Measurable Selection Theorem Research Articles

The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions

This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results.

Quasi-projective and finite-time synchronization of fractional-order memristive complex-valued delay neural networks via hybrid control

<abstract><p>We focused on the quasi-projective synchronization (QPS) and finite-time synchronization (FNTS) for a class of fractional-order memristive complex-valued delay neural networks (FOMCVDNNs). Rather than decomposing the complex-valued system into its real and imaginary components, we adopted a more streamlined approach by introducing a lemma associated with the complex-valued sign function. This innovative technique enabled us to design a simpler discontinuous controller. Then, based on the finite-time Lemma, measurable selection theorem, Lyapunov function theory, properties of the Mittag-Leffler function, and the fractional-order Razumikhin theorem, various substantial results were derived using a novel hybrid control scheme. In conclusion, we presented numerical simulations to illustrate the practical effectiveness of our theoretical findings.</p></abstract>

On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities

Abstract A new class of stochastic differential equations (SDEs) is introduced in this article, which is driven by the generalized stochastic mixed variational inequality (GS-MVI). First, the property of the solution sets of the GS-MVI is proved by Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and Aumann’s measurable selection theorem. Next, we obtain the Carathéodory property of the solution set, with which the discussed SDEs can be transformed to stochastic differential inclusions (SDIs). The solution set of the proposed SDEs is proved to be nonempty through the existence of the solutions of the corresponding SDIs by the tools of fixed point theorem.

A Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.

Existence of optimal control for an integro‐differential Bolza problem

SummaryIn the article, we derive an existence theorem for a Bolza problem described by a nonlinear integro‐differential system of Volterra type. We use an approach based on the lower closure theorem for orientor fields due to Cesari and measurable selection theorem of Fillipov type due to Rockafellar. Paper is a continuation of [D. Idczak, S. Walczak, Necessary optimality conditions for an integro‐differential Bolza problem via Dubovitskii‐Miljutin method, DCDS‐B 24 (5) (2019), 2281‐2292; DOI:10.3934/dcdsb.2019095].

L0-Convex Compactness and Random Normal Structure in L0 (F, B)

Let (B, ∥ · ∥) be a Banach space, (Ω, F, P) a probability space, and L0 (F, B) the set of equivalence classes of strong random elements (or strongly measurable functions) from (Ω, F, P) to (B, ∥ · ∥). It is well known that L0 (F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L0 (F, V) the set of equivalence classes of strong random elements from (Ω, F, P) to V. The central purpose of this article is to prove the following two results: (1) L0 (F, V) is L0-convexly compact if and only if V is weakly compact; (2) L0 (F, V) has random normal structure if V is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.

On a class of generalized stochastic Browder mixed variational inequalities

In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.

Finite-Time Cluster Synchronization of Lur’e Networks: A Nonsmooth Approach

This paper is devoted to the finite-time cluster synchronization issue of nonlinearly coupled complex networks which consist of discontinuous Lur’e systems. On the basis of the definition of Filippov regularization process and the measurable selection theorem, the discontinuously nonlinear function is mapped into a function-valued set, then a measurable function is accordingly selected from the Filippov set to ensure the existence of the solution for the discontinuous system. By designing the finite-time pinning controller, some sufficient conditions are obtained for cluster synchronization of the identical and nonidentical Lur’e networks, respectively. In addition, the settling time for achieving the cluster synchronization is estimated by applying the finite-time stability theory. And finally, a numerical example is presented to illustrate the validity of theoretical analysis.

Conditional convex orders and measurable martingale couplings

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.

Measurable solutions for stochastic evolution equations without uniqueness

A method for obtaining measurable solutions to stochastic evolution equations in which there is no uniqueness for the corresponding non-stochastic equation is presented. It involves a technique based on a measurable selection theorem for set-valued functions. No assumptions are needed on the underlying probability space. An application is given to the stochastic Navier–Stokes problem in arbitrary dimensions. We also show the existence of measurable solutions to stochastic ordinary differential equations in which there is no uniqueness. A finite-dimensional generalization is given to adapted solutions in the case of a normal filtration and path uniqueness.

