Asymptotics of a Difference Equation and Zeroes of Monotone Operators

Abstract

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation: where A is a maximal monotone operator in a real Hilbert space H, and {c n } is a positive sequence of real numbers. We do not assume that 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 zeros of monotone operators, as well as unify and extend previously known results in [2, 3, 7, 10, 18, 22, 23, 25] by considering much weaker conditions on the coefficients {c n }. In particular, our new strong ergodic theorem extends the results of [7] and [23, Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem.