Class Of Inclusions Research Articles

Primal-Dual Extrapolation Methods for Monotone Inclusions Under Local Lipschitz Continuity

In this paper, we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone, whereas the other is locally Lipschitz continuous. We propose primal-dual (PD) extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of [Formula: see text] and [Formula: see text], measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an ε-residual solution of strongly and nonstrongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity [Formula: see text]. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an ε-KKT or ε-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under local Lipschitz continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods. Funding: This work was partially supported by the National Science Foundation [Grant IIS-2211491], the Office of Naval Research [Grant N00014-24-1-2702], and the Air Force Office of Scientific Research [Grant FA9550-24-1-0343].

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.

Properties of the controllability set of one class of differential inclusions

Properties of the controllability set of one class of differential inclusions

Nonsmooth Integral Guiding Potentials in the Problem of the Asymptotic Behavior of Trajectories of Some Classes of Functional Differential Inclusions

Nonsmooth Integral Guiding Potentials in the Problem of the Asymptotic Behavior of Trajectories of Some Classes of Functional Differential Inclusions

Optimal Solutions for a Class of Impulsive Differential Problems with Feedback Controls and Volterra-Type Distributed Delay: A Topological Approach

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model.

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

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.

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.

Convergence of a double step scheme for a class of second order Clarke subdifferential inclusions

In this paper we deal with a second order evolution inclusion involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a double step time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem.

A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f∈X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1

This paper considers a class of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1 < α < 2 with impulses in a Hilbert space. We study the existence of the mild solution for the cases when the multi-valued map has convex and non-convex values. The results are obtained by combining fixed-point theorems with the fractional order cosine family, semigroup theory, and stochastic techniques. A new set of sufficient conditions is developed to demonstrate the approximate controllability of the system. Finally, an example is given to illustrate the obtained results.

Comprehensive Approach of an Implicit Hybrid Urysohn–Stieltjes Integral Inclusions via Fractal Differential Constraint

This investigation establishes the solvability of an implicit hybrid nonlinear Urysohn–Stieltjes type (IHU‐S) integral inclusion. Some sufficient conditions are assumed to prove qualitative features for the solution for this class of inclusions. A comprehensive discussion and analysis, including an example and an application, is presented to illustrate our results.

On the Filippov-Ważewski relaxation theorem for a certain class of fractional differential inclusions

The purpose of this text is to propose an attempt of an extension of the Filippov-Ważewski Relaxation Theorem for a certain class of fractional differential inclusions. The classical result devoted to ordinary differential inclusions is a part of the qualitative theory: a description of the relationship between the solutions to the differential inclusion and the convexified differential inclusion was given. There exist several generalizations of that result and in this note a method is proposed so that the range of some parameters is extended. An example of a possible application may arise in control theory and the question is whether it is possible to have the same reachable set economizing the set of controls.

Existence Results for a Fractional Differential Inclusion of Arbitrary Order with Three-Point Boundary Conditions

This paper studies existence of solutions for a new class of fractional differential inclusions of arbitrary order with three-point fractional integral boundary conditions. Our results are based on Bohnenblust-Karlin’s fixed point theorem.

Pseudo-almost periodic $C^{0}$-solution for evolution inclusion with mixed nonlocal plus local initial conditions

This paper is devoted to the study of a class of evolution inclusion in Banach spaces with nonlocal plus local mixed initial conditions. Under some mild assumptions, a unique solvability result to the multivalued evolution problem is obtained via the arguments of fixed point principle and the theory of $C^0$-semigroup.

Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Abstract We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.

Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type $$(P,\kappa ,{r},\gamma )$$ and Martelli’s fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained.

Existence results for Dirichlet double phase differential inclusions

"In this paper we consider a class of double phase differential inclusions of the type $$\left\{ \begin{array}{ll} -{\rm div\;}\left(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u\right) \in \partial_C^2 f(x,u) , &amp; \mbox{ in }\Omega,\\ u=0, &amp; \mbox{ on }\partial\Omega, \end{array} \right.$$ where $\Omega \subset \mathbb{R}^N$, with $N\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\Omega$ and locally Lipschitz w.r.t. the second variable and $\partial_C^2 f(x,\cdot)$ stands for the Clarke subdifferential of $t\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\partial_C^2 f(x,\cdot)$ satisfies an appropriate growth condition."

Coupled systems of subdifferential type with integral perturbation and fractional differential equations

This paper is mainly devoted to study a class of first-order differential inclusions governed by time-dependent subdifferential operators involving an integral perturbation. Employing then the constructive method used there, we also handle the associated second-order differential inclusion.&#x0D; Our final topic, accomplished in infinite-dimensional Hilbert spaces, is to develop some variants related to coupled systems by such differential inclusions and fractional differential equations.

Gradient estimates for the insulated conductivity problem: The case of m-convex inclusions

We consider an insulated conductivity model with two neighboring inclusions of m-convex shapes in Rd when m ≥ 2 and d ≥ 3. We establish pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order ɛ−1/m+β with β=[−(d+m−3)+(d+m−3)2+4(d−2)]/(2m)∈(0,1/m) as the distance ɛ between these two insulators tends to zero. In particular, the optimality of the blow-up rate is also demonstrated for a class of axisymmetric m-convex inclusions.