Asymptotics of a General Second-Order Difference Equation and Approximation of Zeroes of Monotone Operators. (1st September 2016)

• Record Type: Journal Article
• Authors: Djafari Rouhani, Behzad; Khatibzadeh, Hadi
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 9(2016)
• Page Start: 1107
• Page End: 1130
• Publication Date: 2016-09-01
• Subjects: Asymptotic behavior -- maximal monotone operator -- second-order difference equation -- strong convergence -- weak convergence; 39A12 -- 39A11 -- 47H05; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1200613
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equation where A is a maximal monotone operator in a real Hilbert space H and { c n } and {θ n } are positive real sequences. We do not assume A −1 (0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients { c n } and {θ n }.