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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391301, 11 pages
doi:10.1155/2012/391301
Research Article

Duality Fixed Point and Zero Point Theorems and Applications

Qingqing Cheng, Yongfu Su, and Jingling Zhang

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 22 May 2012; Accepted 29 June 2012

Academic Editor: Rudong Chen

Copyright © 2012 Qingqing Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The following main results have been given. (1) Let 𝐸 be a 𝑝 -uniformly convex Banach space and let 𝑇 ∢ 𝐸 β†’ 𝐸 βˆ— be a ( 𝑝 βˆ’ 1 ) - 𝐿 -Lipschitz mapping with condition 0 < ( 𝑝 𝐿 / 𝑐 2 ) 1 / ( 𝑝 βˆ’ 1 ) < 1 . Then 𝑇 has a unique generalized duality fixed point π‘₯ βˆ— ∈ 𝐸 and (2) let 𝐸 be a 𝑝 -uniformly convex Banach space and let 𝑇 ∢ 𝐸 β†’ 𝐸 βˆ— be a π‘ž - 𝛼 - inverse strongly monotone mapping with conditions 1 / 𝑝 + 1 / π‘ž = 1 , 0 < ( π‘ž / ( π‘ž βˆ’ 1 ) 𝑐 2 ) π‘ž βˆ’ 1 < 𝛼 . Then 𝑇 has a unique generalized duality fixed point π‘₯ βˆ— ∈ 𝐸 . (3) Let 𝐸 be a 2 -uniformly smooth and uniformly convex Banach space with uniformly convex constant 𝑐 and uniformly smooth constant 𝑏 and let 𝑇 ∢ 𝐸 β†’ 𝐸 βˆ— be a 𝐿 -lipschitz mapping with condition 0 < 2 𝑏 / 𝑐 2 < 1 . Then 𝑇 has a unique zero point π‘₯ βˆ— . These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.