Monotone convergence theorems for Henstock-Kurzweil integrable functions and applications

In this paper we prove monotone convergence theorems for Henstock-Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock-Kurzweil integrable functions.     

Stochastic dynamics and Boltzmann hierarchy. I

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. III

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. II

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Well-posedness results for differential and integral equations containing Henstock–Kurzweil integrable vector-valued functions

In this paper, we prove existence, uniqueness and comparison results for solutions of differential and integral equations in Banach spaces containing Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space.     

Global asymptotic stability of solutions of a functional integral equation

Using the technique of measures of noncompactness we prove a theorem on the existence and global asymptotic stability of solutions of a functional integral equation. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A few realizations of the result obtained are indicated.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

On a generalized proximal point method for solving equilibrium problems in Banach spaces

We introduce a regularized equilibrium problem in Banach spaces, involving generalized Bregman functions. For this regularized problem, we establish the existence and uniqueness of solutions. These regularizations yield a proximal-like method for solving equilibrium problems in Banach spaces. We prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex. Moreover, we prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable. We prove, under additional assumptions, that the proximal sequence converges weakly to a solution.     

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.     

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.     

Approximation of lipschitz functions by Δ-convex functions in banach spaces

In this paper we give some result about the approximation of a Lipschitz function on a Banach space by means of Δ-convex functions. In particular, we prove that the density of Δ-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of Δ-convex functions.     

Non-linear partial differential equations with discrete state-dependent delays in a metric space

We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor.     

On local attractivity of solutions of a functional integral equation of fractional order with deviating arguments

In this paper, we study the existence of solutions of a nonlinear functional integral equation of fractional order with deviating arguments. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive. An example is provided to illustrate the theory.     

On BV-Solutions of Some Nonlinear Integral Equations

In this paper we prove uniqueness theorems for bounded variation (shortly: BV) solutions and continuous BV-solutions of the Hammerstein and the Volterra-Hammerstein integral equations. We investigate real-valued functions and functions with values in a Banach space. Submitted: August 16, 2001?Revised: September 13, 2002     

Extremal solutions and comparison principle for nonlinear integro-differential equations in a Banach space

In this paper, we consider an initial value problem for nonlinear integro-differential equations in a Banach space. First, we give a comparison result between the under and over functions and some comparison principles. Then, using these results and the Kuratowski measure of noncompactness, we establish the existence theorem of extremal solutions between the under and over functions, and prove that there exists a unique solution between the lower and upper solutions under an additional Lipschitz's condition. Project supported by the Natural Science Foundation of Shandong Province.     

Correlation functions evolution for the Glauber dynamics in continuum

We construct a correlation functions evolution corresponding to the Glauber dynamics in continuum. Existence of the corresponding strongly continuous contraction semigroup in a proper Banach space is shown. Additionally we prove the existence of the evolution of states and study their ergodic properties.     

Applications of Measure of Noncompactness and Darbo’s Fixed Point Theorem to Nonlinear Integral Equations in Banach Spaces

We prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable.     

Existenz und erste Einschließung positiver Lösungen bei superlinearen Randwertaufgaben zweiter Ordnung

Summary Some classes of second-order superlinear boundary value problems are considered. Using M. A. Krasnoselskii's fixed point theorem for operators expanding a cone in a real Banach space we can prove new existence theorems for positive solutions for nonlinearities not necessarily satisfying a condition of Nagumo-type. At the same time, we give first bounds for the solutions.     

BVφ-solutions of nonlinear integral equations

In this paper we investigate solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray-Schauder alternative for contractions.