Differential Inclusions Research Articles

Generalized Halanay inequality and its application to delay differential inclusions

In this work, by using a generalized Halanay inequality, a regular Lyapunov function and the theory of nonsmooth analysis, we investigate the uniformly ultimate boundedness of the zero solution for nonautonomous delay differential inclusions. The sufficient condition is presented in integral form, which is different from the ones given in some related literature. We also provide different characterizations of the sufficient conditions on the stability, the asymptotic stability, and the exponential stability of the zero solution for nonautonomous delay differential inclusions. Finally, one example is given to illustrate the effectiveness of the theoretical results obtained.

Differential equations with maximal monotone operators

The paper deals with multivalued differential equations in abstract spaces. Nonlocal conditions are assumed. The model includes an m-dissipative multioperator which generates an equicontinuous, not necessarily compact, semigroup. The regularity of the nonlinear term also depends on the Hausdorff measure of noncompactness. The existence of integral solutions is discussed, with a topological index argument. A transversality condition is required. The results are applied to a partial differential inclusion in a bounded domain in Rn with nonlocal integral conditions. The model also includes an m-dissipative but not necessarily compact semigroup generated by a suitable subdifferential operator.

A Distributed k-Winners-Take-All Model With Binary Consensus Protocols.

This article concentrates on solving the k -winners-take-all (k WTA) problem with large-scale inputs in a distributed setting. We propose a multiagent system with a relatively simple structure, in which each agent is equipped with a 1-D system and interacts with others via binary consensus protocols. That is, only the signs of the relative state information between neighbors are required. By virtue of differential inclusion theory, we prove that the system converges from arbitrary initial states. In addition, we derive the convergence rate as O(1/t) . Furthermore, in comparison to the existing models, we introduce a novel comparison filter to eliminate the resolution ratio requirement on the input signal, that is, the difference between the k th and (k+1) th largest inputs must be larger than a positive threshold. As a result, the proposed distributed k WTA model is capable of solving the k WTA problem, even when more than two elements of the input signal share the same value. Finally, we validate the effectiveness of the theoretical results through two simulation examples.

Fuzzy fixed point techniques for analyzing differential inclusions

Fuzzy fixed point techniques for analyzing differential inclusions

Socio‐Legal Production of the Tourist‐Seasonal Labourer for the Finnish Berry Industry

The article investigates the phenomenon of precarious labour within the Finnish wild berry industry, focusing on the socio‐legal dimensions that enable short‐term “just‐in‐time” migration, primarily from Thailand, for the berry season. Since the initial 2005 recruitment of Thai citizens to engage in forest berry picking for the Finnish berry industry, the industry has become heavily reliant on migrant labour. At the same time, the pickers’ situation exemplifies a case of unregulated labour, as pickers are categorised as a group outside of labour laws in Finland. By asking how this “non‐work”—berry picking without labour rights—has repeatedly been justified on a policy level, the article provides a case study that unpacks the creation of a racialised migrant labour force through a statecraft of differential inclusion, in an arrangement regarded to advance rural economies. Empirically, the article draws on an analysis of policy documents through which a particular kind of temporary migration corridor is administered.

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (Eα,β), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models.

New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities

The aim of this work is to study the stability for a class of fuzzy fractional differential variational inequalities modeled by variational inequalities coupled with fuzzy fractional differential inclusions. By introducing the parametric problem with the help of two parameters in the involved mapping and constraint set, we first establish an existence result for the parametric fuzzy fractional differential variational inequality (PFFDVI), and we also establish the compactness and continuous dependence on the initial value for the PFFDVI. Furthermore, when the parameters are perturbed, we obtain the closedness of solution mapping for the PFFDVI. Then we further investigate the upper semicontinuity of solution mapping for the PFFDVI under some mild conditions. Finally, in order to illustrate the obtained results, we give a numerical example at the end of this paper.

