A multivariate convergence theorem for positive sequences of functions

In this paper we explore the structure of certain generalized isometries of the special orthogonal group SO(n) which are transformations that leave any member of a large class of generalized distance measures invariant.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions

We provide sufficient conditions for the convergence of the Newton-like methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. Consequently, some important convergence theorems follow from our main result in this paper.     

A composition theorem for weighted pseudo-almost automorphic functions and applications

In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.     

Kurzweil integral for Riesz space-valued functions: Uniform convergence theorem

Our objective here is to prove that the uniform convergence of a sequence of Kurzweil integrable functions implies the convergence of the sequence formed by its corresponding integrals.     

A theorem on zeta functions associated with polynomials

Let be an -tuple of non-negative integers and be polynomials in such that for all and the series

is absolutely convergent for Re . We consider the zeta functions

All these zeta functions and are analytic functions of when Re is sufficiently large and they have meromorphic analytic continuations in the whole complex plane.

In this paper we shall prove that

As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.

     

A polynomial Hutton theorem with applications

Extending a classical linear result due to Hutton to a nonlinear setting, we prove that a continuous homogeneous polynomial between Banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators. Among other consequences, we apply this result to generalize a classical result due to Aron and Schottenloher about the approximation property on spaces of polynomials and a recent result due to Çaliskan and Rueda about the quasi-approximation property on projective symmetric tensor products.     

Sandwich theorem for quasiconvex functions and its applications

In convex programming, sandwich theorem is very important because it is equivalent to Fenchel duality theorem. In this paper, we investigate a sandwich theorem for quasiconvex functions. Also, we consider some applications for quasiconvex programming.     

A poisson convergence theorem for a particle system with dependent constant velocities

Consider an infinite collection of particles travelling in d-dimensional Euclidean space and let X_n denote the initial position of the n^th particle. Assume that the n^th particle has through all time the random velocity V_n and that {V_n} is a sequence of dependent random variables. Let X_n(t) = X_n + V_nt denote the position of the n^th particle at time t. Conditions are obtained for the convergence of {X_n(t)} to a Poisson process as t→∞. Essentially they require that the dependence in the V_n-sequence decrease with increasing distance between the initial positions and that the conditional distribution of V_n given the initial positions of all the particles and V_nk≠n be absolutely continuous with respect to Lebesgue measure.     

A unifying theorem for Newton’s method on spaces with a convergence structure

We present a semilocal convergence theorem for Newton’s method (NM) on spaces with a convergence structure. Using our new idea of recurrent functions, we provide a tighter analysis, with weaker hypotheses than before and with the same computational cost as for Argyros (1996, 1997, 1997, 2007) [1], [2], [3] and [5], Meyer (1984, 1987, 1992) [13], [14] and [15]. Numerical examples are provided for solving equations in cases not covered before.