Hussain, Nawab (2001) *SOME TYPES OF BEST APPROXIMATION AND THEIR APPLICATIONS.* PhD thesis, Bahauddin Zakariya University, Multan.

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## Abstract

The study of abstract fixed point theory of single valued maps has evolved as a natural extension of the corresponding classical theory on Euclidean spaces. The celebrated Banach contraction principle has become a vigorous tool for studying nonlinear volterra integral equations and nonlinear functional differential equations in Banach spaces. The development of the geometric fixed point theory for multivalued maps was initiated by Nadler [90] in 1969. Using the concept of Hausdorff metric, he established the multi valued contraction principle containing the Banach contraction principle as a special case. The theory of multivalued maps has applications in control theory, convex optimization and economics. The field of approximation theory has become so vast that it intersects with every other branch of analysis and plays an important role in applications in the applied sciences and engineering. Fixed point theorems have been used in many instances in approximation theory (e.g. Al-Thagafi [2], Brosowski [18], Habinaik [42], Prolla [105] and Sehgal and Singh [116]). In the subject of approximation theory one often wishes to know whether some useful properties of the function being approximated are inherited by the approximating function. Meinardus [85] was the first who employed a fixed point theorem to achieve this goal. Random fixed points and random approximations are the stochastic generalizations of the deterministic fixed point and approximation results. The study of the random fixed points and random best approximation of random operators of different types is an active area of investigation lying at the intersection of nonlinear analysis and probability theory. In the setting of Banach spaces, Sehgal and Waters, Sehgal and Singh and Papageorgiou initiated the study of random approximations and obtained stochastic version of Ky Fan's approximation theorem. More work and the inter-play between random fixed point theory and random approximation theory have been carried out by Beg and Shahzad, Khan, Lin and Tan and Yaun. The main purpose of this thesis is to establish approximation theorems for continuous and non-continuous maps in the setting of certain topological vector spaces and cite applications of some types of best approximation in fixed point theory. We divide this thesis into four chapters. Chapter 1 is essentially introductory in nature. Here we fix our notations, recall some basic definitions and summarize some of the familiar classical and recent results about fixed points and best approximations for our need in the sequel. In chapter 2, common fixed point theorems for commuting as well as non-commuting maps in the setup of Hausdorff locally convex spaces are proved. As applications, we derive certain Brosowski-Meinardus type results on invariant best approximation. Similar results in the context of convex metric spaces for R-sub commutative maps are established by using a recent common fixed point theorem of Pant [96]. The latter part of this chapter deals with results on invariant approximation in the setting of locally bounded topological vector spaces where the set of best approximations need not be even star shaped. Finally, we approximate fixed point of a non-expansive map through the iterates of its bisection map in metrizable topological vector spaces. The results of section 2.2 will appear in [48] and [63]. The results of section 2.3 will appear in [61] and those of section 2.4 have appeared in [58] and [64]. In chapter 3, we discuss mainly the inter-play between random fixed point theory and random best approximation theory. In section 3.2, a number of random fixed point theorems for single valued non-expansive random operators defined on a non-convex domain (not even star shaped) in locally bounded topological vector space are obtained and the new results are applied to derive Brosowski-Meinardus type theorems regarding invariant random approximation. A similar result for condensing and demicompact random operators on a stars shaped domain is also presented. In Theorem 3.2.18, a multivalued extended analogue of a well-known result of Singh [131] in Frechet spaces is proved. The class of *-nonexpansive multivalued mappings is different from that of continuous maps. We utilize in section 3.3, the best approximation operator as a selector to find random fixed point and random approximation results for *-nonexpansive multivalued random operators in various settings. The condition (A), introduced by Shahzad and Latif [126] generalizes hemicompactness and the concept of condensing operators. In section 3.4 random fixed point theorems for continuous and *-nonexpansive multivalued random maps satisfying condition (A) defined on unbounded domain in Banach spaces and Frechet spaces are proved. Consequently, a random approximation result for *-nonexpansive maps on unbounded domain is obtained which in turn leads to a random fixed point theorem under a number of boundary conditions. The results of section 3.2 will appear in [65] and [68], while the results of section 3.3 have appeared in l59) and [60] and those of section 3.4 will appear in [7] and [57]. Fan's best approximation theorem provides a connection between approximation theory and fixed point theory. A number of deterministic and stochastic generalizations of this result have appeared in the literature; in particular contributions made by Beg and shahzad, Carbone, Park, Prolla, Sehgal and Singh and Sine are worth mentioning. Section 4.2 concerns the existence of deterministic and random best approximation results for upper semi continuous maps through the notion of a *-nonexpansive map in the setup of Banach spaces and Frechet spaces. In section 4.3, we follow Regan [95] to establish the existence and approximation of solutions in the sense that there exists a bounded sequence Xn and a point Xo in C such that Xn Xo and Xo is a solution to the nonlinear inclusion operator y E Ty where T is a *-nonexpansive multivalued map defined on suitable subset C of a Banach space. As a consequence we obtain random approximation results on unbounded domain in hyperconvex Banach spaces. In section 4.4, we establish Ky Fan type approximation results for continuous and *-nonexpansive maps defined on a compact and non-compact subset of a metrizable topological vector space and hyperconvex space. As an application, random approximation result for stochastic domain is established. The results of earlier part of section 4.2 have appeared in [56] and the latter part may appear in [62]. The results of section 4.3 may appear in [6].

Item Type: | Thesis (PhD) |
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Uncontrolled Keywords: | Fixed Point, Best Approximation, Hyperconvex Spaces, Invariant approximation, topological vector spaces |

Subjects: | Physical Sciences (f) > Mathematics(f5) |

ID Code: | 670 |

Deposited By: | Mr. Muhammad Asif |

Deposited On: | 22 Sep 2006 |

Last Modified: | 04 Oct 2007 21:02 |

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