On convergence criteria of generalized proximal point algorithms

Abstract

We analyze some generalized proximal point algorithms which include the previously known proximal point algorithms as special cases. Weak and strong convergence of the proposed proximal point algorithms are proved under some mild conditions.

A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces

“…Remark 3.5. (1) Our results extend those of [3,16,21,31,33,34] from Hilbert spaces to Banach spaces.…”

“…(2) We remove the conditions that lim n→∞ |r n+1 − r n | = 0 and 0 < lim inf n→∞ λ n ≤ lim sup n→∞ λ n < 1 in Theorem 3.3 of Yao-Noor [34] and the conditions that ∞ n=1 λ n < ∞, lim n→∞ 1 r n+1 − 1 r n = 0 and ∞ n=1 |α n+1 −α n | r n+1 < ∞ in Theorem 1 of Boikanyo-Moroşanu [3].…”

A generalization of the regularization proximal point method

“…Just as in the case of the scheme (1.3), different sets of conditions on the control parameters α n , λ n , γ n and β n have been used to prove strong convergence of the iterative process (1.4), see [2,15,3]. Another proximal method which generates strongly convergent sequences is the prox-Tikhonov method of Lehdili and Moudafi [6] which was extended by Xu [14] in the following way…”

“…where again u, x 0 ∈ H are given, α n ∈ (0, 1), λ n , γ n ∈ [0, 1] with α n + λ n + γ n = 1, and β n ∈ (0, ∞), which was introduced by Yao and Noor [15] also converges strongly (under appropriate assumptions) to the solution of problem (1.1) which is nearest to u. Just as in the case of the scheme (1.3), different sets of conditions on the control parameters α n , λ n , γ n and β n have been used to prove strong convergence of the iterative process (1.4), see [2,15,3].…”

Finding a solution of split null point of the sum of monotone operators without prior knowledge of operator norms in Hilbert spaces

“…This problem was introduced by Martinet [24], and later it has been studied by many authors. It is well-known that the popular iteration method that was used for solving the problem (1.1) is the following proximal point algorithm: for a given x 1 ∈ H, x n+1 = J B λ n x n , ∀n ∈ N, where {λ n } ⊂ (0, ∞) and J B λ n = (I + λ n B) −1 is the resolvent of the considered maximal monotone operator B corresponding to λ n , see also [5,17,22,37,39] for more details.…”