Document Type: Research Paper
Author
- Ebrahim Soori ^{}
Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran.
Abstract
Let $C$ be a nonempty closed convex subset of a real Banach space $E$, let $B: C \rightarrow E $ be a nonlinear map, and let $u, v$ be positive numbers. In this paper, we show that the generalized variational inequality $V I (C, B)$ is singleton for $(u, v)$-cocoercive mappings under appropriate assumptions on Banach spaces. The main results are extensions of the Saeidi's Propositions for finding a unique solution of the variational inequality for $(u, v)$-cocoercive mappings in Banach spaces.
Keywords
- Variational inequality
- Nonexpansive mapping
- $(u
- v)$-cocoercive mapping
- Metric projection
- Sunny nonexpansive retraction
Main Subjects
[1] R.P. Agarwal, D. Oregan, and D.R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, in: Topological Fixed Point Theory and its Applications, vol. 6, Springer, New York, 2009.
[2] R.E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific. J. Math., 47 (1973), pp. 341-356.
[3] R.E. Bruck, Nonexpansive retract of Banach spaces, Bull. Amer. Math. Soc., 76 (1970), pp. 384-386.
[4] R.E. Bruck, Properties of fixed-point sets of nonexpansive mapping in Banach spaces, Trans. Amer. Math. Soc., 179 (1973), pp. 251-262.
[5] T. Ibarakia and W. Takahashi, A new projection and convergence theorems for the projections in Banach spaces, J. Approx. Theory , 149 (2007), pp. 1-14.
[6] S. Saeidi, Comments on relaxed $(gamma, r)$-cocoercive mappings, Int. J. Nonlinear Anal. Appl., 1 (2010), pp. 54-57.
[7] W. Takahashi, Convergence theorems for nonlinear projections in Banach spaces (in Japanese), RIMS Kokyuroku, 1396 (2004), pp. 49-59.
[8] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.