Differential Inclusions Research Articles

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

On some spectral properties of nonlocal boundary-value problems for nonlinear differential inclusion

UDC 517.9 We study the solutions to the Sturm–Liouville boundary-value problem for a nonlinear differential inclusion with nonlocal conditions. The maximal and minimal solutions are demonstrated. The analysis of eigenvalues and eigenfunctions is performed. It is discussed whether multiple solutions may exist for the inhomogeneous Sturm–Liouville boundary-value problem for differential equation with nonlocal conditions.

Unique continuation for differential inclusions

We consider the following question arising in the theory of differential inclusions: Given an elliptic set \Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of \Gamma and touches \Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where \Gamma \subset \mathbb{R}^{2\times 2} is an elliptic, smooth, closed curve. More precisely, we prove that the distance of \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

Erratum: Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions

Erratum: Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions

Fast Convex Optimization via Time Scale and Averaging of the Steepest Descent

In a Hilbert setting, we develop a gradient-based dynamic approach for fast solving convex optimization problems. By applying time scaling, averaging, and perturbation techniques to the continuous steepest descent (SD), we obtain high-resolution ordinary differential equations of the Nesterov and Ravine methods. These dynamics involve asymptotically vanishing viscous damping and Hessian-driven damping (either in explicit or implicit form). Mathematical analysis does not require developing a Lyapunov analysis for inertial systems. We simply exploit classical convergence results for SD and its external perturbation version, then use tools of differential and integral calculus, including Jensen’s inequality. The method is flexible, and by way of illustration, we show how it applies starting from other important dynamics in optimization. We consider the case in which the initial dynamic is the regularized Newton method, then the case in which the starting dynamic is the differential inclusion associated with a convex lower semicontinuous potential, and finally we show that the technique can be naturally extended to the case of a monotone cocoercive operator. Our approach leads to parallel algorithmic results, which we study in the case of fast gradient and proximal algorithms. Our averaging technique shows new links between the Nesterov and Ravine methods. Funding: The research of R.I. Boţ and D.-K. Nguyen was supported by the Austrian Science Fund (FWF), projects W 1260 and P 34922-N.

A note on the approximate controllability of damped second-order differential inclusions with state-dependent delay

In this paper, we use Bohnenblust–Karlin's fixed point theorem and the theory of strongly continuous cosine families of bounded linear operators to study the approximate controllability of damped second-order differential inclusions with state-dependent delay in Banach spaces. In particular, we obtain a new set of sufficient conditions for the approximate controllability of damped second-order differential inclusions with state-dependent delay under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate our main results.

Recursive utility optimization problem described by forward–backward stochastic differential inclusions

The main result of the paper deals with the existence of a solution of a recursive utility optimization problem described by systems of the forward–backward stochastic differential inclusion of the form: X t − X s ∈ ∫ s t F ( τ , X τ ) d τ + ∫ s t ( G ∘ X ) d B τ and Y t ∈ E [ H ( X T ) + ∫ t T Φ ( τ , X τ , Y τ ) d τ | F t ] , considered in the author paper [Kisielewicz. Weak compactness of weak solutions to forward–backward stochastic differential inclusions. Stoch Anal Appl. 2024;42(2):306–326], where the existence of weak solutions and weak compactness in distribution of the set W ( F , H , Φ , G , μ ) of all its weak solutions have been proved.

Continuous selections of solution sets of semilinear differential inclusions of fractional order

continuous selections of solution sets of semilinear differential inclusions of fractional order

Complete Stability of Neural Networks With Extended Memristors.

The article considers a large class of delayed neural networks (NNs) with extended memristors obeying the Stanford model. This is a widely used and popular model that accurately describes the switching dynamics of real nonvolatile memristor devices implemented in nanotechnology. The article studies via the Lyapunov method complete stability (CS), i.e., convergence of trajectories in the presence of multiple equilibrium points (EPs), for delayed NNs with Stanford memristors. The obtained conditions for CS are robust with respect to variations of the interconnections and they hold for any value of the concentrated delay. Moreover, they can be checked either numerically, via a linear matrix inequality (LMI), or analytically, via the concept of Lyapunov diagonally stable (LDS) matrices. The conditions ensure that at the end of the transient capacitor voltages and NN power vanish. In turn, this leads to advantages in terms of power consumption. This notwithstanding, the nonvolatile memristors can retain the result of computation in accordance with the in-memory computing principle. The results are verified and illustrated via numerical simulations. From a methodological viewpoint, the article faces new challenges to prove CS since due to the presence of nonvolatile memristors the NNs possess a continuum of nonisolated EPs. Also, for physical reasons, the memristor state variables are constrained to lie in some given intervals so that the dynamics of the NNs need to be modeled via a class of differential inclusions named differential variational inequalities.

