Abstract
Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E and \(\{T_{n}\}_{n=1}^{\infty}\) be a sequence of L n -Lipschitzian mappings of K into itself with L n ≥1, \(\sum_{n=1}^{\infty}(L_{n}-1)\allowbreak <\infty\) . Let \(\bigcap_{n=1}^{\infty}F(T_{n})\neq\emptyset\) . Convergence theorems to common fixed points of the family \(\{T_{n}\}_{n=1}^{\infty}\) are proved using the Halpern-type iteration process. Corollaries of our theorems present significant improvement of some important recent results (e.g., the results of Aoyama et al. in Nonlinear Anal. 67:2350–2360, 2008).
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The third author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
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Chidume, C.E., Chidume, C.O. & Shehu, Y. Strong convergence theorems for a Mann-type iterative scheme for a family of Lipschitzian mappings. J. Appl. Math. Comput. 35, 251–261 (2011). https://doi.org/10.1007/s12190-009-0354-2
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DOI: https://doi.org/10.1007/s12190-009-0354-2
Keywords
- Lipschitzian maps
- Modulus of convexity
- Reflexive real Banach spaces
- Fixed point
- Uniformly Gâteaux differentiable norm