Set-valued Case Research Articles

Some Properties on Sensitivity, Transitivity and Mixing of Set-Valued Dynamical Systems

In this paper, we study the dynamical properties of set-valued dynamical systems. Specifically, we focus on the sensitivity, transitivity and mixing of set-valued dynamical systems. Under the setting of set-valued case, we define sensitivity and investigate its properties. We also study the transitivity and mixing of set-valued dynamical systems that have been defined. We show that both transitivity and mixing are invariant under topological conjugacy. We also discuss some implication results on the product set-valued function constructed from two different set-valued functions equipped with various transitivity and mixing conditions.

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

Jensen's inequalities for set-valued and fuzzy set-valued functions

Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Štrboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries.

Fractional-like Hukuhara derivatives in the theory of set-valued differential equations

Abstract In this paper a fractional-like Hukuhara-type derivative is introduced for a set of equations. Connections between the new defined notion and the classical Hukuhara derivative are established and, in addition, some applications to the theory of set-valued differential equations are discussed. Namely, for a family of equations with fractional-like Hukuhara-type derivatives of the set of states: (a) a version of the comparison principle is proposed; (b) local existence conditions are derived; (c) an estimate of the deviation of the approximate solution from the exact one is given. With this research we extend the theory of fractional-like differential equations to the set-valued case.

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.

Dynamical properties of shift maps on inverse limits with a set valued function

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

Set-valued and interval-valued stationary time series

Stationarity is a key tool in classical time series. In order to analyze the set-valued time series, it must be extended to the set-valued case. In this paper, stationary set-valued time series is defined via D p metric of set-valued random variables. Then, estimation methods of expectation and auto-covariance function of stationary set-valued time series are proposed. Unbiasedness and consistency of the expectation estimator and asymptotic unbiasedness of the auto-covariance function estimator are justified. After that, a special case of the set-valued time series, known as interval-valued time series, is considered. Two forecast methods of the stationary interval-valued time series are explicitly presented. Furthermore, the interval-valued time series is contextualized in the Box–Jenkins framework: an interval-valued autoregression model, along with its parameter estimation method, is introduced. Finally, experiments on both simulated and real data are presented to justify the efficiency of the parameters estimation method and the availability of the proposed model.

Some new fixed point theorems for single and set-valued admissible mappings in Menger PM-spaces

In this paper, the new concepts of \(\alpha \)-admissible mappings with respect to \(\eta \) in single-valued case and \(\alpha _{*}\)-admissible mappings with respect to \(\eta _{*}\) in set-valued case in Menger PM-spaces are introduced, and some new fixed point results for both single-valued and set-valued mappings under certain contractive conditions are studied, which extend and generalize some results in corresponding literatures. Some examples are provided to illustrate the validity of our main results, and an application is also given to discuss the existence of solution for Volterra type integral equations.

A Characterization Theorem for Aumann Integrals

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be written as the Aumann integral with respect to some $\sigma$-finite measure. These conditions include the analog of the conlinearity and monotone convergence properties of the classical Daniell-Stone theorem for the Lebesgue integral, and three geometric properties that are peculiar to the set-valued case as they are redundant in the one-dimensional setting.

Fixed points of set-valued Caristi-type mappings on semi-metric spaces via partial order relations

Abstract In this paper, we present some fixed-point theorems that are related to a set-valued Caristi-type mapping. The main results extend the recent work which was presented by Jiang and Li (Fixed Point Theory Appl. 2013:74, 2013) from a single-valued setting to a set-valued case. Further, the presented results also improve essentially many results that have appeared, because we have removed some conditions from the auxiliary function. Meanwhile, we give some partial answers to an important problem which was raised by Kirk (Colloq. Math. 36:81-86, 1976). MSC:47H10, 37C25.

Set-valued average value at risk and its computation

New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first ’regulator’ version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both versions over finite probability spaces. Several examples illustrate various features of the theoretical constructions.

Set-Valued Lebesgue and Riesz Type Theorems

Abstract In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions

Set-valued functions are an important mathematical notion and play a crucial role in several practical areas. At the same time, pseudo-analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations. This paper focuses on pseudo-integration of set-valued functions, which is generalization of Aumann's research, and corresponding extensions of the Jensen and Chebyshev integral inequalities to the set-valued case. Since the integral inequalities in question are widely used in various aspects of mathematics, the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set-valued functions with pseudo-analysis.

Set-valued stochastic integral equations driven by martingales

We consider a notion of set-valued stochastic Lebesgue–Stieltjes trajectory integral and a notion of set-valued stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of set-valued stochastic integral equations. The existence and uniqueness of the solution to such the equations is proven. As a generalization of set-valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales.

Pseudo-Convergences of Sequences of Measurable Functions on Monotone Multimeasure Spaces

In this paper, we study different types of pseudo-convergences of se- quences of measurable functions with respect to set-valued non-additive monotonic set functions and we establish some pseudo-versions of Egoroff theorem in the set-valued case. We also characterize important structural properties of monotone multimeasures. Mathematics Subject Classication

Continuity properties and Alexandroff theorem in Vietoris topology

In several previous papers, we introduced and studied various continuity properties in the set-valued case. Precisely, we established results concerning increasing/decreasing convergence, exhaustivity, order continuity and regularity in Hausdorff topology for closed-set-valued monotone set multifunctions on a real normed space.In this paper, taking as starting point of a previous work, we introduce and study from another point of view, corresponding continuity properties in Vietoris topology for more general monotone set multifunctions on a Hausdorff linear topological space. Thus, we obtain generalizations of some previous results, which in turn, generalize to the set-valued case, well known results from classical fuzzy (i.e., monotone) measure theory. An Alexandroff theorem in Vietoris topology and its converse are also established.

Optimality and Duality for Non-Lipschitz Multiobjective Optimization Problems

Nowadays set-valued optimization means set-valued analysis and its application to optimization, and it is an extension of continuous optimization to the set-valued case. In this research area, one investigates optimization problems with constraints and/or an objective function described by set-valued maps, or investigations in set-valued analysis are applied to standard optimization problems. In this article, we are concerned with a set-valued optimization problem (P). Using a notion of approximation derived from Jourani and Thibault, we give necessary and sufficient optimality conditions for (P). Based on necessary optimality conditions given by Amahroq and Gadhi [2] (see Theorem 8), our approach consists of formulating the Mond–Weir dual problem (D) and establishing duality theorems for (P) and (D) without any constraint qualification.

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.