Geodesic
Extensions of Contractive Mappings and Edelstein's Iterative Test
Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 185-192

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A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 ...) as it is to find the fixed point directly. 
Bryant, