Fatou's Lemma Research Articles

Time‐optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion with Clarke's subdifferential

In this article, we discuss the optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion of order with Clarke's subdifferential. Firstly, we establish the existence of mild solutions for this class of equations by employing the Krasnoselskii's fixed‐point theorem, without using Lipschitz conditions. Subsequently, we develop set of sufficient assumptions, utilizing Cesari property and Filippov theorem, to ensure the existence of feasible pairs for the feedback control systems. Eventually, using the Mazur theorem and Fatou's lemma, we derive the optimal state–control pair. The study also introduces and explores the problem of time‐optimal feedback control. In conclusion, an application is provided to enhance the theoretical findings.

Stationary distribution and extinction of a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process

In the past few decades, the human immunodeficiency virus (HIV), as a deadly pathogen, has imposed great harmful effects on human health. In addition, from the viewpoint of the microscopic, the interference of random factor exists in the process of virus replication. However, there are few works devoted to the theoretical studies of viral infection models with biologically reasonable stochastic effects yet. To study the effects of stochastic perturbations on the pathogenesis process of HIV/AIDS, in this paper, we develop and study a stochastic HLIV model with viral production and Ornstein–Uhlenbeck process, in which we assume that the conversion rate from an eclipse-phase cell to an infected cell satisfies the Ornstein–Uhlenbeck process. Firstly, by using the existence and uniqueness theory of solutions, we prove that there is a unique global solution to the stochastic model for any initial value. Then we use stochastic Lyapunov function methods and Fatou lemma to establish sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of the disease. In addition, by adopting the comparison theorem of one-dimensional stochastic differential equation and the theory of ergodicity, we obtain sufficient conditions for complete eradication of the infected cells and free viruses. Our theoretical results show that the bigger speed of reversion or the smaller intensity of volatility is conducive to clearing infected cells and free viruses while the larger intensity of volatility or the smaller speed of reversion is not. We also find that this way of introducing stochastic perturbations is reasonable from the perspective of mathematics and biology. Finally, numerical simulations are carried out to demonstrate our main results. The modeling and methods can be applied to study other multigroup viral infection models or epidemic model with Ornstein–Uhlenbeck process, such as S-DI-A model, DS-I-A model, DS-DI-A model, SEI epidemic model, et al.

Mathematical analysis of a SIPC age-structured model of cervical cancer.

Human Papillomavirus (HPV), which is the main causal factor of cervical cancer, infects normal cervical cells on the specific cell's age interval, i.e., between the G1 to S phase of cell cycle. Hence, the spread of the viruses in cervical tissue not only depends on the time, but also the cell age. By this fact, we introduce a new model that shows the spread of HPV infections on the cervical tissue by considering the age of cells and the time. The model is a four dimensional system of the first order partial differential equations with time and age independent variables, where the cells population is divided into four sub-populations, i.e., susceptible cells, infected cells by HPV, precancerous cells, and cancer cells. There are two types of the steady state solution of the system, i.e., disease-free and cancerous steady state solutions, where the stability is determined by using Fatou's lemma and solving some integral equations. In this case, we use a non-standard method to calculate the basic reproduction number of the system. Lastly, we use numerical simulations to show the dynamics of the age-structured system.

An Age-Structured Model for Tilapia Lake Virus Transmission in Freshwater with Vertical and Horizontal Transmission.

This paper proposes a mathematical model for tilapia lake virus (TiLV) transmission in wild and farmed tilapias within freshwater. This model takes into account two routes of transmission: vertical and horizontal. This latter route integrates both the direct and indirect transmission. We define an explicit formula for the reproductive number [Formula: see text] and show by means of the Fatou's lemma that the disease-free equilibrium is globally asymptotically stable when [Formula: see text]. Furthermore, we find an explicit formula of the endemic equilibria and study its local stability as well as the uniform persistence of the disease when [Formula: see text]. Finally, a numerical scheme to solve the model is developed and some parameters of the model are estimated based on biological data. The numerical results illustrate the role of routes of transmission on the epidemic evolution.

Fatou's Lemma in Its Classical Form and Lebesgue's Convergence Theorems for Varying Measures with Applications to Markov Decision Processes

The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. In the present paper, we provide sufficient conditions when Fatou's lemma holds in its classical form for a sequence of weakly converging measures. The functions can take both positive and negative values. Similar results for sequences of setwise converging measures are also proved. We also put forward analogies of Lebesgue's and the monotone convergence theorems for sequences of weakly and setwise converging measures. The results obtained are used to prove broad sufficient conditions for the validity of optimality equations for average-cost Markov decision processes.

Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

This paper describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to the...

