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2008•Scientiae mathematicae JaponicaeRequires access

APPROXIMATING COMMON FIXED POINTS OF NONEXPANSIVE SEMIGROUPS IN BANACH SPACES BY METRIC PROJECTIONS

Sachiko Atsushiba, Wataru Takahashi

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Abstract

In this paper, we prove a strong convergence theorem by the hybrid method for nonexpansive semigroups in Banach spaces. Using this theorem, we obtain some strong convergence theorems in Banach spaces. Let H be a real Hilbert space with inner product �· , ·� and norm �·� and let C be a nonempty closed convex subset of H. Then, a mapping T : C → C is called nonexpansive (5) ifTx − Ty �≤� x − yfor all x, y ∈ C. We denote by F (T ) the set of fixed points of T. We know iteration procedures for finding a fixed point of a nonexpansive mapping; see, for instance, (11, 14). In 2003, Nakajo and Takahashi (13) studied the following iteration procedure of finding a fixed point of a nonexpansive mapping in a Hilbert space by using the hybrid method in mathematical programming:

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In this paper, we prove a strong convergence theorem by the hybrid method for nonexpansive semigroups in Banach spaces. Using this theorem, we obtain some strong convergence theorems in Banach spaces. Let H be a real Hilbert space with inner product �· , ·� and norm �·� and let C be a nonempty closed convex subset of H. Then, a mapping T : C → C is called nonexpansive (5) ifTx − Ty �≤� x − yfor all x, y ∈ C. We denote by F (T ) the set of fixed points of T. We know iteration procedures for finding a fixed point of a nonexpansive mapping; see, for instance, (11, 14). In 2003, Nakajo and Takahashi (13) studied the following iteration procedure of finding a fixed point of a nonexpansive mapping in a Hilbert space by using the hybrid method in mathematical programming:

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Available abstract

In this paper, we prove a strong convergence theorem by the hybrid method for nonexpansive semigroups in Banach spaces. Using this theorem, we obtain some strong convergence theorems in Banach spaces. Let H be a real Hilbert space with inner product �· , ·� and norm �·� and let C be a nonempty closed convex subset of H. Then, a mapping T : C → C is called nonexpansive (5) ifTx − Ty �≤� x − yfor all x, y ∈ C. We denote by F (T ) the set of fixed points of T. We know iteration procedures for finding a fixed point of a nonexpansive mapping; see, for instance, (11, 14). In 2003, Nakajo and Takahashi (13) studied the following iteration procedure of finding a fixed point of a nonexpansive mapping in a Hilbert space by using the hybrid method in mathematical programming:

Key concepts: Mathematics, Banach space, Fixed point, Hilbert space, Regular polygon, Metric space, Discrete mathematics, Uniformly convex space

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APPROXIMATING COMMON FIXED POINTS OF NONEXPANSIVE SEMIGROUPS IN BANACH SPACES BY METRIC PROJECTIONS — Research Paper | ScholarLens