The minimization of functionals of the form \[ \Phi(f) = \| Kf-g \| ^2 + |\!|\!|f|\!|\!|_{W_1,p_1}^{p_1} + |\!|\!|f|\!|\!|_{W_2,p_2}^{p_2} + \cdots + |\!|\!|f|\!|\!|_{W_n,p_n}^{p_n}, \quad f \in \mathcal{H}, \] is considered, where \( K: \mathcal{H} \to \mathcal{H}^\prime \) is a bounded linear operator between Hilbert spaces \( \mathcal{H} \) and \( \mathcal{H}^\prime \), and \( g \in \mathcal{H}^\prime \). In addition we have \( 1 \leq p_i \leq 2 \) for \( i = 1,2,\dots, n \), and the notation \( |\!|\!|f|\!|\!|_{W_i,p_i} = (\sum_{\gamma\in \Gamma} w_{i,\gamma} | \langle f, \varphi_\gamma \rangle |^{p_i})^{1/p_i} \) for \( i = 1,2,\dots, n \) is used, where \( \{\varphi_\gamma\}_{\gamma \in \Gamma} \) denotes an orthonormal basis of the Hilbert space \( \mathcal{H} \), and \( W_i = \{w_{i,\gamma}\}_{\gamma \in \Gamma} \) denotes a system of nonnegative weights (\(i = 1,2,\dots, n\)). Under the assumption \( \| K \| < 1 \) and some other conditions, the author considers an iterative process of the form \( f^m = S(f^{m-1}+K^*(g-Kf^{m-1})), \;m = 1,2,\dots \), where \( f^0 \in \mathcal{H} \) is arbitrarily chosen, and \( K^* \) denotes the adjoint operator of \( K \), and \( S: \mathcal{H} \to \mathcal{H} \) is a nonexpansive operator specified in the paper. It is shown that this iterative process converges strongly to a minimizer of the considered functional \( \Phi \). In addition, a multiparameter Tikhonov type regularization of the form \[ \Phi_{\alpha,g}(f) = \| Kf-g \| ^2 + \alpha_1 |\!|\!|f|\!|\!|_{W_1,p_1}^{p_1} + \alpha_2 |\!|\!|f|\!|\!|_{W_2,p_2}^{p_2} + \cdots + \alpha_n |\!|\!|f|\!|\!|_{W_n,p_n}^{p_n}, \quad f \in \mathcal{H}, \] is considered, where \( \alpha_1, \dots, \alpha_n \) are nonnegative regularization parameters, and the notation \( \alpha = (\alpha_1, \dots, \alpha_n) \) is used here. It is shown that under the conditions \( \lim_{\epsilon \to 0} \alpha_i(\epsilon) = 0, \;\lim_{\epsilon \to 0} \frac{\epsilon^2}{\alpha_i(\epsilon)} = 0 \) and \( \lim_{\epsilon \to 0} \frac{\alpha_i(\epsilon)}{\alpha_j(\epsilon)} = 1 \) for each \( 1 \leq i, j \leq n \) and some other conditions specified in the paper, a regularizing scheme is obtained, i.e., for each \( f_0 \in \mathcal{H} \) one has \[ \lim_{\epsilon \to 0} \sup_{\| g - Kf_0\| < \epsilon} \| f_{\alpha(\epsilon),g}^*-f^\dagger \| = 0 . \] Here \( f^\dagger \) denotes the unique minimizer of the functional \(|\!|\!|\cdot|\!|\!|_{W_1,p_1}^{p_1} + \cdots + |\!|\!|\cdot|\!|\!|_{W_n,p_n}^{p_n} \) on \( N(K)+f_0 = \{f \in \mathcal{H} : Kf = Kf_0 \} \), and \( f_{\alpha,g}^* \) denotes the minimizer of the functional \( \Phi_{\alpha,g} \). Finally some applications are presented.