Iterative Algorithms for Nonlinear Operators

By Hong-Kun Xu. 2002

1,837 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b848cafc5e3dd5a4f1eb44352508f69c11e2c144

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Citations in the corpus

Modified Tseng's extragradient methods for solving pseudo-monotone variational inequalities (2019, methodology): “The next technical lemma is very useful and used by many authors, for example Liu [21] and Xu [36].”
A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality (2014, background): “4 [23] Assume {an} is a sequence of nonnegative real numbers such that anþ1 ð1 nÞan þ n, where { n} is a sequence in (0, 1) and { n} is a sequence such that (1) P1 n1⁄41 n 1⁄4 1; (2) lim supn!1 n/ n 0 or P1 n1⁄41 j nj51.”
A new iterative method for the split common fixed point problem in Hilbert spaces (2017, neutral): “5 [12]: Assume (an) is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn)an + γnδn, n ≥ 0,”
Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems (2018, background): “9 [34,35]: Let {an} be sequences of nonnegative real numbers such that: an+1 ≤ (1 − αn)an + αnbn,”
An iterative method for solving split monotone variational inclusion and fixed point problems (2015, neutral): “5 [27,29] Assume {an} is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn)an + γnδn, n ≥ 0,”
Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings (2008, background): “Lemma 2.3 [ 6 ] Assume {an} is a sequence of nonnegative real numbers such that”
Effective metastability of Halpern iterates in CAT(0) spaces (2011, background): “Another partial answer in the case of uniformly smooth Banach spaces was obtained by Xu [53] for (λn) satisfying (C1), (C3) and (C6) (which is weaker than Lions’ (C5)).”
A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem (2019, positive): “Lemma 2.11 (Xu 2002) Let {an} be a sequence of nonnegative real numbers satisfying the following relation: an+1 ≤ (1 − αn)an + αnσn + γn, n ≥ 1, where (i) {αn} ⊂ [0, 1], ∑∞n=1 αn = ∞, (ii) lim supn→∞ σn ≤ 0, (iii) γn ≥ 0, (n ≥ 1) and ∑∞n=1 γn ∞.”