Set-valued Mappings Research Articles

Synchronization robustness analysis of memristive-based neural networks with deviating arguments and stochastic perturbations

<abstract><p>In this article, we investigate the robustness of memristive-based neural networks (MNNs) with deviating arguments (DAs) and stochastic perturbations (SPs). Based on the set-valued mapping method, differential inclusion theory and Gronwall inequalities, we derive the upper bounds for the width of DAs and the intensity of SPs. When the DAs and SPs are smaller than these upper bounds, the MNNs maintains exponential synchronization. Finally, several specific simulation examples demonstrate the effectiveness of the results.</p></abstract>

Answers to questions on Kannan's fixed point theorem in strong $ b $-metric spaces

<abstract><p>Our purpose of this paper is to answer several open questions posed by Doan (AIMS Math., 6 (2021), 7895–7908). First, we present two fixed point theorems, which are positive answers to Doan's questions. Second, we establish a new type of Riech's fixed point theorem to improve a result of Doan. Finally, we offer a straightforward example illustrating that a set-valued mapping satisfying the conditions of our fixed point theorem may has more than one fixed point.</p></abstract>

New results on finite-time projective synchronization for memristor-based hybrid delayed BAM neural networks with applications to DNA image encryption

<abstract> <p>With the popularization of digital image technology, image information has inevitably developed to involved the disclosure of personal privacy; in this study, a color image encryption algorithm was designed to encrypt and decrypt images by using chaotic sequences of a class of memristor-based hybrid delayed bidirectional associative memory neural networks (MHDBAMNNs) to protect images from illegal acquisition and use. Additionally, the discontinuity problem of the right-hand side of the Filippov system due to the hopping property of the memristor has been treated by using differential inclusion and set-valued mapping theories, and a sufficient criterion for guaranteeing the synchronization of finite-time projections derived based on the drive-response concept, Lyppunov stability theorem, and inequality technique. To improve the security performance, a color image encryption algorithm based on a combination of Chen's hyperchaotic system and a DNA codec operation was adopted, also, the robustness and validity of our proposed approach was demonstrated through image performance analysis. Furthermore, the potential application of the model in secure transmission has been explored.</p> </abstract>

Higher-order Mond-Weir duality of set-valued fractional minimax programming problems

In this paper, we consider a set-valued fractional minimax programming problem (abbreviated as SVFMPP) (MFP), in which both the objective and constraint maps are set-valued. We use the concept of higher-order ?-cone arcwisely connectivity, introduced by Das [1], as a generalization of higher-order cone arcwisely connected setvalued maps. We explore the higher-order Mond-Weir (MWD) form of duality based on the supposition of higher-order ?-cone arcwisely connectivity and prove the associated higher-order converse, strong, and weak theorems of duality between the primary (MFP) and the analogous dual problem (MWD).

On a Hilfer type fractional integro-differential inclusion

A boundary value problem associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. The existence of solutions is established in the case when the set-valued map has nonconvex values.

On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t,x,x˙)∋0 with the constraint x˙(t)∈B(t), t∈[a,b], on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×Rn×Rn×Rm⇉ is superpositionally measurable, and B:[a,b]⇉Rn is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form f(t,x,x˙,u)=0, x˙(t)∈B(t), u(t)∈U(t,x,x˙), t∈[a,b].

Interior of the Integral of a Set-Valued Mapping and Problems with a Linear Control System

The dependence of the radius of a ball centered at zero inscribed in the values of the integral of a set-valued mapping on the upper integration limit is studied. For some types of integrals, exact asymptotics of the radius with respect to the upper limit are found when the upper limit tends to zero. Examples of finding this radius are considered. The results obtained are used to derive new sufficient conditions for the uniformly continuous dependence of the minimum time and solution-point in the linear minimum time control problem on the initial data. We also consider applications in some algorithms with a reachability set of a linear control system

On Almost Everywhere K-Additive Set-Valued Maps

Abstract Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2 Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ 1 in X and ℐ 2 in X 2, we prove that if F is ℐ 2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2 Y \{∅} such that F = G ℐ 1-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].

Role Of ψ-Operator in the Study of Minimal Open Sets

The operator ψ has been introduced as an associated set-valued set function. Although it has importance for the study of minimal open sets as well as minimal I-open sets. As a result of this study, we introduce minimal I^*-open sets . In this study, several characterizations of minimal I^*-open sets are also investigated. This study also discusses the role of minimal I^*-open sets in the *-locally finite spaces. In an aspect of topological invariant, the homeomorphic images of minimal I^*-open set has been discussed here.

Quantized iterative learning control for impulsive differential inclusion systems with data dropouts

This paper studies the quantized iterative learning control with encoding–decoding mechanism of a class of impulsive differential inclusion systems with random data dropouts. First, the set-valued mappings in the differential inclusion systems are transformed into single-valued mappings by using the Steiner-type selector. Then, a learning algorithm based on the intermittent update principle is designed to address the data asynchronism problem caused by two-sided data dropouts. If the data are successfully transmitted at the actuator and measurement sides, then the control input is effectively updated. Furthermore, a suitable scaling sequence is introduced to ensure the system output to achieve zero-error tracking performance for a desired trajectory. An upper bound of the quantization level is determined such that the quantization error is always bounded. The results show that the quantization method reduces the burden of network communication at the cost of increasing the amount of computation, and the learning algorithm does not require the data dropouts to satisfy a certain probability distribution. Finally, the effectiveness of the learning algorithm is verified by numerical simulations of the switched reluctance motor system.

