CEU Electronic Theses and Dissertations, 2011
| Author | Boikanyo, Oganeditse Aaron |
|---|---|
| Title | Iterative Processes for Solving Nonlinear Operator Equations |
| Summary | As a method of approximating zeros of a given maximal monotone operator $A$ in a real Hilbert space, the proximal point algorithm (PPA) which was initiated by B. Martinet (1979) was considered in a more general setting by R. T. Rockafellar (1976), who proved that it converges weakly to a solution of $0\in Ax$ when the sequence of errors is summable in norm. After O. G\"{u}ler (1991) showed that the PPA fails in general to converge strongly, modifications of the PPA, among them the inexact \lq Halpern-type\rq \: iterative process which was introduced by H. K. Xu (2002) and the regularization method of N. Lehdili and A. Moudafi (1996), were obtained in order to enforce strong convergence, still under the summability condition on errors. Definitely this condition is too strong from a computational point of view. We obtain in this thesis other strong convergence results associated with these methods as well as their generalizations under the general condition that errors converge to zero in norm. These results are proved under new sets of conditions on the control parameters involved, which are either weaker than the ones previously used by other authors or are distinct alternative sets of conditions. Other strongly convergent sequences of proximal iterates, such as the method of alternating resolvents and the viscosity approximation method are also constructed. Some illustrations on how these methods can be used to approximate minimum values and/or minimizers of certain convex functionals are given. Apart from addressing the two important problems in the theory of proximal point algorithms -- that of strong convergence instead of weak convergence and the one concerning acceptable errors -- the results presented in this thesis improve, generalize and refine many existing results in the literature. |
| Supervisor | Gheorghe Morosanu |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2011/boikanyo_oganeditse.pdf |
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