Fixed point iterations for nonlinear Hammerstein equation involving nonexpansive and accretive mappings (Q1120114)

The solution of the nonlinear Hammerstein operator equation \[ x+KNx=f, \] where K and N are nonexpansive and accretive mappings, and K also satisfies a monotonicity condition is approximated in a Hilbert space by means of fixed point iteration processes. The main result: Theorem: Let H be a separable Hilbert space and let C be a nonempty bounded closed convex subset of H. Suppose: (a) N: \(C\to C\) is a nonlinear nonexpansive monotone map; (b) K: \(C\to C\) is a nonexpansive monotone map; such that for some \(\mu >0\) and each \(x\in H:<Kx,x>\geq \| Kx\|^ 2\). Define S: \(C\to C\) by \(Sx=f-KNx\). Let \(\{x_ n\}\) be a sequence defined iteratively by \(x_ 0\in C\), \[ x_ n=(1-a_ n)x_ n+a_ nSy_ n;\quad y_ n=(1-b_ n)x_ n+b_ nSx_ n,\quad n\geq 0, \] where \(\{a_ n\}\) and \(\{b_ n\}\) are real sequences satisfying the following conditions: (i) \(0\leq a_ n,b_ n\leq 1\) for all n; (ii) \(\limsup b_ n<1\); (iii) \(\sum_{n}a_ nb_ n=\infty\). Then \(\{x_ n\}\) converges weakly to the unique solution of \(x+KNx=f.\)