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We study existence and asymptotic behavior of a bounded solution of the following Tikhonov regularized second-order difference equation (): where, A is a multivalued maximal monotone operator defined on a Hilbert space and are positive real parameters. We first prove under condition existence of unique bounded solution of (). For asymptotic behavior, we use a suitable assumption on cn and to prove strong convergence of un to the element of minimal norm of Some applications are thereafter discussed with respect to minimization and saddle-point problems. Specially, we study the rate of convergence of optimal values in convex minimization and convex-concave problems. We end the paper by concluding remarks and noticing some research perspectives. [ABSTRACT FROM AUTHOR]

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equationwhereAis a maximal monotone operator in a real Hilbert spaceHand {cn} and {θn} are positive real sequences. We do not assumeA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing 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 {cn} and {θn}. [ABSTRACT FROM AUTHOR]

We investigate asymptotic behavior of solutions to the following second order difference equation where A is a maximal monotone operator in a real Hilbert space H and {c i } is a positive real sequence. We show weak and strong convergence of solutions to an element of A −1(0), under appropriate assumptions on A and the sequence {c i }. Our results extend and improve previous results by Morosanu [21] and Mitidieri-Morosanu [18] (see also [19, pp. 156–168]) under more general assumptions on c i . [ABSTRACT FROM AUTHOR]