On the Approximation of Zeros of Monotone Operators by Solutions to a Difference Equation

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: { un+1 − 2un + un−1 ∈ cnAun; n ≥ 1 u0 ∈ H, supn≥0 |un| < +∞, where A is a maximal monotone operator in a real Hilbert space H, and {cn} 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 {cn}. 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.