Countable Random Sets: Uniqueness in Law and Constructiveness

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: First, the study of generators for \sigma-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We will prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.

Stochastic Monge–Kantorovich problem and its duality

In this article, we prove the existence of a stochastic optimal transference plan for a stochastic Monge–Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels is also discussed.

On measurable minimax selectors

In this note we consider the upper value of a zero-sum game with payoff function depending on a state variable. We provide a new and much simpler proof of a measurable minimax selection theorem established 25 years ago by the author in Nowak (1985) [19]. A discussion of the basic assumptions and relations with the literature on stochastic games and (minimax) control models is also included.

Dynamic programming for stochastic target problems and geometric flows

Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed.

Equilibrium points of random generalized games

In this paper, the concepts of random maximal elements, random equilibria and random generalized games are described. Secondly by measurable selection theorem, some existence theorems of random maximal elements for Lc‐majorized correspondences are obtained. Then we prove existence theorems of random equilibria for non‐compact one‐person random games. Finally, a random equilibrium existence theorem for non‐compact random generalized games (resp., random abstract economics) in topological vector spaces and a random equilibrium existence theorem of non‐compact random games in locally convex topological vector spaces in which the constraint mappings are lower semicontinuous with countable number of players (resp., agents) are given. Our results are stochastic versions of corresponding results in the recent literatures.

Holomorphic Processes in Banach Spaces and Banach Algebras

The main result is that if F is an analytic multifunction and B t is a complex Brownian motion, then F( B t ) is a subholomorphic process. It has previously been shown that such processes enjoy many interesting sample-path properties. As special cases of the theorem above, we recover f holomorphic ⇒ f(B t) is a local conformal martingale, φ subharmonic ⇒ φ(B t) is a local submartingale. We also prove a stochastic form of Radó′s theorem, and a holomorphic selection theorem for convex-valued subholomorphic processes of a nature quite different from the usual type of measurable selection theorem.

On Deterministic and Random Fixed Points

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for $F(\omega , \cdot )$ produces a random fixed point theorem for $F$ provided the $\sigma$-algebra $\Sigma$ for $\Omega$ is a Suslin family and $F$ has a measurable graph (in particular, when $F$ is random continuous with closed values and $X$ is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

On deterministic and random fixed points

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for F ( ω , ⋅ ) F(\omega , \cdot ) produces a random fixed point theorem for F F provided the σ \sigma -algebra Σ \Sigma for Ω \Omega is a Suslin family and F F has a measurable graph (in particular, when F F is random continuous with closed values and X X is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time

Dalang, Morton and Willinger (1990) have proved a beautiful version of the Fundamental Theorem of Asset Pricing which pertains to the case of finite discrete time: In this case the absence of arbitrage opportunities already characterizes the existence of an equivalent martingale measure. The purpose of this paper is to give an elementary proof of this important theorem which relies only on orthogonality arguments. In contrast, the original proof of Dalang . uses heavy functional analytic machinery, in particular measurable selection and measure-decomposition theorems. We feel that the theorem (as well as its proof) should be accessible to a wider public and we therefore made an effort to keep the arguments as selfcontained as possible. In a final chapter we review and prove the necessary tools for our presentation of the theorem.

Monotone enclosure for elliptic and parabolic sytems with nonmonotone nonlinearities

Monotone iteration techniques for weakly coupled systems having nonmonotone nonlinearities have been developed recently by M. Khavanin and V. Lakshmikantham and by S. Carl. In this paper an existence and enclosing theorem is given in terms of upper and lower solutions and a new monotone iteration technique, which in a natural way extends the monotone iteration technique used in the “monotone case” to the “nonmonotone case” considered here, is proposed. The maximum and minimum functions that occur in the iteration scheme allow one to work under low regularity conditions; particularly, no local Lipschitz conditions on the nonlinearities are required for the iteration process. A crucial role in the analysis is played by a measurable selection theorem which is usually known from optimization theory. Furthermore, conditions which ensure that the iteration process converges to the (uniquely) defined solution are given.