Convergence theorem for solving a new concept of the split variational inequality problems and application
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Convergence theorem for solving a new concept of the split variational inequality problems and application Anchalee Sripattanet1 · Atid Kangtunyakarn1 Received: 18 March 2020 / Accepted: 13 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we propose the split of modified variational inequality problems (SMVIP), by using the concept of the modified generalized system of variational inequalities (MGSV). Then, we prove the strong convergence theorem for solving fixed point problems of nonlinear mappings and two variational inequality problems and solving the SMVIP. Applying our main result, we prove strong convergence theorems of the split minimization problem and the split variational inequality problem. In support of our main result, a numerical example is also presented. Keywords Variational inequality · The intermixed algorithm · Strong convergence theorem · Fixed point Mathematics Subject Classification 47H09 · 47H10 · 90C33
1 Introduction The variational inequality problem (VIP) is to find a point u ∈ C such that Au, v − u ≥ 0, for all v ∈ C. The set of all solutions of the variational inequality is denoted by V I (C, A). Historically the variational inequality was introduced by Stampachhia [34] in 1964. After that variational inequalities become interested in various topics such as optimization, physic and applied sciences (see, for instance, [5–15,17,22,29,31]). In 2012, Kangtunyakarn [24] modified the set of variational inequality problems VI(C,A) as follows: V I (C, a A + (1 − a)B) = {x ∈ C : y − x, (a A + (1 − a)B)x ≥ 0, ∀y ∈ C, a ∈ (0, 1)},
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Atid Kangtunyakarn [email protected] Anchalee Sripattanet [email protected]
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Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand 0123456789().: V,-vol
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A. Sripattanet, A. Kangtunyakarn
where A and B are the mappings of C into H . He also proved the strong convergence theorem of a new iterative scheme for finding a common element of the set of fixed point problems of infinite family of κi -pseudo-contractive mappings and the set of solutions of equilibrium problem and two sets of solutions of variational inequality problem as follows: Theorem 1 Let C be a closed convex subset of Hilbert space H and let F : C × C → R be a bifunction satisfying (A1 )–(A4 ), let A, B : C → H be α and β-inverse strongly ∞ be κ -pseudo-contractive mappings of C into itself with monotone, respectively. Let {Ti }i=1 i j j j j j j k = sup ki and ρi = (α1 , α2 , α3 ) ∈ I × I × I , where I = [0, 1], α1 + α2 + α3 = 1, j j j j j α1 + α2 ≤ b < 1 and α1 , α2 , α3 ∈ (κ, 1) for all j = 1, 2, . . .. For every n ∈ N, let Sn and S be S-mapping generated by Tn , Tn−1 , . . . , T1 and ρn , ρn−1 , . . . , respectively. Assume that ∞ F(T ) ∩ E P(F) ∩ V I (C, A) ∩ V I (C, B) = ∅ and let {x } and {u } be generated F = ∩i=1 i n n by x1 , u ∈ C and F(u n , y) + r1n y − u n , u n − xn ≥ 0, ∀y ∈ C, (1) xn+1 = αn u + (1 − αn )Sn PC (I − γ (a A + (1 − a)B))u n , ∀n ≥ 1
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