A modified contraction method for solving certain class of split monotone variational inclusion problems with application

Abstract

The main purpose of this paper is to propose a new modified contraction method for solving a certain class of split monotone variational inclusion problems in real Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a solution of the aforementioned problem. Our strong convergence result is obtained when the underline operator is monotone and Lipschitz continuous, and the knowledge of its Lipschitz constant is not required. As application, we solved the split linear inverse problems, for which we also considered a special case of this problem, namely, the LASSO problem. We also give some numerical illustrations of the proposed method in comparison with other methods in the literature to further show the applicability and advantage of our results. The results obtained in this paper generalize and improve many recent results in this direction. Keywords Armijor-like search rule • Contraction method • Split monotone variational inclusion problems • Variational inequality problems • Strong convergence Mathematics Subject Classification 47H09 • 47H10 • 49J20 • 49J40 Communicated by Baisheng Yan.

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“…Lemma 2 (see [33,35]). Let H be a real Hilbert space with μ, ] ∈ H and α ∈ R, then the following holds…”