A modified contraction method for solving certain class of split monotone variational inclusion problems with applicatio
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A modified contraction method for solving certain class of split monotone variational inclusion problems with application C. Izuchukwu1
· J. N. Ezeora2 · J. Martinez-Moreno3
Received: 22 March 2020 / Revised: 31 May 2020 / Accepted: 6 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
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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C. Izuchukwu [email protected]; [email protected] J. N. Ezeora [email protected]; [email protected] J. Martinez-Moreno [email protected]
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School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
2
Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
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Department of Mathematics, Faculty of Experimental Science, University of Jaén, Jaén, Spain 0123456789().: V,-vol
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1 Introduction Let X 1 and X 2 be two vector spaces, T : X 1 → X 2 be a linear operator, and I P1 and I P2 be two inverse problems in X 1 and X 2 respectively. A Split Inverse Problem (SIP) (see Censor et al. 2012; Gibali 2017) is formulated as follows: Find x ∗ ∈ X 1 that solves I P1 ,
(1.1)
y ∗ = T x ∗ ∈ X 2 solves I P2 .
(1.2)
such that
The first known case of the SIP is the following Split Feasibility Problem (SFP) introduced and studied by: Censor and Elfving (1994): Find x ∗ ∈ C such that y ∗ = T x ∗ ∈ Q,
(1.3)
where C and Q are nonempty closed and convex subsets of R N and R M , respectively, and T ∈ R M×N is a real matrix. The SFP is known to have wide applications in many fields such as phase retrieval, medical image reconstruction, signal processing, radiation therapy treatment planning, among others (for example, see Byrne 2004; Censor et al. 2006; Censor and Elfving 1994; Censor et al. 2005; Senakka and Cholamjiak 2016 and the references therein). Another form of the SIP which is more general than the SFP is the
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