The simplest convergence theorem

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

For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10

formula omitted]-splittings of matrices

We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which \begin{document}$ 0 $\end{document} is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczynski (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.

Weak Orlicz space and its convergence theorems

Convergence theorems for Choquet integrals with generalized autocontinuity

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x*, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.

Several convergence theorems for sequences of (S)fuzzy integral are discussed, such as monotone decreasing convergence theorem, pseudo-almost everywhere convergence theorem, and so on. The necessary and sufficient conditions for pseudo-almost convergence theorem are derived, and a weak necessary condition for the monotone decreasing convergence theorem is also obtained.

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.

By exploiting the natural setting of a convergence theorem of Burkholder, a direct and elementary proof of the theorem is given. This proof is also new for the martingale convergence theorem and the martingale transform convergence theorem which are corollaries to the above-mentioned convergence theorem.

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.

In this paper, we extend the \({\mathcal{B}}\)-integral on \(\mathbb{R}\) to one on \(\mathbb{R}^{m}\). After proving a weaker version of the equiintegrability theorem, the equivalence of five generalized convergence theorems will be established, which include the equiintegrability theorem, two versions of the generalized controlled convergence theorem which are based on \(UACG^\nabla\) and \(UACG_{\mathcal{B}}^{\ast\ast}\), respectively, the generalized variational convergence theorem and the uniformly \(MGSRS_{\mathcal{B}}\) (modified \(GSRS\) with respect to \({\mathcal{B}}\)) convergence theorem.

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

Fixed point theorems for Lipschitzian type mappings in CAT(0) spaces

CHAPTER 7 - Conditional Expectation and Martingale Theory