A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed
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(2019) 38:77
A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces A. Taiwo1 · L. O. Jolaoso1 · O. T. Mewomo1 Received: 16 October 2018 / Revised: 12 March 2019 / Accepted: 18 March 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we introduce a modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and split-equality fixed-point problem for Bregman quasi-nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We introduce a generalized step size such that the algorithm does not require a prior knowledge of the operator norms and prove a strong convergence theorem for the sequence generated by our algorithm. We give some applications and numerical examples to show the consistency and accuracy of our algorithm. Our results complement and extend many other recent results in this direction in literature. Keywords Split feasibility problem · Minimization problem · Proximal operator · Bregman quasi-nonexpansive · Split equality problem · Fixed point problem Mathematics Subject Classification 47H10 · 47J25 · 47N10 · 65J15 · 90C33
1 Introduction Let E 1 and E 2 be Banach spaces and let C and Q be nonempty closed convex subsets of E 1 and E 2 , respectively. We denote the dual of E 1 and E 2 by E 1∗ and E 2∗ , respectively. Let A: E 1 → E 2 ba a bounded linear operator. The split feasibility problem (SFP) can be formulated as:
Communicated by Gabriel Haeser.
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O. T. Mewomo [email protected] A. Taiwo [email protected] L. O. Jolaoso [email protected]; [email protected]
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School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
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find x ∈ C such that Ax ∈ Q.
(1.1)
The notion of SFP was first introduced by Censor and Elfving (1994) in the framework of Hilbert spaces for modeling inverse problems which arise from phase retrievals and medical image reconstruction. The SFP has attracted much attention due to its applications in modeling real-world problems such as inverse problem in signal processing, radiation therapy, data denoising and data compression (see Ansari and Rehan 2014; Bryne 2002; Censor et al. 2005, 2006; Mewomo and Ogbuisi 2018; Shehu and Mewomo 2016 for details). A very popular algorithm constructed to solve the SFP in real Hilbert spaces was the following CQ-algorithm proposed by Bryne (2002). Let x1 ∈ C and compute xn+1 = PC (xn − μA∗ (I − PQ )Axn ), n ≥ 1,
(1.2)
where A∗ is the adjoint of A, PC and PQ are the metric projections of C and Q, respectively, μ ∈ (0, λ2 ) with λ being the spectral radius of the operator A∗ A. The sequence generated by (1.2) was shown to converge weakly to a solution of the SFP (1.1). Schöpfer et al. (2008) studied the problem (1.1) in p-uniformly convex real Banach spaces which are also uniformly smooth and proposed the f
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