities. -7.1 Introduction. -7.2 Preliminaries. -7.3 Convergence Theorems. -7.4 Further Convergence Theorems. -7.4.1 Convergence Theorems. -7.5 The case of Lp spaces, 1 2. -7.6 Historical remarks. 8 Iterative Methods for Zeros of F -Accretive-Type Operators. -8.1 Introduction and preliminaries. -8.2 Some remarks on accretive operators. -8.3 Lipschitz strongly accretive maps. -8.4 Generalized F -accretive self-maps. -8.5 Generalized F -accretive non-self maps. -8.6 Historical remarks. -9 Iteration Processes for Zeros of Generalized F -Accretive Mappings. -9.1 Introduction. -9.2Uniformly continuous generalized F -hemi-contractive maps. -9.3 Generalized Lipschitz, generalized F -quasi-accretive mappings. -9.4 Historical remarks. -10 An Example Mann Iteration for Strictly Pseudo-contractive Mappings. -10.1 Introduction and a convergence theorem. -10.2 An example. -10.3 Mann iteration for a class of Lipschitz pseudo-contractive maps. -10.4 Historical remarks. -11 Approximation of Fixed Points of Lipschitz Pseudo-contractive Mappings. -11.1 Lipschitz pseudo-contractions. -11.2 Remarks. -12 Generalized Lipschitz Accretive and Pseudo-contractive Mappings. -12.1 Introduction. -12.2 Convergence theorems. -12.3 Some applications. -12.4 Historical remarks. -13 Applications to Hammerstein Integral Equations. -13.1 Introduction. -13.2 Solution of Hammerstein equations. -13.2.1 Convergence theorems for Lipschitz maps. -13.2.2 Convergence theorems for bounded maps. -13.2.3 Explicit algorithms. -13.3 Convergence theorems with explicit algorithms. -13.3.1 Some useful lemmas. -13.3.2 Convergence theorems with coupled schemes for the case of Lipschitz maps. -13.3.3 Convergence in Lp spaces, 1 2: . -13.4 Coupled scheme for the case of bounded operators. -13.4.1 Convergence theorems. -13.4.2 Convergence for bounded operators in Lp spaces, 1 2:. -13.4.3 Convergence theorems for generalized Lipschitz maps. -13.5 Remarks and open questions. -13.6 Exercise. -13.7 Historical remarks. -14 Iterative Methods for Some Generalizations of Nonexpansive Maps. -14.1 Introduction. -14.2 Iteration methods for asymptotically nonexpansive mappings. -14.2.1 Modified Mann process. -14.2.2 Iteration method of Schu. -14.2.3 Halpern-type process. -14.3 Asymptotically quasi-nonexpansive mappings. -14.4 Historical remarks. -14.5 Exercises. -15 Common Fixed Points for Finite Families of Nonexpansive Mappings. -15.1 Introduction. -15.2 Convergence theorems for a family of nonexpansive mappings. -15.3 Non-self mappings. -16 Common Fixed Po

Convergence theorems for Choquet integrals with generalized autocontinuity

An elementary application of Fatou’s lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration. 1 The convergence from below theorem Three famous convergence-related results appear in most introductory courses on measure and integration: the monotone convergence theorem, Fatou’s lemma and the dominated convergence theorem. In teaching this material it is common to follow the approach taken in, for example, [1, Chapter 1]. There Rudin begins by proving the monotone convergence theorem and then deduces Fatou’s lemma. Finally, he deduces the dominated convergence theorem from Fatou’s lemma. The result which we call the convergence from below theorem (Theorem 1.2 below) is essentially distilled from this proof of the dominated convergence theorem ([1, pp. 26-27]). We do not claim originality for this result, or for the related Theorem 1.3. They are presumably known, although we know of no explicit references for them. However, we wish to make a case that that they should be better known than they are. In particular, we suggest that Theorem 1.2 deserves a name and a place in the syllabus when this material is taught. Throughout we discuss results concerning pointwise convergence. In the usual way, there are versions of all these results in terms of almost-everywhere convergence instead. For convenience, we shall use the following terminology. Let X be a set, let (fn) be a sequence of functions from X to [0,∞] and let f be another function from X to [0,∞]. We say that the functions fn converge to f from below on X if the functions fn tend to f pointwise on X and fn(x) ≤ f(x) (n ∈ N, x ∈ X). We say that the functions fn converge to f monotonely from below on X if the functions fn tend to f pointwise on X and, for all x ∈ X, we have f1(x) ≤ f2(x) ≤ f3(x) ≤ · · ·. We begin by recalling the statement of the monotone convergence theorem.

The interchange of the ‘limit of an integral’ with the ‘integral of a limit’ for sequenc- es of functions is crucial in relevant applications, such as Fourier series for decom- posing periodic functions into sinusoidal components, and Fubini’s theorem for changing the order of integration of multivariable functions. This expository paper reviews three classical results in real analysis for cases where the limit of an integral of a sequence of functions equals the integral of the limiting function: (1) Mono- tone Convergence Theorem, (2) Uniform Convergence Theorem, and the broad- est result, (3) Dominated Convergence Theorem. While proofs of (2) are typically studied in undergraduate analysis, the proofs of (1) and (3) are usually reserved for graduate-level measure theory, where they are taught in a more general context. The purpose of this paper is to summarize and adapt W. A. J. Luxembourg’s un- dergraduate-friendly proof [7] of (3) Arzel`a’s Dominated Convergence Theorem, to demonstrate the nontrivial direction of (1) Monotone Convergence Theorem for Riemann Integrals. Our aim is to demystify the hidden logic involved in these well-established theorems, making them more accessible for undergraduate analysis.

Generalized Lebesgue integrals of fuzzy complex valued functions

I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence it can be seen that in certain situations nonlinear approximations are more powerful than linear approximations. The recent notion of branched continued fraction is introduced in the multivariate section and is later used for the construction of multivariate rational interpolants. II. Pade Approximants. A survey of the theory of these local rational approximants for a given function is presented, including the problems of existence, unicity and computation. Also considered are the convergence of sequences of Pade approximants and the continuity of the Pade operator which associates with a function its Pade approximant of a certain order. A special section is devoted to the multivariate case. III. Rational Interpolants. These rational functions fit a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and Pade approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. IV. Applications. The previous types of rational approximants are used here to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques, and it is seen that nonlinear methods are very useful in situations involving singularities. Subject Index.

Markovian imprecise jump processes: Extension to measurable variables, convergence theorems and algorithms

Chapter 5 - Standard Convergence Theorems, The Fubini Theorem

Chapter 13 - The Henstock—Kurzweil Integral

The convergence theorems we start with yield sufficient conditions that the integral can be interchanged with pointwise convergence of functions. These theorems clearly require continuity of the set function. First the Monotone Convergence Theorem derives from the special case proved in Chapter 1. It is valid very generally for monotone set functions. The other important theorem, Lebesgue’s Dominated Convergence Theorem, is first proved for the classical case of measures and later on it is generalized for subadditive set functions in applying the classical case with Lebesgue measure on the distribution functions. For this purpose we weaken pointwise convergence to stochastic convergence and further to convergence in distribution. These different types of convergence are important, too, in probability theory and statistics. By the way we get some information on measurability of the limit function even if the set function is not continuous.

This section gives a summary of some elementary facts used frequently throughout this book, and can be regarded as an appendix. In particular, we consider sufficient conditions for the interchange of integration and limit operations. In detail, we discuss a result on uniform convergence, the dominated convergence theorem, the bounded convergence theorem, Fatou's lemma, and the monotone convergence theorem from the points of view of both Lebesgue integration theory and Riemann integration theory. Note that these are well-known results; hence we will be brief in details. For the proof of the monotone convergence theorem and Fubini's theorem we merely refer to the appropriate literature.

There are several types of nonlinear integrals with respect to nonadditive measures, such as the Choquet, Sipos, Sugeno, and Shilkret integrals. In order to put those integrals into practical use and aim for application to various fields, it is indispensable to establish convergence theorems of such nonlinear integrals. However, they have individually been discussed for each of the integrals up to the present. In this article, several important convergence theorems of nonlinear integrals, such as the monotone convergence theorem, the bounded convergence theorem, and the Vitali convergence theorem, are formulated in a unified way regardless of the types of integrals.

The present study introduces the concepts of ideal convergence (I-convergence), ideal Cauchy (I-Cauchy) sequences, I *-convergence, and I *-Cauchy sequences in intuitionistic fuzzy metric spaces. It defines I-limit and I-cluster points as a sequence in these spaces. Afterward, it examines some of their basic properties. Lastly, the paper discusses whether phenomena should be further investigated.

Measure theory is a branch of mathematics. Length, area, volume and weight are instances of measure concept. The emphasis in this chapter is mainly on the concept of measure, Borel set, measurable function, Lebesgue integral, Lebesgue-Stieltjes integral, monotone class theorem, Carathéodory extension theorem, measure continuity theorem, product measure theorem, monotone convergence theorem, Fatou’s lemma, Lebesgue dominated convergence theorem, and Fubini theorem. The main results in this chapter are well-known. For this reason the credit references are not given. This chapter can be omitted by the readers who are familiar with the basic concepts and theorems of measure and integral.

Chapter 2: Martingales and quasimartingales - Basic inequalities and convergence theorem - Application to stochastic algorithms