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MEAN ERGODIC THEOREMS FOR A SEQUENCE OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES

Masanori Akatsuka, Koji Aoyama, Wataru Takahashi

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Abstract

Let C be a closed convex subset of a Hilbert space and {Tn} a sequence of nonexpansive self-mappings of C. Then we consider the following iterative sequence {zn}: x1 = x ∈ C, xn+1 = Tnxn, and zn =1 /n n=1 xk for n ∈ . In this paper, we obtain a weak convergence theorem for such a sequence {zn}. Using our result, we get a nonlinear ergodic theorem which is a generalization of Baillon (2). Further we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings.

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What this paper is about

Let C be a closed convex subset of a Hilbert space and {Tn} a sequence of nonexpansive self-mappings of C. Then we consider the following iterative sequence {zn}: x1 = x ∈ C, xn+1 = Tnxn, and zn =1 /n n=1 xk for n ∈ . In this paper, we obtain a weak convergence theorem for such a sequence {zn}. Using our result, we get a nonlinear ergodic theorem which is a generalization of Baillon (2). Further we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings.

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OpenAlex reports 5 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

Let C be a closed convex subset of a Hilbert space and {Tn} a sequence of nonexpansive self-mappings of C. Then we consider the following iterative sequence {zn}: x1 = x ∈ C, xn+1 = Tnxn, and zn =1 /n n=1 xk for n ∈ . In this paper, we obtain a weak convergence theorem for such a sequence {zn}. Using our result, we get a nonlinear ergodic theorem which is a generalization of Baillon (2). Further we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings.

Key concepts: Ergodic theory, Mathematics, Hilbert space, Sequence (biology), Regular polygon, Generalization, Fixed point, Countable set

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