Remark on the Yosida approximation iterative technique for split monotone Yosida variational inclusions
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Remark on the Yosida approximation iterative technique for split monotone Yosida variational inclusions Monairah Alansari1 · M. Dilshad2
· M. Akram3
Received: 25 February 2020 / Revised: 28 May 2020 / Accepted: 16 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In an attractive article, Rahman et al. introduced the split monotone Yosida variational inclusions (SMYVI) and estimate the approximate solution of the split monotone Yosida variational inclusions using nonexpansive property of operators. The main result of this paper has flaw and not correct in the present form. We modify the SMYVI and give the strong convergence theorem under some new assumptions. We also give a weak convergence theorem to solve modified split Yosida variational inclusion problem using properties of averaged operators with three new supporting lemmas. Keywords Split monotone Yosida variational inclusions · Inverse strongly monotone operator · Averaged operator · Nonexpansive operator Mathematics Subject Classification 47H05 · 47H09 · 47J25
1 Introduction
The evolution equation x (t) + A(x) = 0, x(0) = x0 is the mathematical model of several physical problems of practical utilizations. It is not easy to solve if the function A is not continuous. To overcome this obstacle, Yosida introduced an idea to find a sequence of Lipschitz functions that approximate A in some sense. On the other hand, it is noted that a monotone operator in Hilbert spaces can be regularized into a single-valued Lipschitzian
Communicated by Baisheng Yan.
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M. Dilshad [email protected] Monairah Alansari [email protected] M. Akram [email protected]
1
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia
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Department of Mathematics, University of Tabuk, Tabuk 71491, Kingdom of Saudi Arabia
3
Department of Mathematics, Islamic University of Madinah, 170, Madinah, Kingdom of Saudi Arabia 0123456789().: V,-vol
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monotone operators by means of Yosida approximation. The Yosida approximation operators are advantageous to estimate the solution of variational inclusion problem using resolvent operators. In recent time, many authors applied Yosida approximation operators to solve variational inclusion and system of variational inclusion problems, see (Cao 2003; Lan 2013; Ahmad et al. 2017; Akram et al. 2018) and references therein. Let H be a Hilbert space and B be a multi-valued monotone operator, and then, the Yosida approximation of B is defined by JλB = λ1 [I − RλB ], where RλB = [I +λB]−1 be the resolvent of B, λ > 0 and the Yosida inclusion problem is to find x ∈ H , such that: 0 ∈ JλB (x) + B(x). A similar Yosida inclusion problem and a system of Yosida inclusion problems have been investigated in Ahmad et al. (2017) and Akram et al. (2018) with generalized monotone mappings. The above Yosida inclusion problem has been also studied in the setting of Hadamard manifold (Dilshad 2019). Suppose that H1 , H2 be Hilbert spa
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