Reintroducing Time, Money and Constraints: Viability to Bridge the Economic and Monetary Theories

Why use a viabilistic approach to model economy? Because it is urgent to re-assess the modeling of the business cycle that integrates dynamic money creation and that has at its core the notion of constraints. The purpose of this paper is thus to propose an evolutionary economic model using a mathematical framework derived from set-valued analysis, differential inclusions and viability theory. As this approach is dynamical, the model will not use two concepts on which neo-classical economics are built i.e., (1) the existence of static repeated equilibria, (2) the optimization by an economic agent of a utility function. Furthermore, we will not use any self-organization process or “invisible hand”, as in our viabilist framework the economic viability will be maintained by explicit choices of the economic agents and of the lender of last resort over the budgetary and monetary rules to apply. Indeed, even simple constraints on the second-derivative of the means of payments already induce four-phased business cycles. Coupling these constraints over the creation of means of payment with economic evolutions of prices and commodities through a budgetary constraint allows for a single formalism, tackling jointly the question of the viability of economic and monetary evolutions.

On a class of evolution problems driven by maximal monotone operators with integral perturbation

The present paper is dedicated to the study of a first-order differential inclusion driven by time and state-dependent maximal monotone operators with integral perturbation, in the context of Hilbert spaces. Based on a fixed point method, we derive a new existence theorem for this class of differential inclusions. Then, we investigate an optimal control problem subject to such a class, by considering control maps acting in the state of the operators and the integral perturbation.

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Qualitative Study on ψ−Caputo fractional differential inclusion with non-local conditions and feedback control

Qualitative Study on ψ−Caputo fractional differential inclusion with non-local conditions and feedback control

On Diferential Inclusions Arising from Some Discontinuous Systems

The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known definitions of the solution of discontinuous systems, then the motion of an object while being on some surface can be described in terms of differential inclusions. With the help of the previously developed apparatus for solving differential inclusions, a method is constructed for finding the trajectories of a system moving in a such a mode. Since some of frequently used discontinuous controls contain nonsmooth functions of phase variables, the paper pays special attention to study the differential properties of such systems. At the end of the paper controls of a slightly different, in contrast to the classical, type are considered which have useful differential properties, and a method is constructed for solving systems with such controls considered both before hitting the required surface and moving in its vicinity.

Hilfer-Katugampola fractional stochastic differential inclusions with Clarke sub-differential

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of stochastic Hilfer-Katugampola fractional differential inclusions (SHKFDIs) with non-instantaneous impulsive (NIIs) that is strengthened by Brownian motion (BM) and the Clarke sub-differential. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using stochastic analysis, the Clarke sub-differential's characteristics, and the multi-valued fixed point theorem. Subsequently, by employing Balder's theorem, the existence of optimal control pairs for the considered system is investigated. Eventually, an example is provided to validate the obtained results.

On an Initial Value Problem for Nonconvex-Valued Fractional Differential Inclusions in a Banach Space

On an Initial Value Problem for Nonconvex-Valued Fractional Differential Inclusions in a Banach Space

Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions

Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions

On Impulsive Fractional Differential Inclusions with a Nonconvex-valued Multimap in Banach Spaces

On Impulsive Fractional Differential Inclusions with a Nonconvex-valued Multimap in Banach Spaces

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.

On an m-dimensional system of quantum inclusions by a new computational approach and heatmap

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.

Application of the Mittag-Leffler kernel in stochastic differential systems for approximating the controllability of nonlocal fractional derivatives

This research focuses on utilizing the Mittag-Leffler kernel within stochastic differential systems to estimate the controllability of nonlocal Atangana–Baleanu fractional derivatives. By assuming the automatic control of the corresponding linear system, a novel set of necessary and sufficient conditions for the approximate controllability of the fractional stochastic differential inclusions of Atangana–Baleanu is established. Furthermore, the study explores the approximate controllability of the proposed system with infinite delay. The investigation relies on the fixed-point theorem for multivalued operators and fractional calculus to derive these outcomes. Lastly, an illustrative example is provided to highlight the practical implications of the research findings.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems

We establish that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of a discrete-time stochastic system as well as the set of their mathematical expectations converge to a non-random, convex and compact set. We apply these results to the averaging a hybrid system with a slow continuous-time component and a fast discrete-time component. It is shown that the solutions of the hybrid system are approximated by the solutions of a differential inclusion. The novelty of our results is that we allow the state-control space of the fast component to be non-denumerable.