Abstract
In this paper, we present two iterative algorithms involving Yosida approximation operators for split monotone variational inclusion problems ( S p MVIP ). We prove the weak and strong convergence of the proposed iterative algorithms to the solution of S p MVIP in real Hilbert spaces. Our algorithms are based on Yosida approximation operators of monotone mappings such that the step size does not require the precalculation of the operator norm. To show the reliability and accuracy of the proposed algorithms, a numerical example is also constructed.
Highlights
Variational inequality which was brought into existence by Hartman and Stampacchia [1] plays an important role as mathematical model in physics, economics, optimization, networking structural analysis, and medical images
Λ1 and λ1 are the resolvent and Yosida approximation operators of V1 + G1, respectively, for λ1 > 0, following are equivalent: (i) x∗ ∈ H1 is the solution of ðV1 + G1Þ−1ð0Þ
The proof is trivial which is an immediate consequence of definitions of resolvent and Yosida approximation operator of maximal monotone mapping V1 + G1.☐
Summary
Variational inequality which was brought into existence by Hartman and Stampacchia [1] plays an important role as mathematical model in physics, economics, optimization, networking structural analysis, and medical images. Kazmi and Rizwi [9] proposed the following iterative method for approximating the common solutions of SpVIP and fixed point problem of a nonexpansive mapping: h i yn = RGλ 1 xn + γA∗ RGλ 2 − I Axn , ð6Þ xn+1 = αn f ðxnÞ + ð1 − αnÞSyn, where f is a contraction and S is nonexpansive mapping. Sitthithakerngkiet et al [10] studied the common solutions of SpVIP and a fixed point of an infinite family of nonexpansive mappings and introduced the following iterative method:. Due to the fact that the zero of Yosida approximation operator associated with monotone operator G is the zero of inclusion problem 0 ∈ GðxÞ and inspired by the work of Moudafi, Byrne, Kazmi, and Dilshad et al, our motive is to propose two iterative methods to solve SpMVIP.
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