Asymptotics of a General Second-Order Difference Equation and Approximation of Zeroes of Monotone Operators

作者：Rouhani Behzad Djafari^*; Khatibzadeh Hadi

来源：Numerical Functional Analysis and Optimization, 2016, 37(9): 1107-1130.

DOI：10.1080/01630563.2016.1200613

• Asymptotic behavior
• maximal monotone operator
• second-order difference equation
• strong convergence
• weak convergence

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equation
{u(n+1) - (1 + theta(n))u(n) + theta(u)u(n-1) is an element of c(n)Au(n), n >= 1 u(0) = x, sup(n >= 0) vertical bar un vertical bar < + infinity
where A is a maximal monotone operator in a real Hilbert space H and {c(n)} and {(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 {theta(n)}.

• 出版日期2016