We provide a new proof technique to obtain strong convergence of the sequences generated by viscosity iterative methods for a Rockafellar-type iterative algorithm and a Halpern-type iterative algorithm to a zero of an accretive operator in Banach spaces.
Due to its property of large sparsity, the nonsymmetric saddle-point problem (1.1) is suitable for solving by iterative methods.
Equations are solved by iterative methods and explicit time integration.
(B.2) and (B.3), the latter is generally faster to solve by iterative methods.
Problem (9) then can be solved by fixed point iterative methods for solving nonlinear equations or any other suitable nonlinear equation solver.
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [15] and iterative methods for the approximation of fixed points of such types of non-Lipschitzian mappings were studied by Bruck et al. [15], Agarwal et al. [19], Chidume et al. [20], Kim and Kim [21] and many others.
In this paper, motivated by the above mentioned results, we consider viscosity iterative methods for the Rockafellar-type iterative algorithm (1.1) and Halpern-type iterative algorithm (1.2).
The study on iterative methods for strict pseudocontractive mappings was initiated by Browder and Petryshyn [1] in 1967, but the iterative methods for strict pseudocontractive mappings are far less developed than those for nonexpansive mappings.
By utilizing Lemma 4.1, we suggest and analyze the following explicit projection iterative methods for solving problem (11).
In [11], Xu proved that the sequence { x n } generated by the iterative method for a nonexpansive mapping S presented below with the initial guess x 0 ∈ H chosen arbitrary: x n + 1 = α n b + ( I − α n A ) S x n, ∀ n ≥ 0, (1.2). converges strongly to the unique solution of the minimization problem (1.1) provided the sequence { α n } satisfies certain conditions.
