In this paper, we introduce a new general alternative regularization algorithm for solving split equilibrium and fixed point problems in real Hilbert spaces. The proposed method does not require a prior estimate of the norm of the bounded linear operator nor a fixed stepsize for its convergence. Instead, we employ a line search technique and prove a strong convergence result for the sequence generated by the algorithm. A numerical experiment is given to show that the proposed method converges faster in terms of number of iteration and CPU time of computation than some existing methods in the literature.
In this paper, we consider the iteration method called 'Picard-Mann hybrid iterative process' for finding a fixed point of continuous functions on an arbitrary interval. We give a necessary and sufficient condition for convergence of this iteration for continuous functions on an arbitrary interval. Also, we compare the rate of convergence of the Picard-Mann hybrid iteration with the other iterations and prove that it is better than the others under the same computational cost. Moreover, we present numerical examples.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let {T n } : C → H be a sequence of nearly nonexpansive mappings such thatC → H be a γ-Lipschitzian mapping and F : C → H be a L-Lipschitzian and η-strongly monotone operator. This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence {x n } converges strongly to x * ∈ F which is also the unique solution of the following variational inequality:As a special case, this projection method can be used to find the minimum norm solution of above variational inequality; namely, the unique solution x * to the quadratic minimization problem: x * = ar g mi n x∈F x 2 . The results here improve and extend some recent corresponding results of other authors.
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