Existence and Asymptotic Behavior of Second-Order Difference Equation with Tikhonov Regularization. (10th December 2018) - Balhag, Aicha; Chbani, Zaki; Riahi, Hassan
Abstract: We study existence and asymptotic behavior of a bounded solution of the following Tikhonov regularized second-order difference equation (D S x, ε ):u n + 1 − 2 u n + u n − 1 ∈ c n A u n + ε n c n u n ; for n ≥ 1, with ; u 0 = x . where, A is a multivalued maximal monotone operator defined on a Hilbert space andc n, ε n are positive real parameters. We first prove under conditionA − 1 ( 0 ) ≠ ∅, existence of unique bounded solution of (D S x, ε ). For asymptotic behavior, we use a suitable assumption on cn andε n to prove strong convergence of un to the element of minimal norm ofA − 1 ( 0 ) . 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.