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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 [{\it T. Suzuki}, J. Math. Anal. Appl. 340, No. 2, 1088--1095 (2008; Zbl 1140.47041)], a mapping $T :C\rightarrow X$ is said to satisfy condition $(C)$ on $C$ if, for all $x,y\in C$, $\frac{1}{2}$ $\|x-Tx\|\leq\|x-y\|$ implies $\|Tx-Ty\|\leq \|x-y\|$. 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 $\mu \ge 1$, the mapping $T:C\rightarrow X$ is said to satisfy condition $(E_\mu)$ on $C$ if, for all $x,y\in C$, $\|x-Ty\|\leq \mu\|x-Tx\|+\|x-y\|$. Moreover, $T$ is said to satisfy condition $(E)$ on $C$ whenever $T$ satisfies $(E_\mu)$ for some $\mu \geq 1$. For $\lambda \in(0,1)$, the mapping $T$ is said to satisfy condition $(C_\lambda)$ on $C$ if, for all $x,y\in C$, $\lambda \|x-Tx\|\leq \|x-y\|$ implies $\|Tx-Ty\|\leq \|x-y\|\,\|x-Ty\|$. The authors study for both the classes of mappings, satisfying condition $(E)$ or $(C_\lambda)$, the existence of fixed points and their asymptotic behavior.

##### MSC:
 47H09 Mappings defined by “shrinking” properties 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
 [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) · Zbl 1066.47055 [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