Optimal Control of a Roll-to-Roll Dry Transfer Process With Bounded Dynamics Convexification

Abstract For efficient roll-to-roll (R2R) production of flexible electronic components, a precise R2R transfer peeling process is essential, requiring accurate modeling and control. This paper introduces a novel approach to confining the dynamics of a nonlinear R2R mechanical peeling system within a convex set known as a norm-bounded linear differential inclusion (NLDI). This method utilizes constraints on uncertain system variables to create a tighter NLDI representation compared to other convexification techniques. Moreover, it offers drastically reduced computational cost compared to previous methods applied to convexify the R2R peeling system. The NLDI is employed to generate an H∞-optimal controller for the R2R peeling system, and both simulations and experiments demonstrate better dynamic performance compared to other controllers for R2R transfer.

Discussion on the Approximate Controllability of Second-Order Neutral Differential Inclusions with Damping

Discussion on the Approximate Controllability of Second-Order Neutral Differential Inclusions with Damping

Properties of the controllability set of one class of differential inclusions

Properties of the controllability set of one class of differential inclusions

Topological structure of the solution sets to neutral evolution inclusions driven by measures

Abstract This study is concerned with topological structure of the solution sets to evolution inclusions of neutral type involving measures on compact intervals. By using Górniewicz-Lassonde fixed-point theorem, the existence of solutions and the compactness of solution sets for neutral measure differential inclusions are obtained. Second, based on the R δ {R}_{\delta } -structure equivalence theorem, by constructing a continuous function that can make the solution set homotopic at a single point, the R δ {R}_{\delta } -type structure of the solution sets of this kind of differential inclusion is obtained.

Towards a sociolinguistics of deglobalization

Abstract This article advocates for a ‘sociolinguistics of deglobalization’ that focuses on the sociolinguistic impacts of major shifts in the world system that have taken place since 2008 and have become particularly salient amidst the covid pandemic. Drawing on case studies of China, Indonesia, Ethiopia, and Nigeria, we describe the period after 2008 as an emerging post-neoliberal epoch characterized by the intensifying of state power, constrained resistance, differential inclusion, and organized abandonment. Our article explores the theoretical, methodological, and ethical challenges of researching the sociolinguistic impacts of these developments. We also argue that our discipline's dominant approach to global-scale analysis—the sociolinguistics of globalization—has impeded our ability to perceive the emerging dynamics of deglobalization. Our call for a sociolinguistics of deglobalization is offered as both a provocation and invitation to our discipline to engage with the rapidly changing nature of world politics. (Deglobalization, sovereignty, neoliberalism, coloniality, securitization)*

Differential inclusion of NEB exons 143 and 144 provides insight into NEB-related myopathy variant interpretation and disease manifestation

Biallelic pathogenic variants in the gene encoding nebulin (NEB) are a known cause of congenital myopathy. We present two brothers with congenital myopathy and compound heterozygous variants (NC_000002.12:g.151692086G>T; NM_001271208.2: c.2079C>A; p.(Cys693Ter) and NC_000002.12:g.151533439T>C; NM_001271208.2:c.21522+3A>G ) in NEB. Transcriptomic sequencing on affected individual muscle revealed that the extended splice variant c.21522+3A>G causes exon 144 skipping. Nebulin isoforms containing exon 144 are known to be mutually exclusive with isoforms containing exon 143, and these isoforms are differentially expressed during development and in adult skeletal muscles. Affected individuals MRI patterns of muscle involvement were compared to the known pattern of relative abundance of these two isoforms in muscle. We propose that the pattern of muscle involvement in these affected individuals better fits the distribution of exon 144-containing isoforms in muscle than with previously published MRI findings in NEB-related disease due to other variants. Our report introduces disease pathogenesis and manifestation as a result of alteration of isoform distributions in muscle.

On set-valued fixed points, their geometry and solution of boundary value problem for a differential inclusion

On set-valued fixed points, their geometry and solution of boundary value problem for a differential inclusion

Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.

New exploration on approximate controllability of nondensely defined Hilfer neutral-type delayed nonlinear differential inclusion system with non-instantaneous impulse

This manuscript aims to investigate the existence and approximate controllability results for integral solution of nondensely defined Hilfer neutral differential inclusion system of Sobolev-type with infinite delay and non-instantaneous impulse in a Hilbert space. By utilizing the semigroup theory of bounded linear operators, fixed point technique, fractional calculus, and the theory of multivalued maps, the principal discussions are demonstrated. The sufficient conditions for the existence and approximate controllability are demonstrated under the consideration that the linear operator A is defined nondensely and satisfies the Hille–Yosida condition. Finally, an example is given to support the validity of the study.

Multidimensional Fractional Calculus: Theory and Applications

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications.