| The main contents of the thesis:In Chapter1, firstly, we give out an overview of the backgrounds and developments of the Fixed Point Theory. Secondly, we provide some descriptions of the main results in the thesis. Thirdly, we point out some basic concepts and lemmas which are needed in the proof of our results.In chapter2, Nakajo and Takahashi proved an strong convergent theorem in an closed convex subset of a Hilbert space for the nonexpansive mapping in2003, but they didn’t give out the explicit expression of the algorithm. The purpose in this chapter is that we find a new subsequence algorithm and proved the convergence of a fixed point of the mappings. So our result extend that of Nakajo’s form nonexpansive mappings to asymptotically mappings. This algorithm can give out the explicit expression of iterative approximation. The results here extend many relevant results of the authorsNakajo and Takahashi.In chapter3, We studied the viscosity approximation of implicit iterative sequence of the fixed point on asymptotically nonexpansive mapping in Banach spaces. The methods is as same as the chapter2but the the iterative sequences are different from chapter2. Therefore,we extend some results of chapter2from Hilbert spaces to Banach spaces.In chapter4, in1967, Halpern introduced a iterative sequence which now is called Halpern iteration. He pointed out some necessary conditions of the Halpern iteration. In2004, Xu introduced a new iterative sequence which is called implicit iteration. In the implicit iteration if we let f(xn)≡y∈K, then we can get Mann iteration. Motivated and inspired by the schemes of Mann and Xu, we introduce a new iterative scheme and prove the strong convergence of the sequence by the accordance property of asymptoti -cally nonexpansive mapping in a real Banach space E with a uniformly differentiable norm. The results extend and improve the related results of the articles, such as the papers of Halpern, Xu, Chidume, etc. |
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