Iterative methods for solving split common fixed point problems in Hilbert spaces

The split common fixed point problems (SCFPP) attracted and continued to attract the attention of many researchers; this is due to its applications in many branches of mathematics both pure and applied. Further, SCFPP provides us with a unified structure to study large number of nonlinear mappings. Our interest here is to apply these mappings to propose some algorithms for solving split common fixed point problems and its variant forms, in the end, we prove the convergence results of these algorithms. In other words, we construct parallel and cyclic algorithms for solving the split common fixed point problems for strictly pseudocontractive mappings and prove the convergence results of these algorithms. We also suggest some iterative methods for solving the split common fixed point problems for the class of total quasi asymptotically nonexpansive mappings and prove the convergence results of the proposed algorithms. As a special case of this split common fixed point problems, we consider the split feasibility problem and prove its convergence results. To solve the split common fixed point problems, one needs to estimate the norm of the bounded linear operator. To determine the norm of this bounded linear operator is a tough task. In this regard, we consider an algorithm for solving such a problem which does not need any prior information on the norm of the bounded linear operator and establish the convergence results of the proposed algorithm. These were done by considering the class of demicontractive mappings. We also formulate and analyse algorithms for solving the split common fixed point equality problems for the class of finite family of quasi-nonexpansive mappings. Furthermore, we propose another problem namely split feasibility and fixed point equality problems and suggest some new iterative methods and prove their convergence results for the class of quasi-nonexpansive mappings. Finally, as a special case of the split feasibility and fixed point equality problems, we consider the split feasibility and fixed point problems and propose Ishikawa-type extragradients algorithms for solving these split feasibility and fixed point problems for the class of quasi-nonexpansive mappings in Hilbert spaces. In the end, we prove the convergence results of the proposed algorithms. Results proved in this thesis continue to hold for different types of problems, such as; convex feasibility problem, split feasibility problem and multiple-set split feasibility problems. For more details, see Chapter 3 Corollaries 3.4.1, 3.4.2, 3.4.3 and 3.4.4.

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