A fixed point theorem for contraction mappings

V. M. Sehgal

International Journal of Mathematics and Mathematical Sciences, 1982, vol. 5, 1-4

Abstract:

Let S be a closed subset of a Banach space E and f : S → E be a strict contraction mapping. Suppose there exists a mapping h : S → ( 0 , 1 ] such that ( 1 − h ( x ) ) x + h ( x ) f ( x ) ∈ S for each x ∈ S . Then for any x 0 ∈ S , the sequence { x n } in S defined by x n + 1 = ( 1 − h ( x n ) ) x n + h ( x n ) f ( x n ) , n ≥ 0 , converges to a u ∈ S . Further, if ∑ h ( x n ) = ∞ , then f ( u ) = u .

Date: 1982
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DOI: 10.1155/S0161171282000271

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