Fatou's lemma for weakly converging measures under the uniform integrability condition

В статье устанавливаются лемма Фату и теорема Лебега о мажорируемой сходимости для слабо сходящихся последовательностей конечных мер и равномерно интегрируемых по этим мерам функций. Заметка содержит также новые формулировки равномерной леммы Фату, равномерной теоремы Лебега о мажорируемой сходимости, теоремы Данфорда-Петтиса и фундаментальной теоремы о мерах Янга, основанные на эквивалентности условия равномерной интегрируемости и кажущегося более слабым условия асимптотической равномерной интегрируемости для последовательностей функций и конечных мер.

Transformation of the pseudo-integral and related convergence theorems

In this paper a transformation of the pseudo-integral into another pseudo-integral is considered. Using the transformed pseudo-integral, monotone convergence theorems for the pseudo-integral are obtained. Based on the reverse Fatou's lemma for the pseudo-integral, the dominated convergence-type theorem is proven.

Minimizers for nonconvex variational problems in the plane via convex/concave rearrangements

Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma. In this paper we point out that such an approach was considered in the late 1930's in a series of papers by E.J. McShane for problems satisfying the usual coercivity condition. In addition, we will update some hypotheses that McShane made by making use of a result due to T.S. Angell, concerning property (D) on the avoidance of the Lavrentiev phenomenon.

Existence of stationary bargaining equilibria

This paper addresses the question of existence of stationary Markov perfect equilibria in a class of dynamic games that includes many known bargaining models and models of coalition formation. General sufficient conditions for existence of equilibria are currently lacking in a number of interesting environments, e.g., models with non-convexities, consumption lower bounds, or an evolving state variable. The main result establishes existence of equilibrium under compactness and continuity conditions, without the structure of convexity or strict comprehensiveness used in the extant literature. The proof requires a precise selection of voting equilibria following different proposals using a generalization of Fatou's lemma.

Uniform Fatou's lemma

Fatou's lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou lemma, and it holds under the minor assumption that all the integrals in the inequality are well-defined. The uniform Fatou lemma improves the classic Fatou lemma in the following directions: the uniform Fatou lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou lemma may indeed provide a more accurate inequality than the classic Fatou lemma; (b) the uniform Fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence.

On the Brezis–Lieb lemma without pointwise convergence

Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of the gap between the integral for a sequence and the integral for its pointwise limit. This note studies the question if such gap can be evaluated when there is no a.e. convergence. In particular, it gives the same lower bound for the gap in L^p as the gap in the Brezis-Lieb lemma (including the case vector-valued functions) provided that p is greater or equal than 3 and the sequence converges both weakly and weakly in the sense of a duality map. It also shows that the statement is false if p<3. An application is given in form of a Brezis-Lieb lemma for gradients.

Fatou's Lemma for Weakly Converging Probabilities

Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This paper describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.

Weak sequential convergence in [formula omitted] and an exact version of Fatouʼs lemma

The class of nonatomic finite measure spaces with the saturation property, as developed in Maharam (1942) and Hoover–Keisler (1984), is characterized by the Fatou (and Lebesgue) property of a well-dominated sequence of multifunctions taking values in a Banach space. With multifunctions reduced to functions, this Fatou characterization also extends to a variant of the closure property found in optimal control theory. The results are developed through a considered overview of the relevant literature on the exact and approximate Fatou lemma phrased in terms of Bochner integration.

Minimal supersolutions of convex BSDEs

We study the nonlinear operator of mapping the terminal value $\xi$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.

Some properties and convergence theorems of set-valued Choquet integrals

The paper aims at discussing the set-valued Choquet integral, which is the integral of set-valued random variables with respect to capacities. We mainly present representation theorems of the set-valued random variable by using a sequence of Choquet integrable selections and then we investigate some properties of set-valued Choquet integrals, especially subadditive property and inequality of the metric of set-valued Choquet integrals. We also prove Fatou's Lemmas, Lebesgue dominated convergence theorem and monotone convergence theorem of set-valued Choquet integrals under the weaker conditions than that in previous works.

Liapunov-type inequality for universal integral

The Choquet integral and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. In this paper, previous results of Hong (Nonlinear Analysis 2011 74:7296–7303) are improved by relaxing some of their requirements. Carlson's, Sandor's, Bushell–Okrasinski's type inequalities and Fatou's lemma for universal integral are studied in a rather general form, thus generalizing some recent results. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.

A HÖLDER–YOUNG–LIEB INEQUALITY FOR NORMS OF GAUSSIAN WICK PRODUCTS

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite-dimensional framework.

Fatou’s lemma for Sugeno integral

Fatou’s lemma plays an important role in classical probability and measure theory. Non-additive measure is a generalization of additive probability measure. Sugeno’s integral is a useful tool in several theoretical and applied statistics which have been built on non-additive measure. In this paper, a Fatou-type lemma for Sugeno integral is shown. The studied inequality is based on the classical Fatou lemma for Lebesgue integral. To illustrate the proposed inequalities some examples are given.

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

AbstractWe investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping numberwhich is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants ofFσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.