Strong Convergence Theorems for Variational Inequalities and Split Equality Problem
Joint Authors
Wu, Yu Jing
Chen, Ru Dong
Shi, Luo Yi
Source
Journal of Function Spaces and Applications
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-7, 7 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-10-23
Country of Publication
Egypt
No. of Pages
7
Main Subjects
Abstract EN
Let H1, H2, and H3 be real Hilbert spaces, let C⊆H1, Q⊆H2 be two nonempty closed convex sets, and let A:H1→H3, B:H2→H3 be two bounded linear operators.
The split equality problem (SEP) is to find x∈C, y∈Q such that Ax=By.
Let H=H1×H2; consider f:H→H a contraction with coefficient 0<α<1, a strongly positive linear bounded operator T:H→H with coefficient γ̅>0, and M:H→H is a β-inverse strongly monotone mapping.
Let 0<γ<γ̅/α, S=C×Q and G:H→H3 be defined by restricting to H1 is A and restricting to H2 is -B, that is, G has the matrix form G=[A,-B].
It is proved that the sequence {wn}={(xn,yn)}⊆H generated by the iterative method wn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)] converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 and 〈Mw̃,w-w̃〉≥0 for all w∈S.
Moreover, if we take M=G*G:H→H, γn=0, then M is a β-inverse strongly monotone mapping, and the sequence {wn} generated by the iterative method wn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn) converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 for all w∈S.
American Psychological Association (APA)
Wu, Yu Jing& Chen, Ru Dong& Shi, Luo Yi. 2013. Strong Convergence Theorems for Variational Inequalities and Split Equality Problem. Journal of Function Spaces and Applications،Vol. 2013, no. 2013, pp.1-7.
https://search.emarefa.net/detail/BIM-449919
Modern Language Association (MLA)
Wu, Yu Jing…[et al.]. Strong Convergence Theorems for Variational Inequalities and Split Equality Problem. Journal of Function Spaces and Applications No. 2013 (2013), pp.1-7.
https://search.emarefa.net/detail/BIM-449919
American Medical Association (AMA)
Wu, Yu Jing& Chen, Ru Dong& Shi, Luo Yi. Strong Convergence Theorems for Variational Inequalities and Split Equality Problem. Journal of Function Spaces and Applications. 2013. Vol. 2013, no. 2013, pp.1-7.
https://search.emarefa.net/detail/BIM-449919
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-449919