We consider the constrained convex minimization problem: min_x∈C f(x) we will present gradient projection method which generates a sequence x^k according to the formula x^(k+1) = P_c(x^k − £\_k∇f(x^k)), k= 0, 1, ¡P ¡P ¡P , our ideal is rewritten the formula as a xed point algorithm: x^(k+1) = T_(£\k)x^k, k = 0, 1, ¡P ¡P ¡P is used to solve the minimization problem. In this paper, we present the gradient projection method(GPM) and different choices of the stepsize to discuss the convergence of gradient projection method which converge to a solution of the concerned problem.

In this paper I extend the method of mixed monotony, to construct monotone sequences that converge to the unique solution of an initial value delay differential equation. / En este artículo se prueba una generalización del método de monotonía mixta, para construir sucesiones monótonas que convergen a la solución única de una ecuación diferencial de retraso con valor inicial.

In this paper we discuss iterative algorithms for solving the split feasibility problem (SFP). We study the CQ algorithm from two approaches: one is an optimization approach and the other is a fixed point approach. We prove its convergence first as the gradient-projection algorithm and secondly as a fixed point algorithm. We also study a relaxed CQ algorithm in the case where the sets C and Q are level sets of convex functions. In such case we present a convergence theorem and provide a different and much simpler proof compared with that of Yang [7].

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

Cette thèse est consacrée à la résolution d'un problème fondamental de l'analyse variationnelle qu'est la recherchede zéros d'opérateurs maximaux monotones dans un espace de Hilbert. Nous nous sommes tout d'abord intéressés au cas de l'opérateur somme étendue de deux opérateurs maximaux monotones; la recherche d'un zéro de cet opérateur est un problème dont la bibliographie est peu fournie: nous proposons une version modifiée de l'algorithme d'éclatement forward-backward utilisant à chaque itération, l'epsilon-élargissement d'un opérateur maximal monotone,afin de construire une solution. Nous avons ensuite étudié la convergence d'un nouvel algorithme de faisceaux pour construire ID zéro d'un opérateur maximal monotone quelconque en dimension finie. Cet algorithme fait intervenir une double approximation polyédrale de l'epsilon-élargissement de l'opérateur considéré / This thesis is devoted to solving a basic problem of variational analysis which is the search of zeros of maximal monotone operators in a Hilbert space. First of aIl, we concentrate on the case of the extended som of two maximal monotone operators; the search of a zero of this operator is a problem for which the bibliography is not abondant: we purpose a modified version of the forward-backward splitting algorithm using at each iteration, the epsilon-enlargement of a maximal monotone operator, in order to construet a solution. Secondly, we study the convergence of a new bondie algorithm to construet a zero of an arbitrary maximal monotone operator in a finite dimensional space. In this algorithm, intervenes a double polyhedral approximation of the epsilon-enlargement of the considered operator

This paper offers an existence result for finite dimensional stochastic differential inclusions with maximal monotone drift and diffusion terms. Kravets studied only set-valued drifts in [5], whereas Motyl [4] additionally observed set-valued diffusions in an infinite dimensional context. In the proof we make use of the Yosida approximation of maximal monotone operators to achieve stochastic differential equations which are solvable by a theorem of Krylov and Rozovskij [7]. The selection property is verified with certain properties of the considered set-valued maps. Concerning Lipschitz continuous set-valued diffusion terms, uniqueness holds. At last two examples for application are given.

This paper offers an existence result for finite dimensional stochastic differential inclusions with maximal monotone drift and diffusion terms. Kravets studied only set-valued drifts in [5], whereas Motyl [4] additionally observed set-valued diffusions in an infinite dimensional context. In the proof we make use of the Yosida approximation of maximal monotone operators to achieve stochastic differential equations which are solvable by a theorem of Krylov and Rozovskij [7]. The selection property is verified with certain properties of the considered set-valued maps. Concerning Lipschitz continuous set-valued diffusion terms, uniqueness holds. At last two examples for application are given.

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

In the work we modify the well-known minimal surface problem to a very special form, where the exponent two is replaced by a general positive parameter. To the modified problem we define four notions of solution in nonreflexive Sobolev space and in the space of functions of bounded variation. We examine the relationships between these notions to show that some of them are equivalent and some are weaker. After that we look for assumptions needed to prove the existence of solution to the problem in the sense of definitions provided. We outline that in the setting of spaces of functions of bounded variation the solution exists for any positive finite parameter and that if we accept some restrictions on the parameter then the solution exists in the Sobolev space, too. We also provide counterexample indicating that if the domain is non-convex, the solution in Sobolev space need not exist. Powered by TCPDF (www.tcpdf.org)