New Accuracy Criteria for Modified Approximate Proximal Point Algorithms in Hilbert Spaces

Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings

“…Later many authors studied its convergence in a Hilbert space or a Banach space. See for instance, [7][8][9][10][11][12] and the references therein. Kamimura and Takahashi [13] have been recently introduced and studied the proximal-type algorithm for finding an element of T −1 0 in a uniformly smooth and uniformly convex Banach space E, which is an extension of Solodov and Svaiter's proximal-type algorithm.…”

Iterative Scheme with Errors for Common Zeros of Finite Accretive Mappings and Nonlinear Elliptic Systems

“…(iii) The error sequence { } is considered in the iterative scheme (A). (iv) Recall that, in [2], Lemma 8 is a key tool to prove the convergence of { } generated by (10). In particular, to obtain the main result, they imposed an additional condition on the function in Lemma 8 that ( ) ≤ / max{1, 2 1 }, where 1 > 0 is a constant satisfying some conditions.…”

“…It was proved that, under some conditions, the sequence { } produced by (5) converges weakly to a point in ( ). Later, many mathematicians try to combine the ideas of proximal method and Mann iterative method to approximate the zeros of -accretive mappings; see, for example, [7][8][9][10][11][12][13][14] and the references therein.…”

Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods

“…It was shown that, if Recently, many authors studied the problems of modifying Rockafellar's inexact proximal point method 1.3 in order to strong convergence to be guaranteed. In 2008, Ceng et al 4 gave new accuracy criteria to modified approximate proximal point algorithms in Hilbert spaces; that is, they established strong and weak convergence theorems for modified approximate proximal point algorithms for finding zeros of maximal monotone operators in Hilbert spaces. In the meantime, Cho et al 5 proved the following strong convergence result.…”