Modified Iterative Schemes for a Fixed Point Problem and a Split Variational Inclusion Problem

Abstract

In this paper, we alter Wang’s new iterative method as well as apply it to find the common solution of fixed point problem (FPP) and split variational inclusion problem (SpVIP) in Hilbert space. We discuss the weak convergence for (SpVIP) and strong convergence for the common solution of (SpVIP) and (FPP) using appropriate assumptions. Some consequences of the proposed methods are studied. We compare our iterative schemes with other existing related schemes.

Modified Mann-Type Algorithm for Two Countable Families of Nonexpansive Mappings and Application to Monotone Inclusion and Image Restoration Problems

“…Notice that lim n→∞ δ n = 1, we obtain δ n i j y n i j y as j → ∞. Applying ( 28), (29) and Lemma 3, we obtain that y ∈ Fix(S) ∩ Fix(T). The characterization of metric projection via Lemma 4 yields that −x * , y − x * = 0 − x * , y − x * ≤ 0 which lead to a contradiction.…”

“…Recently, Akutsah et al [28] introduced a new algorithm for finding a common fixed point of generalized nonexpansive mappings. Moreover, the fixed point problems have been studied in many aspects, see for instance [29][30][31][32][33]. Inspired by Bot et al [16], Artsawang and Ungchittrakool [25] introduced the inertial Mann-type algorithm for finding a fixed point of a nonexpansive mapping and applying to monotone inclusion and image restoration problems as follows:…”

Halpern-Type Inertial Iteration Methods with Self-Adaptive Step Size for Split Common Null Point Problem

“…Comparison: Furthermore, we compare our proposed methods to the methods in Byrne et al [8], Kazmi [9], Dilshad et al [21] and Akram et al [11]. We select κ = 0.15 for Byrne et al [8], Kazmi [9]; κ = 0.5 for Akram et al [11]; µ = µ 1 = 1, µ 2 = 2 for Byrne et al [8], Kazmi [9], Dilshad et al [21] and Akram et al [11]; h(x) = x 2 , T(x) = x for Kazmi [9] and Akram et al [11]. We consider the following cases: ; It is noticed that our schemes are easy to implement and the choosing step size is free from the pre-calculation of ∥A∥.…”

“…The stopping criteria for our computation is D n < 10 −15 , where D n = ∥z n+1 − z n ∥. We compare our proposed methods to the methods in Byrne et al [8], Kazmi [9], Dilshad et al [21] and Akram et al [11].…”

“…Later, Dilshad et al [10] investigated the common solution of (S p VI s P) and the fixed point of a finite collection of nonexpansive mappings. Recently, an alternative method was suggested by Akram et al [11] to explore the common solution of (S p VI s P) and (FPP) as follows:…”