The mann and ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings

On the Existence and Iterative Approximation Problems of Solutions for Set-Valued Variational Inclusions in Banach Spaces

“…By using Michael's selection theorem [16] and Nadler's theorem [18] some existence theorems and some iterative algorithms for solving this kind of set-valued variational inclusions in Banach spaces are established and suggested. The results presented in this paper generalize, improve, and unify the corresponding results of Noor [19][20][21][22][23][24], Ding [8], Huang [10], Kazmi [12], Jung and Morales [11], Liu [14], Hassouni and Mouafi [9], Zeng [28], and Chang [2][3][4][5].…”

“…(ii) Theorems 4.1-4.4 generalize, improve, and unify the corresponding recent results of Noor [19][20][21][22][23][24], Liu [14], Ding [8], Huang [10], Kazmi [12], Chang [2][3][4][5], Jung and Morales [11], Hassouni and Moudafi [9], and Zeng [28].…”

“…In some special cases, it was also considered in Noor [21]. Recently this problem with N being some special case was also considered in the setting of Banach spaces (see Chang [4]). …”

On convergence of random iterative schemes with errors for strongly pseudo-contractive Lipschitzian maps in real Banach spaces

“…With the developments in random fixed point theory, there has been a renewed interest in random iterative schemes [2,3,7,8,10]. In linear spaces, Mann and Ishikawa iterative schemes are two general iterative schemes which have been successfully applied to fixed point problems [1,5,6,13,14,16,19,26,28,37]. Recently, many stability and convergence results of iterative schemes have been established, using Lipschitz accretive pseudo-contractive) and Lipschitz strongly accretive (or strongly pseudo-contractive) mappings in Banach spaces [9,10,12,13,22,23,24,32,37].…”

An example on the Mann iteration method for Lipschitz pseudocontractions

“…Under the following additional assumptions: (i) lim c n = 0; (ii) ∞ n=0 c n = ∞, the sequence {x n } generated from (1.2) is generally referred to as the Mann sequence in light of [10]. The recursion formula (1.2) has also been used to approximate solutions of numerous nonlinear operator equations and nonlinear variational inclusions in Banach spaces (see, e.g., [3,4,6,11]). A class of nonlinear mappings more general than and including the nonexpansive mappings is the class of pseudocontractions.…”