Four parameter proximal point algorithms

A generalization of the regularization proximal point method

“…Theorem 3.1 refines [2, Theorem 5] and [3,. Note that when T is the identity operator, then we recover many other results announced recently [1,5,6,12,13,14].…”

“…We have shown in the proof of Theorem 5 [2] that the exact iterative process of (1.6), namely, the sequence (v n ) defined by…”

“…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…”

“…Therefore, the results already proved for the contraction proximal point algorithm also hold for the regularization method and vice versa. The authors [2] generalized the regularization method as…”

“…The purpose of this paper is to investigate if the method used in [12] can be applied to the scheme (1.6) (which is different from (1.4) except when λ n = 0 for all n and T = I, the identity operator [2]) in order to get a strong convergence result of a sequence generated by it under minimal assumptions on the control parameters α n and β n , thereby refining previously obtained results associated with the iterative process (1.6).…”