Asymptotic Behavior of Generalized Nonexpansive Sequences and Mean Points

H. K. Pathak, Donal O’Regan, Mohd Shariq Khan, Ravi P. Agarwal · Georgian Mathematical Journal · 2004

Abstract Let 𝐸 be a real Banach space with norm ‖ · ‖ and let {𝑥𝑛}𝑛≥0 be a generalized nonexpansive sequence in 𝐸 (i.e., ‖ 𝑥𝑖+1 – 𝑥𝑗+1 ‖2 ≤ ‖𝑥𝑖 – 𝑥𝑗‖2 + (ε(𝑖+1, 𝑗+1) – ε(𝑖, 𝑗))2 for all 𝑖, 𝑗 ≥ 0, where the series of nonnegative terms is convergent). Let . We deal with the mean point of concerning a Banach limit μ. If 𝐸 is reflexive and 𝑑 = 𝑑(0, 𝐾), then we show that and there exists a point 𝑧0 with ‖ 𝑧0 ‖= 𝑑 such that . In the sequel, this result is applied to obtain the weak and strong convergence of .

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