1 Erratum to: Optim Lett DOI 10.1007/s11590-012-0516-2

The original publication of the article contains errors which need to be amended as mentioned below:

  1. (1)

    In Section 2, “Formulations and basic facts”, lines 2 and 3, “Let \(T,g,h:\mathcal H \rightarrow \mathcal H \) be three nonlinear single-valued operators” should be replaced by “Let \(T,g,h:\mathcal H \rightarrow \mathcal H \) be three nonlinear single-valued operators such that \(g\) be an onto operator”.

  2. (2)

    In the problem (2.1), “\(\rho \varphi (u)\)” must be changed to “\(\rho \varphi (h(u))\)”.

  3. (3)

    In Section 6, part “(III)” must be edited as below: (III) Let the operators \(T\) and \(g\) be linear and suppose that the inverses of \(T\) and \(g\), that is, \(T^{-1}\) and \(g^{-1}\) exist. Then (6.1) can be written as follows:

    $$\begin{aligned} Th^{-1}J_{\varphi }^{\rho }z+\rho ^{-1}R_{\varphi }z=0&\Leftrightarrow T(g^{-1}(z-R_{\varphi }z))+\rho ^{-1}R_{\varphi }z=0\\&\Leftrightarrow g^{-1}(z-R_{\varphi }z)=T^{-1}(-\rho ^{-1}R_{\varphi }z)\\&\Leftrightarrow z-R_{\varphi }z=g(-\rho ^{-1}T^{-1}R_{\varphi }z)\\&\Leftrightarrow z=R_{\varphi }z-\rho ^{-1}gT^{-1}R_{\varphi }z\\&\Leftrightarrow z=(I-\rho ^{-1}gT^{-1})R_{\varphi }z. \end{aligned}$$
  4. (4)

    In Algorithm 6.5, the iterative process “\(z_{n+1}=(1-\alpha _n)z_n+\alpha _n(I-\rho ^{-1}hT^{-1}) R_{\varphi }z_n\)”, must be replaced by “\(z_{n+1}=(1-\alpha _n)z_n+ \alpha _n(I-\rho ^{-1}gT^{-1})R_{\varphi }z_n\)”.

  5. (5)

    In Algorithm 6.8, the iterative process “\(z_{n+1}=(1-\alpha _n)z_n+\alpha _n(I-\rho ^{-1}hT^{-1})Q_Kz_n\)”, should be replaced by “\(z_{n+1}=(1-\alpha _n)z_n+\alpha _n(I-\rho ^{-1}gT^{-1})Q_Kz_n\)”. All the assertions are valid with these corrections.