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2006•Taiwanese Journal of MathematicsRequires access

THE SET OF COMMON FIXED POINTS OF A ONE-PARAMETER CONTINUOUS SEMIGROUP OF NONEXPANSIVE MAPPINGS IS $F(\frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}))$ IN STRICTLY CONVEX BANACH SPACES

Tomonari Suzuki

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Abstract

In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly continuous semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.

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What this paper is about

In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly continuous semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.

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Available abstract

In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly continuous semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.

Key concepts: Mathematics, Banach space, Semigroup, Regular polygon, Combinatorics, Convex function, Fixed point, Discrete mathematics

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THE SET OF COMMON FIXED POINTS OF A ONE-PARAMETER CONTINUOUS SEMIGROUP OF NONEXPANSIVE MAPPINGS IS $F(\frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}))$ IN STRICTLY CONVEX BANACH SPACES — Research Paper | ScholarLens