Approximation of Fixed Point of Pseudocontractive Map in a Real Banach Pace

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2009-08-25
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Fixed point theory has been one of most outstanding fields that contributes immensely to the growth of mathematics. This thesis seeks to answer a question that was a consequence of a research undertaken in this field. Recently, Zhou and Jin proved the following result: let E be a real Banach sf)ace with / a uniformly convex dual E*, and let K be a non-empty closed convex and bounded subset of E. Assume that T: K K is a continuous and strong pseudocontraction, the Ishikwa iteration sequence {x„ converges strongly to the unique fixed point of T. However, when T is continuous strongly pseudocontractive mapping, one question arises naturally: if T neither is lipschitzian nor has the bounded range, whether or not the Ishikawa iterative sequence {xn generated converges strongly to the unique fixed point of T. It is therefore the purpose of this thesis to solve the above question. This thesis is arranged in four chapters. The first chapter which is the introductory chapter gives a broad overview of the history and the significance of the study of the fixed point theory. The chapter emphasizes on the spaces, operators and iterative sequences involved in the study of fixed point theory. The second chapter discusses the basic concepts involved in the study of fixed point theory. These concepts include sets, maps, operators and iterative sequences. This chapter gives explicit definitions of terminologies, states theorems and lemmas along with their vivid proofs, which are related to these concepts. The discussion in this chapter would be useful in the analysis of the main work of the thesis. The third chapter provides an explicit analysis on how the Ishikawa iterative sequence strongly converges to the unique fixed point of pseudocontractive operator defined in a uniformly smooth Banach space. This operator is defined such that it is neither lipschitzian nor has the bounded range. The concluding chapter draws a number of special cases of the main work of this thesis from the analysis in the third chapter and ends by advocating for more research to be undertaken in fixed point theory. 
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A thesis submitted to the College of Science, Department of Mathematics, 2009
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