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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:
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(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”.
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(2)
In the problem (2.1), “\(\rho \varphi (u)\)” must be changed to “\(\rho \varphi (h(u))\)”.
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(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)
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 “\(z_{n+1}=(1-\alpha _n)z_n+ \alpha _n(I-\rho ^{-1}gT^{-1})R_{\varphi }z_n\)”.
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(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.
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The online version of the original article can be found under doi:10.1007/s11590-012-0516-2.
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Balooee, J., Cho, Y.J. Erratum to: Algorithms for solutions of extended general mixed variational inequalities and fixed points. Optim Lett 7, 1957–1958 (2013). https://doi.org/10.1007/s11590-013-0627-4
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DOI: https://doi.org/10.1007/s11590-013-0627-4