The goal of the paper is to yield progress in elastic-net regularization coming from statistics introduced and established in the inverse problems community by H. Zou and T. Hastie [“Regularization and variable selection via the elastic net”, J. R. Stat. Soc., Ser. B, Stat. Methodol. 67, No. 2, 301–320 (2005; Zbl 1069.62054)], B.-T. Jin, D. A. Lorenz and S. Schiffler [“Elastic-net regularization: error estimates and active set methods”, Inverse Probl. 25, No. 11, Article ID 115022 (2009; Zbl 1188.49026)], and C. De Mol, E. De Vito and L. Rosasco [“Elastic-net regularization in learning theory”, J. Complexity 25, No. 2, 201–230 (2009; Zbl 1319.62087)]. For an ill-posed linear operator equation \(Kx=y\) with a smoothing forward operator \(K:\ell^2(\Gamma) \to H\) (with \(H\) a Hilbert space), motivated by imaging problems and sparsity effects, penalized least squares with penality term \(p_\omega(x)=2 \omega \|x\|_{\ell^1}+\|x\|_{\ell^2}^2\) are exploited for obtaining stable approximate solutions. The used linear combination of \(\ell^1\) and \(\ell^2\) terms with weight parameter \(\omega>0\) can be adapted to different situations in a wide range of applications. In contrast to the standard case of \(\ell^2\) regularization, the handling of \(\ell^1\) terms requires special numerical algorithms for computing the regularized solutions. On the one hand, this paper makes contributions to such algorithms. On the other hand, the paper contains functional analytic studies on the convergence and on the asymptotic behaviour of regularized solutions. The authors discuss the interplay of balls having radius \(R>0\) in \(\ell^1\) and the magnitude of \(\omega\), which means that the choice of regularization parameters as always is a delicate problem. It seems to be an aim of the authors to cast in a unified framework the classical Tikhonov regularization, the \(\ell^1\) regularization, and the mixed version under consideration.