Geometric estimation of the solution to $x+Tx=0$ for unbounded densely defined monotone operator $T$ in Hilbert space

Abstract:

In recent papers R. Bruck and J. C. Dunn have given convergent schemes for approximating the solution p of $x + Tx = f$ for T a monotone mapping on a Hilbert space, with T locally bounded. The present paper derives a geometric fact and uses this in a direct manner to give a scheme applicable to densely defined T. The scheme is computable with decreasing error estimates without any assumptions of boundedness. The convergence of the scheme to the solution p is proven under the weak condition that $\left \| {{x_n} + T{x_n}} \right \|$ grow no faster than ${n^{1/2}}$.