Abstract
Some relaxed hybrid iterative schemes for approximating a common element of the sets of zeros of infinite maximal monotone operators and the sets of fixed points of infinite weakly relatively non-expansive mappings in a real Banach space are presented. Under mild assumptions, some strong convergence theorems are proved. Compared to recent work, two new projection sets are constructed, which avoids calculating infinite projection sets for each iterative step. Some inequalities are employed sufficiently to show the convergence of the iterative sequences. A specific example is listed to test the effectiveness of the new iterative schemes, and computational experiments are conducted. From the example, we can see that although we have infinite choices to choose the iterative sequences from an interval, different choice corresponds to different rate of convergence.
Highlights
Throughout this paper, let X be a real Banach space with norm · and X∗ be the dual space of X
0, which implies from Lemma 2.4 that (J + rn,iAi)–1J(xn + en) – → 0, as n → ∞
This ensures that JB1yn – JBkyn → 0 for k = 1, as n → ∞
Summary
Throughout this paper, let X be a real Banach space with norm · and X∗ be the dual space of X. If X is a real reflexive, strictly convex, and smooth Banach space and K is a non-empty closed and convex subset of X, for ∀x ∈ X, there exists a unique point x0 ∈ K such that φ(x0, x) = inf{φ(y, x) : y ∈ K} In this case, we can define the generalized projection mapping K : X → K by K x = x0 for ∀x ∈ X [3]. Klin-eam et al [5] presented the following projection iterative scheme for maximal monotone operator A and two strongly relatively non-expansive mappings B and C in a real uniformly convex and uniformly smooth Banach space X.
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