624 research outputs found
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods
This work presents a comprehensive discretization theory for abstract linear
operator equations in Banach spaces. The fundamental starting point of the
theory is the idea of residual minimization in dual norms, and its inexact
version using discrete dual norms. It is shown that this development, in the
case of strictly-convex reflexive Banach spaces with strictly-convex dual,
gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently,
abstract mixed methods with monotone nonlinearity. Crucial in the formulation
of these methods is the (nonlinear) bijective duality map.
Under the Fortin condition, we prove discrete stability of the abstract
inexact method, and subsequently carry out a complete error analysis. As part
of our analysis, we prove new bounds for best-approximation projectors, which
involve constants depending on the geometry of the underlying Banach space. The
theory generalizes and extends the classical Petrov-Galerkin method as well as
existing residual-minimization approaches, such as the discontinuous
Petrov-Galerkin method.Comment: 43 pages, 2 figure
Proof mining in metric fixed point theory and ergodic theory
In this survey we present some recent applications of proof mining to the
fixed point theory of (asymptotically) nonexpansive mappings and to the
metastability (in the sense of Terence Tao) of ergodic averages in uniformly
convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page
Existence and approximation of fixed points of right Bregman nonexpansive operators
We study the existence and approximation of fixed points of right Bregman nonexpansive operators in reflexive Banach space. We present, in particular, necessary and sufficient conditions for the existence of fixed points and an implicit scheme for approximating them
Operator inclusions and operator-differential inclusions
In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable.
In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear system are also given.
In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions
x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e.
in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions
Iterative algorithms for solutions of nonlinear equations in Banach spaces.
Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
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