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Fixed point theory for a class of generalized nonexpansive mappings. (English) Zbl 1214.47047

Let C be a nonempty subset of a Banach space X. In [T. Suzuki, J. Math. Anal. Appl. 340, No. 2, 1088–1095 (2008; Zbl 1140.47041)], a mapping T:CX is said to satisfy condition (C) on C if, for all x,yC, 1 2 x-Txx-y implies Tx-Tyx-y. The class of mappings satisfying condition (C) is larger than the class of nonexpansive mappings. In the paper under review, the authors define two new classes of generalized nonexpansive mappings, which contain properly the class of mappings satisfying condition (C). For μ1, the mapping T:CX is said to satisfy condition (E μ ) on C if, for all x,yC, x-Tyμx-Tx+x-y. Moreover, T is said to satisfy condition (E) on C whenever T satisfies (E μ ) for some μ1. For λ(0,1), the mapping T is said to satisfy condition (C λ ) on C if, for all x,yC, λx-Txx-y implies Tx-Tyx-yx-Ty.

The authors study for both the classes of mappings, satisfying condition (E) or (C λ ), the existence of fixed points and their asymptotic behavior.

47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
[1]Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. math. Anal. appl. 340, 1088-1095 (2008) · Zbl 1140.47041 · doi:10.1016/j.jmaa.2007.09.023
[2]Goebel, K.: Concise course on fixed point theorems, (2002)
[3]Goebel, Kazimierz; Kirk, W. A.: Iteration processes for nonexpansive mappings, Contemp. math. 21, 115-123 (1983) · Zbl 0525.47040
[4]Dhompongsa, S.; Inthakon, W.; Kaewkhao, A.: Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, J. math. Anal. appl. 350, 12-17 (2009) · Zbl 1153.47046 · doi:10.1016/j.jmaa.2008.08.045