For solving ill-posed problems of the form \(Ax=y\), \(A: X\to Y\), nonlinear iteration processes are considered (1) \(x^{k+1}=P_ kU_ kx^ k\), (2) \(x^{k+1}=[\lambda P_ k+(1-\lambda)U_ k]x^ k\), \(x^ 0\in X\), \(k=0,1,\dots\), where \(P_ k\) defined by the set \(Q=\{x\in X: g_ j(x)\leq 0\), \(j=1,2,\dots,m\}\). Theorem: Let \(\{U_ k\}\), \(\{P_ k\}\), \(U_ k,P_ k: X\to X\) satisfy a) for every \(k\) \(U_ k\in P_ m=\{T: \| Tx-z\|^ 2\leq \| x-z\|^ 2-\nu_ k\| Tx-x\|^ 2\) \(\forall z\in M=\{z: Tz=z\}, x\in X\}\), where \(\nu_ k\geq \nu\geq 0\); b) if \(z_{k_ i}\rightharpoonup x\) weakly and \(z_{k_ i}-U_{k_ i}z_{k_ i}\to 0\) then \(x\in M\); c) for every \(k\) \(P_ k\in P_ Q=\{T:\| Tx-z\|^ 2\leq \| x-z\|^ 2-\nu_ k'\| Tx- x\|^ 2\) \(\forall z\in Q=\{z: Tz=z\}\neq\emptyset, x\in X\}\), where \(\nu_ k' \geq \nu'>0\): d) if \(z_{k_ i} \rightharpoonup x\) weakly and \(z_{k_ i}-P_{k_ i}z_{k_ i}\to 0\) then \(x\in Q\).
Then for every \(x^ 0 \in X\) for (1) and (2) we have 1. \(x^ k \to x \in M\cap Q\); 2. \(\inf_ y \{\lim_ k \| x^ k- y\|: y\in M\cap Q\} =\lim_ k \| x^ k-x\| = d\geq 0\); 3. either \(\| x^{k+1}- x\| < \| x^ k-x\|\) for every \(k\), or \(\{x^ k\}\) is stationary starting from \(k \geq k_ 0\); 4. \(\lim_ k\| x^{k+1}- x^ k\| = 0\) and moreover for every \(y \in M\cap Q\), \(\sum^ \infty_{k=0}\| x^{k+1}-x^ k \|^ 2 \leq (\| x^ 0- y\|^ 2-d^ 2)/ \hbox{min} \{\nu,\nu'\}\).