A modified iterative method for split problem of variational inclusions and fixed point problems
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A modified iterative method for split problem of variational inclusions and fixed point problems P. Majee1 · C. Nahak1
Received: 2 March 2017 / Accepted: 16 February 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, we study a new split problem of variational inclusions and fixed point problems. With the help of averaged mappings, we introduce a new iterative method to solve the stated split problem. Strong convergence results are obtained under mild conditions. Also, we present some preliminary numerical examples to demonstrate the convergence and efficiency of our proposed algorithms. Keywords Nonexpansive mapping · Averaged mapping · Split variational inclusion problem · Split feasibility problem · Fixed point problem Mathematics Subject Classification 47H10 · 49J40
1 Introduction Let H1 and H2 be two real Hilbert spaces with inner product ·, · and norm · . Let C and Q be two nonempty closed and convex subset of H1 and H2 , respectively. A mapping T : H1 → H1 is said to be nonexpansive if T x − T y ≤ x − y, ∀x, y ∈ H1 . The set of all fixed points of T is defined as Fix(T ) = {x ∈ H1 : T x = x}. Split feasibility problem was first introduced by Censor and Elfving (1994) in the finite dimensional space, which could be formulated as follows: find x ∈ H1 such that x ∈ C and Ax ∈ Q,
(1)
Communicated by José Mario Martínez.
B
C. Nahak [email protected] P. Majee [email protected]
1
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
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P. Majee, C. Nahak
where A : H1 → H2 is a mapping. This problem has got much attention due to its several applications in signal processing, image reconstruction, intensity-modulated radiation therapy, approximation theory, control theory, biomedical engineering and geophysics (see Byrne 2002; Censor et al. 2006; Censor and Elfving 1994 and the references therein). Also, problem (1) is strongly related to some general problems such as the convex feasibility problem (Bauschke and Borwein 1996), the multiple-set split feasibility problem (Censor et al. 2005), the split equality problem (Moudafi 2013), the split common fixed point problem (Yao et al. 2015). Now, let Γ = {x ∈ C such that Ax ∈ Q}. To find a solution of the split feasibility problem (1), Byrne (2002, 2004) introduced the well-known CQ algorithm, which is defined as follows: for any x0 ∈ H1 define xn+1 = PC (xn − γ A∗ (I − PQ )Axn ), n ≥ 0, where PC and PQ are the projection operator on C and Q, respectively. Later, relaxed CQ algorithm, averaged CQ algorithm and many other algorithms were introduced to tackle the problem (1) (see, Xu 2006; Yang 2004 and references therein). Zhu et al. (2013) introduced the following problem: find x ∈ C ∩ Fix(T ) such that Ax ∈ Q ∩ Fix(S),
(2)
where T and S are nonexpansive mappings on C and Q, respectively. To find a solution of (2), Zhu et al. (2013) introduced the following algorithm: take x 0 ∈ H1 and vn = T PC (xn − γ A∗ (I − S PQ )Axn ), (3) x
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