Continuity properties of the ball hull mapping

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

In this paper, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. By using a new proof method which is different from the one used in Huang et al. (J Optim Theory Appl 162:548---558 2014), a sufficient and necessary condition for the nonemptiness and boundedness of solution sets is established. Basing on this result, some new characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are proved. Compared with the known results in Huang et al. (2014), the key assumption that $$K_\infty \cap (F(K))^{w\circ }_C=\{0\}$$K??(F(K))Cw?={0} is not required in finite dimensional spaces. Furthermore, the corresponding result of Huang et al. (2014) is extended to the case of infinite dimensional spaces. Some examples are also given to illustrated the main results.

Soliton systems can be represented by means of a finite dimensional parameter space based on spectra of the Lax pair. In the presence of perturbation, chaos and/or self-organization appear in these parameter space which may or may not represent behavior of the solution of the original partial differential equation having an infinite dimension. Here, we present some interesting examples of inter-relationship between behaviors of solutions in reduced (finite dimensional space) and unreduced (infinite dimensional space) parameter spaces of optical solitons in fibers.

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

We consider the problem of developing filter banks with the perfect reconstruction property for finite dimensional signals. We are motivated by the discrete, finite dimensional character of digital signals and images, which naturally leads to the study of the discrete counterpart of multiresolution analysis and wavelet series expansions in infinite dimensional spaces such as L/sub 2/ and l/sub 2/. In finite dimensional spaces, all computations can be performed using finite matrix operations. The discrete Fourier transform (DFT) is the natural tool for the harmonic analysis in such spaces, in which the circular convolution operation plays a vital role. There has also been interest in this problem by other authors. However, our approach is distinct: in a sense, it is simpler and more independent of the well-known theory in L/sub 2/.

Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications

This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject. Table of Contents: Background material; Sobolev spaces on $\mathbb{R}^n$; Differentiable measures on linear spaces; Some classes of differentiable measures; Subspaces of differentiability of measures; Integration by parts and logarithmic derivatives; Logarithmic gradients; Sobolev classes on infinite dimensional spaces; The Malliavin calculus; Infinite dimensional transformations; Measures on manifolds; Applications; References; Subject index. (Surv/164)

On a Topological Method for the Analysis of the Asymptotic Behavior of Dynamical Systems and Processes

The Walrasian equilibrium problem with a finite dimensional commodity space has been studied rather extensively in the past. The existence of equilibrium prices in economies with a finite dimensional commodity space has been demonstrated very satisfactorily; see [8,9]. However, a number of economic situations lead naturally to infinite dimensional commodity spaces. In such a case, the mathematical tools employed in the finite dimensional case do not yield similar equilibrium results. Due to the nature of infinite dimensional spaces, questions about compactness of budget sets, continuity of utility and excess demand functions, utility maximization problems, etc. are very subtle. For this very reason, there are no satisfactory results guaranteeing the existence of equilibrium prices in economies with infinite dimensional commodity spaces. However, in spite of these difficulties, considerable progress has been made on the equilibrium problem with infinite dimensional commodity spaces.

We consider a scalar objective minimization problem over the solution set of another optimization problem. This problem is known as a simple bilevel optimization problem and has drawn a significant attention in the last few years. Our inner problem consists of minimizing the sum of smooth and non-smooth functions while the outer one is the minimization of a smooth convex function. We first formulate and give strong convergence analysis of an inertial algorithm for fixed-point problem of a non-expansive operator in an infinite dimensional Hilbert space. Then we convert the simple bilevel optimization problem to a fixed-point problem of a non-expansive operator in finite dimensional space and design the corresponding algorithm and establish its convergence. Our numerical experiments show that the proposed method in this paper outperforms the currently known best algorithm to solve the class of bilevel optimization problem considered.

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Although the division of distributions by real polynomials and real analytic functions (which are nonzero) is always possible in finite dimensional spaces (from classical results of Hrmander and Lojasiewicz respectively), we show that this is not always possible in infinite dimensional locally convex spaces.In particular, we characterize those locally convex spaces where division is always possible.I. Introduction.Let S be a distribution and g a C00 function on an open subset in a locally convex space.We say that g divides S if there exists a distribution T on so that gT = S.This problem was first studied by L. Schwartz [8] who proved that the division by a nonzero holomorphic function is always possible on a connected open subset of C".Later, L. Hrmander [6] and S. Lojasiewicz [7] respectively solved the division by real polynomials and analytic functions in finite dimensional spaces.J. F. Colombeau, R. Gay and B. Perrot [4] extended Schwartz's result to complex locally convex spaces.These works established that if g does not vanish on any open subset of , the division is always possible.Our purpose is to answer a natural question in Colombeau, Gay and Perrot [4] concerning division by real polynomials and real analytic functions in infinite dimensional locally convex spaces.We prove division is always possible by finite type real polynomials and finite type real analytic functions which are nonzero (defined later).But the division by general polynomials is not possible in general.We characterize the spaces where division is always possible and will see that they form a rather hmited class; in particular, the unique infinite dimensional Frchet space where division by nonzero real polynomials or real analytic functions is always possible is RN, and the unique Silva space is R(N).II.Notation.They are classical.We denote by E a Hausdorff real locally convex space and an open subset of E. A mapping/: - R is called a C00 function if, for every convex balanced bounded subset B of E, the restriction of/to n EB (where EB denotes, as usual, the vector space spanned by B and normed with the gauge pB of B) is C in the usual Frchet sense of calculus in normed spaces.(This is the