Application of new strongly convergent iterative methods to split equality problems
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Application of new strongly convergent iterative methods to split equality problems Pankaj Gautam1 · Avinash Dixit1
· D. R. Sahu2 · T. Som1
Received: 18 June 2019 / Revised: 16 May 2020 / Accepted: 23 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we study the generalized problem of split equality variational inclusion problem. For this purpose, we introduced the problem of finding the zero of a nonnegative lower semicontinuous function over the common solution set of fixed point problem and monotone inclusion problem. We proposed and studied the convergence behaviour of different iterative techniques to solve the generalized problem. Furthermore, we study an inertial form of the proposed algorithm and compare the convergence speed. Numerical experiments have been conducted to compare the convergence speed of the proposed algorithm, its inertial form and already existing algorithms to solve the generalized problem. Keywords Split equality problem · Variational inclusion problem · Fixed point problem · Quasi-nonexpansive mapping Mathematics Subject Classification 47J25 · 47H05 · 47H09 · 49J53
1 Introduction In 1994, Censor and Elfving (Censor and Elfving 1994) first introduced the split feasibility problem (SFP) in finite-dimensional spaces. Such problems arise in signal processing, specif-
Communicated by Ernesto G. Birgin.
B
Avinash Dixit [email protected] Pankaj Gautam [email protected] D. R. Sahu [email protected] T. Som [email protected]
1
Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India
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Department of Mathematics, Banaras Hindu University, Varanasi 221005, India 0123456789().: V,-vol
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ically in phase retrieval and other image restoration problems. It has been found that the SFP can also be used in different areas such as computer tomography and intensity-modulated radiation therapy (Censor et al. 2005, 2006, 2007). The split feasibility problem (SFP) is find x ∗ ∈ C such that Ax ∗ ∈ Q,
(1.1)
where C and Q are nonempty closed convex subsets of real Hilbert spaces H1 and H2 , respectively, and A : H1 → H2 is a bounded linear operator. Some works on split feasibility problems in an infinite-dimensional real Hilbert space can be found in Byrne (2002), Censor et al. (2006) and Xu (2006). In 2012, Censor et al. (2012) introduced the following split variational inequality problem: find x ∗ ∈ C such that f (x ∗ ), x − x ∗ ≥ 0 for all x ∈ C, and y ∗ = Ax ∗ ∈ Q that solves g(y ∗ ), y − y ∗ ≥ 0 for all y ∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces H1 and H2 , respectively, A : H1 → H2 is a bounded linear operator and f : H1 → H1 , g : H2 → H2 are the given operators. In 2011, Moudafi (2011) extended the split variational inequality problem (Censor et al. 2012) and proposed the following split monotone variational inclusion problem (SMVIP): find x ∗ ∈ H1 such that f (x ∗ ) + B1 (x ∗ )
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