Algorithms for finding a common element of the set of common fixed points for nonexpansive semigroups, variational inclu
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Algorithms for finding a common element of the set of common fixed points for nonexpansive semigroups, variational inclusions and generalized equilibrium problems Meng Wen1,2 · Changsong Hu3 · Angang Cui4 · Jigen Peng1 Received: 10 October 2019 / Accepted: 13 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this work, we introduced an iterative algorithm for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup, the set of solutions to a variational inclusion and the set of solutions to a generalized equilibrium problem in a real Hilbert space. Furthermore, we proved the convergence theorem of the proposed iterative algorithm under some mild conditions on algorithm parameters. Finally, we gave some numerical examples which support our main theorem at last part. The results presented in this paper generalize the well-known theorems. Keywords Nonexpansive semigroup · Variational inclusion · Equilibrium problem · Inverse-strongly monotone mapping Mathematics Subject Classification 47H09 · 47H10
1 Introduction Throughout the paper unless otherwise stated, let H be a real Hilbert space with norm . and inner product ., .. Let C be a nonempty, closed and convex subset of H . Let A : C → H be a nonlinear mapping. Then A
B B
Meng Wen [email protected] Jigen Peng [email protected]
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School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, People’s Republic of China
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School of Science, Xi’an Polytechnic University, Xi’an 710048, People’s Republic of China
3
Department of Mathematics, Hubei Normal University, Huangshi 435002, People’s Republic of China
4
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China 0123456789().: V,-vol
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(1) is said to be monotone if Ax − Ay, x − y ≥ 0,
∀x, y ∈ C,
(2) is called be η-strongly monotone if there exists a positive constant η such that Ax − Ay, x − y ≥ ηx − y2 ,
∀x, y ∈ C,
(3) is called k-Lipschitzian if there exists a positive constant k such that Ax − Ay ≤ kx − y,
∀x, y ∈ C,
(4) is said to be inverse-strongly monotone if there exists a constant α > 0 such that Ax − Ay, x − y ≥ αAx − Ay2 ,
∀x, y ∈ C.
For such a case, A is also said to be α-inverse-strongly monotone. Let B : H → H be a single-valued nonlinear mapping and let M : H → 2 H be a set-valued mapping. The variational inclusion is to find u ∈ H such that θ ∈ B(u) + M(u),
(1.1)
where θ is a zero vector in H . The set of solutions to variational inclusion (1.1) is denoted by I (B, M). When B = 0, then (1.1) becomes the inclusion problem introduced by Rockafellar [1]. Let F be a bifunction of C × C into R, where R denotes the set of real numbers and let A : C → H be an inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem: find x ∈ C such that F(x, y) + Ax, y − x ≥ 0,
∀y ∈ C.
(1.2)
The solution set of (1.2) is denoted by EP(F,A), i.e., E P(F, A) = {x
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