A new approximation scheme for solving various split inverse problems
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A new approximation scheme for solving various split inverse problems A. Taiwo1 · A. O.-E. Owolabi1 · L. O. Jolaoso1 · O. T. Mewomo1 · A. Gibali2,3 Received: 31 January 2020 / Accepted: 1 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we study the split equality problem for systems of monotone variational inclusions and fixed point problems of set-valued demi-contractive mappings in real Hilbert spaces. A new viscosity algorithm for solving this problem is introduced along with its strong convergence theorem. Several known theoretical applications, such as, split common null point problem for systems of monotone variational inclusions and fixed point problems, split equality saddle-point and fixed point problem are given. Two primary numerical examples which illustrate and compare the behavior of the new scheme, suggest that the method has a potential applicable value besides its theoretical generalization. Our work extends and generalizes some existing works in the literature as well as provide some new direction for future work. Keywords Viscosity approximation algorithm · Variational inclusion problem · Demi-contractive mapping Mathematics Subject Classification 47H10 · 47J25 · 47N10 · 65J15 · 90C33
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A. Gibali [email protected] A. Taiwo [email protected] A. O.-E. Owolabi [email protected] L. O. Jolaoso [email protected] O. T. Mewomo [email protected]
1
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
2
Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel
3
The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel
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A. Taiwo et al.
1 Introduction In various applied sciences, such as quantum mechanics, filtration theory, and more, see e.g., [2], an operator equation which consists of the sum of two monotone operators is considered. Let H be a real Hilbert space with inner product ·, · and norm · and I : H → H be the identity mapping on H . The Variational Inclusion Problem (VIP) defined for the sum of two mappings is defined as finding a point x ∗ ∈ H such that 0 ∈ f x ∗ + F x ∗,
(1)
where f : H → H is an operator and F : H → 2 H is a set-valued operator. The inverse problem model (1) is quite general and stands at the core of many applications such as image recovery, signal processing, and more, see e.g., [5,12,15,25,31]. Recently, VIPs have been studied in the settings of real Hilbert and Banach spaces by many authors, for example [51,53] and the references therein. The problem includes, as special cases, saddle-point problem, equilibrium problem, variational inequality problem, split feasibility problem and minimization problem, see [17–19,21–23,27,30,32,44]. Let H1 and H2 be real Hilbert spaces and C and Q be nonempty closed convex subsets of H1 and H2 respectively. Let A : H1 → H2 be a bounded linear operator with adjoint operator A∗ . The split c
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