Semicontinuity Research Articles

Wong-Zakai approximations and random attractors for nonlocal stochastic Schrödinger lattice systems in weighted spaces

This paper deals with the long-term behavior of nonlocal Schrödinger lattice systems with multiplicative white noise and their Wong-Zakai approximate systems in weighted spaces. We first prove the existence and uniqueness of tempered pullback attractors for the original stochastic systems and the Wong-Zakai approximations. Then, we establish the upper semicontinuity of these attractors for the Wong-Zakai approximate systems as the step length of the Wiener shift approaches zero.

Asymptotic behavior of non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by linear mixed white noise

This paper is concerned with the asymptotic behavior of the non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by the linear mixed white noise, which simultaneously contains linear additive noise and multiplicative noise. For the sake of the long-term behavior of the system we considered, we need to utilize a different Ornstein–Uhlenbeck transformation than the general one. First, the existence and uniqueness of pullback random attractors are demonstrated. Then, we prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Analysis and evaluation of the effectiveness of program codes for particle swarm and ant colony methods for lower semi-continuous functions

In this paper, a comparative analysis of two stochastic methods of global optimization is carried out: the Particle Swarm Optimization method and the Ant Colony Optimization method. The analysis and evaluation of efficiency were carried out using parameters such as the number of calls to the objective function, the rate of convergence, resources, computational efficiency, sensitivity to parameters, and resistance to initial conditions. The article presents the program codes of these methods in the Python environment, designed to calculate the global extremum of semi-continuous functions from below. Well- known test functions were used to construct (by gluing) semi-continuous functions at different values of input parameters.

Asymptotic analysis of the 2D and 3D convective Brinkman–Forchheimer equations in unbounded domains

In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains [Formula: see text] of the Poincaré domain [Formula: see text] such that [Formula: see text] as [Formula: see text]. If [Formula: see text] and [Formula: see text] are the global attractors of CBF equations corresponding to [Formula: see text] and [Formula: see text], respectively, then we show that for large enough [Formula: see text], the global attractor [Formula: see text] enters into any neighborhood [Formula: see text] of [Formula: see text]. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors.

A Lower Semicontinuous Time Separation Function for $$C^0$$ Spacetimes

A Lower Semicontinuous Time Separation Function for $$C^0$$ Spacetimes

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.

Revisited convexity notions for $$L^\infty $$ variational problems

AbstractWe address a detailed study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals, as well as those arising by the $$L^p$$ L p -approximation, as $$p \rightarrow +\infty $$ p → + ∞ of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork underlying further research in the $$L^\infty $$ L ∞ context.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Semicontinuous maps on module varieties

Abstract We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that 𝑔-vectors and 𝐸-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.

Attractors of a thermoelastic Bresse system with mass diffusion

AbstractThis paper concerns the long‐time dynamics of a thermoelastic Bresse system with mass diffusion. We prove the existence of a global attractor by showing that the system is gradient and asymptotically smooth. In addition, the attractor is characterized as an unstable manifold of the set of stationary solutions. The quasi‐stability of the system and the finite fractal dimension of the global attractor are established by a stabilizability inequality. Finally, we prove the upper semicontinuity of the global attractor regarding the parameter in a dense residual set.

The cantor set and inverse limits of upper semi-continuous functions

The cantor set and inverse limits of upper semi-continuous functions

Optimality Conditions for Mathematical Programs with Vanishing Constraints Using Directional Convexificators

This article deals with mathematical programs with vanishing constraints (MPVCs) involving lower semi-continuous functions. We introduce generalized Abadie constraint qualification (ACQ) and MPVC-ACQ in terms of directional convexificators and derive necessary KKT-type optimality conditions. We also derive sufficient conditions for global optimality for the MPVC under convexity utilizing directional convexificators. Further, we introduce a Wolfe-type dual model in terms of directional convexificators and derive duality results. The results are well illustrated by examples.

Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations

Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Some Unique Fixed Point Theorems on Complete Metric Space with Upper Semi Continuous Mapping

Some Unique Fixed Point Theorems on Complete Metric Space with Upper Semi Continuous Mapping

Lower Semicontinuity of Intersections of Set-Valued Maps and Applications on Bilevel Games

Lower Semicontinuity of Intersections of Set-Valued Maps and Applications on Bilevel Games

Two-dimensional Jacobians $$\textrm{det}\hspace{0.56905pt}$$ and $$\textrm{Det}\hspace{0.56905pt}$$ for bounded variation functions and applications

AbstractThe paper deals with the comparison in dimension two between the strong Jacobian determinant $$\textrm{det}\hspace{0.56905pt}$$ det and the weak (or distributional) Jacobian determinant $$\textrm{Det}\hspace{0.56905pt}$$ Det . Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valued U in $$BV(\Omega )^2$$ B V ( Ω ) 2 such that the equality $$\textrm{det}\hspace{0.56905pt}(\nabla U)=\textrm{Det}\hspace{0.56905pt}(\nabla U)$$ det ( ∇ U ) = Det ( ∇ U ) holds either in the distributional sense on $$\Omega $$ Ω , or almost-everywhere in $$\Omega $$ Ω when U is in $$W^{1,1}(\Omega )^2$$ W 1 , 1 ( Ω ) 2 . The key-assumption of the result is the regularity of the Jacobian matrix-valued $$\nabla U$$ ∇ U along the direction of a given non vanishing vector field $$b\in C^1(\Omega )^2$$ b ∈ C 1 ( Ω ) 2 , i.e.$$\nabla U\, b$$ ∇ U b is assumed either to belong to $$C^0(\Omega )^2$$ C 0 ( Ω ) 2 with one of its coordinates in $$C^1(\Omega )$$ C 1 ( Ω ) , or to belong to $$C^1(\Omega )^2$$ C 1 ( Ω ) 2 . Two examples illustrate this new notion of two-dimensional distributional determinant. Finally, we prove the lower semicontinuity of a polyconvex energy defined for vector-valued functions U in $$BV(\Omega )^2$$ B V ( Ω ) 2 , assuming that the vector field b and one of the coordinates of $$\nabla U\, b$$ ∇ U b lie in a compact set of regular vector-valued functions.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

On the complexity of a quadratic regularization algorithm for minimizing nonsmooth and nonconvex functions

In this paper, we consider the problem of minimizing the function f ( x ) = g 1 ( x ) + g 2 ( x ) − h ( x ) over R n , where g 1 is a proper and lower semicontinuous function, g 2 is continuously differentiable with a Hölder continuous gradient and h is a convex function that may be nondifferentiable. This problem has important practical applications but is challenging to solve due to the presence of nonconvexities and nonsmoothness. To address this issue, we propose an algorithm based on a proximal gradient method that uses a quadratic approximation of the function g 2 and a nonconvex regularization term. We show that the number of iterations required to reach our stopping criterion is O ( max { ϵ − β + 1 β , η 2 β ϵ − 2 ( β + 1 ) β } ) . Our approach offers a promising strategy for solving this challenging optimization problem and has potential applications in various fields. Numerical examples are provided to illustrate the theoretical results.