Further remarks on local K-boundedness of K-subadditive set-valued maps

AbstractLet X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map $$F:X\rightarrow \mathcal{C}\mathcal{C}(Y)$$ F : X → C C ( Y ) , for which the functionals $$f_{y^*}(x)=\inf y^*(F(x))$$ f y ∗ ( x ) = inf y ∗ ( F ( x ) ) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional $$y^*$$ y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001).

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps.

Metric regularity properties of monotone mappings

The theory of metric regularity deals with properties of set-valued mappings that provide estimates useful in solving inverse problems and generalized equations. Maximal monotone mappings, which dominate applications related to convex optimization, have valuable special features in this respect that have not previously been recorded. Here it is shown that the property of strong metric subregularity is generic in an almost everywhere sense. Metric regularity not only coincides with strong metric regularity but also implies local single-valuedness of the inverse, rather than just of a graphical localization of the inverse. Consequences are given for the solution mapping associated with a monotone generalized equation.

Some notes on the pan-integrals of set-valued functions

This note answers an open problem which was proposed in the paper Kang et al. (2023) [9]. By means of subadditivity of fuzzy measure we show that the set-valued pan-integral based on a fuzzy measure and the operations pair (+,⋅) possesses the usual form of linearity. Thus, this open problem has a positive solution, and the previous result concerning the pseudo-linearity of set-valued pan-integrals is improved. We provide an example to show that the condition of subadditivity of fuzzy measure is necessary to the linearity of set-valued pan-integrals.The pan-integral of set-valued functions w.r.t. a fuzzy measure and pan-operations pair (⊕,⊗) is introduced in a natural way. It covers the previous set-valued pan-integral based on a fuzzy measure and the arithmetic operations pair (+,⋅), and the set-valued Sugeno integral w.r.t. a fuzzy measure and the logical operations pair (∨,∧).

Conic cancellation laws and some applications in set optimization

We discuss, on finite and infinite dimensional normed vector spaces, some versions of Rådström cancellation law (or lemma) that are suited for applications to set optimization problems. In this sense, we call our results ‘conic’ variants of the celebrated result of Rådström, since they involve the presence of an ordering cone on the underlying space. Several adaptations to this context of some topological properties of sets are studied and some applications to subdifferential calculus associated to set-valued maps and to necessary optimality conditions for constrained set optimization problems are given. Finally, a stability problem is considered.

Existence of solutions for polyhedral convex set optimization problems

Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear programs and multiple objective linear programs provide proper subclasses. In this article, we choose a solution concept for arbitrary polyhedral convex set optimization problems out of several alternatives, show existence of solutions and characterize the existence of solutions in different ways. Two known results are obtained as particular cases, both with proofs being easier than the original ones: The existence of solutions of bounded polyhedral convex set optimization problems and a characterization of the existence of solutions of vector linear programs.

Finite-Time and Fixed-Time Synchronization of Delayed Memristive Neural Networks via Adaptive Aperiodically Intermittent Adjustment Strategy.

This article investigates the finite-time and fixed-time synchronization for memristive neural networks (MNNs) with mixed time-varying delays under the adaptive aperiodically intermittent adjustment strategy. Different from previous works, this article first employs the aperiodically intermittent adjustment feedback control and adaptive control to drive the MNNs to achieve synchronization in finite time and fixed time. First of all, according to the theories of set-valued mappings and differential inclusions, the error MNNs is derived, and its finite-time and fixed-time stability problems are discussed by applying the Lyapunov function method and some LMI techniques. Moreover, by meticulously designing an effective aperiodically intermittent adjustment with adaptive updating law, sufficient conditions that guarantee the finite-time and fixed-time synchronization of the drive-response MNNs are obtained, and the settling time is explicitly estimated. Finally, three numerical examples are provided to illustrate the validity of the obtained theoretical results.

A Comprehensive Study on Advancement in Hybrid Contraction and Graphical Analysis of £-Fuzzy Fixed Points with Application

Hybrid contractions serve as a flexible and versatile framework for establishing fixed-point Theorems and analyzing the convergence of iterative algorithms. This paper demonstrates the adapted form of the admissible hybrid fuzzy Z-contraction in the perspective of £-fuzzy set-valued maps for extended ♭-metric spaces. Sufficient criteria for obtaining £-fuzzy fixed points for this contraction have been established. In addition, the hypotheses of its main result are endorsed by some nontrivial supportive examples featuring graphical illustrations. Consequently, the concept of graphical extended ♭-metric spaces is introduced and a £-fuzzy fixed point result in the context of newly defined space is derived. Illustrative examples, incorporating relevant graphs, are provided with the support of a computer simulation to validate the established results, enhancing the understanding of the underlying notions and investigations. The concepts presented here not only considerably improve, enrich, and extend a number of well-known pre-existing fixed-point results but also assemble and merge several ones in the corresponding domain.

Some results on joint continuity of two variable set-valued mappings

In this paper, we will show that under certain circumstances, a two variable set-valued mapping F:X×Y→2Z is jointly continuous on a set of the form D×{y}, where D is a dense Gδ subset of X and y∈Y. Among some other results, we will show that if X is a Baire space, Y a topological space and Z is a Tychonoff locally weakly k-developable space, then every set-valued mapping F from X×Y into compact subsets of Z, is Vietoris continuous at each point of X×{y0}, where y0∈Y is a W-point, provided that all its x-sections Fx are Vietoris continuous and all its y-sections Fy are Vietoris quasi-continuous. Our results improve some old related results in the literature.

Optimality Conditions for Approximate Solutions of Set Optimization Problems with the Minkowski Difference

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.