System of generalized resolvent equations with corresponding system of variational inclusions

The purpose of this paper is to introduce a new system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between system of generalized resolvent equations and system of variational inclusions. The iterative algorithms for finding the approximate solutions of system of generalized resolvent equations are proposed. The convergence of approximate solutions of system of generalized resolvent equations obtained by the proposed iterative algorithm is also studied.      

Parametric generalized set-valued variational inclusions and resolvent equations

In this paper, we develop the sensitivity analysis for generalized set-valued variational inclusions and generalized resolvent equations. We establish the equivalence between the parametric generalized set-valued variational inclusions and parametric generalized resolvent equations, by using the resolvent operator technique without assuming the differentiability of the given data.     

New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u₀, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u₀, where a>0 is a parameter and u₀ is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen.     

Direct transformation of variational problems into Cauchy systems. I. Scalar-quadratic case

This series of papers addresses three interrelated problems: the solution of a variational minimization problem, the solution of integral equations, and the solution of an initial-valued system of integro-differential equations. It will be shown that a large class of minimization problems requires the solution of linear Fredholm integral equations. It has also been shown that the solution of a linear Fredholm integral equation is identical to the solution of a Cauchy system. In this paper, we bypass the Fredholm integral equations and show that the minimization problem directly implies a solution of a Cauchy system. This first paper in the series looks only at quadratic functionals and scalar functions.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-77-3383.     

An optimal regularized projection method for solving ill-posed problems via dynamical systems method

A new regularized projection method was developed for numerically solving ill-posed equations of the first kind. This method consists of combining the dynamical systems method with an adaptive projection discretization scheme. Optimality of the proposed method was proved on wide classes of ill-posed problems.     

Dynamical systems method (DSM) for unbounded operators

Let be an unbounded linear operator in a real Hilbert space , a generator of a semigroup, and let be a nonlinear map. The DSM (dynamical systems method) for solving equation consists of solving the Cauchy problem , , where is a suitable operator, and proving that i) 0$">, ii) , and iii) .

Conditions on and are given which allow one to choose such that i), ii), and iii) hold.

     

Some existence results for a class of degenerate semilinear elliptic systems

Using a variational approach, we investigate a class of degenerate semilinear elliptic systems with measurable, unbounded nonnegative weights, where the domain is bounded or unbounded. Some existence results are obtained.     

Slow manifolds for dissipative dynamical systems

A class of infinite-dimensional dissipative dynamical systems is defined for which the slow invariant manifolds can be calculated. Large-time behavior of the evolution of such systems is studied.     

An existence theorem for neutral variational problems of Bolza

This note proves an existence theorem for a generalized Bolza-type problem that has time delays in both the state and velocity variables. The assumptions are stated in terms of a modification of the classical Hamiltonian, and extend ideas of Rockafellar to the delay case.     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

Implicit runge-kutta methods for differential inclusions

This paper is concerned with the application of implicit Runge-Kutta methods suitable for stiff initial value problems to initial value problems for differential inclusions with upper semicontinuous right-hand sides satisfying a uniform one-sided Lipschitz condition and a growth condition. The problems could stem from differential equations with state discontinuous right-hand sides. It is shown that there exist methods with higher order of convergence on intervals where the solution is smooth enough. Globally we get at least the order one.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

Pseudomonotone variational inequality problems: Existence of solutions

Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given. This research was partially done while the author was visiting the University of Chile thanks to the support of an ECOS program.