Hypercircle inequalities and sampling theory

In this article the well-known hypercircle inequality is extended to the Riesz bases setting. A natural application for this new inequality is given by the estimation of the truncation error in nonorthogonal sampling formulas. Examples including the estimation of the truncation error for wavelet sampling expansions or for nonorthogonal sampling formulas in Paley–Wiener spaces are exhibited.     

A hybrid steepest-descent method for variational inequalities in Hilbert spaces

A Mann-type hybrid steepest-descent method for solving the variational inequality ?F(u*), v ? u*? ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473–504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185–201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51–69).     

On the norm of the Lp-Dunkl transform

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.     

Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant on the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein–Uhlenbeck operator perturbed by a critical singular potential in a two-dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which are established in the work of Goldstein et al. [Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl Anal. 2012;91(11):2057–2071] and Hauer and Rhandi [A weighted Hardy inequality and nonexistence of positive solutions. Arch Math. 2013;100:273–287] in the non-critical cases, and are also considered as extensions of a result in Cabré and Martel [Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potential singulier. C R Acad Sci Paris Sér I Math. 1999;329:973–978] to the Kolmogorov operator case perturbed by a critical singular potential.     

Optimal control for distributed linear systems subjected to null-controllability

In this article, we apply the notion of hierarchic control on a distributed system in which the state is governed by a parabolic equation. This notion assumes that we have two controls where one will be the Leader and the other, the Follower. The first control is of controllability type subjected to a constraint, while the second expresses that the state does not move too far from a given state. The results are achieved by means of an observability inequality of the Carleman type, which is ‘adapted’ to the constraint.     

Contact problems with friction for hemitropic solids: boundary variational inequality approach

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov–Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.     

Multivariate inequalities based on Sobolev representations

Here we derive very general multivariate tight integral inequalities of Chebyshev–Grüss, Ostrowski types and of comparison of integral means. These are based on well-known Sobolev integral representation of a function. Our inequalities engage ordinary and weak partial derivatives of the involved functions. We also give their applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions.     

Multidimensional weighted Opial inequalities

In this article we produce Opial-type weighted multidimensional inequalities over balls and arbitrary smooth bounded domains. The inequalities are sharp. The functions under consideration vanish on the boundary.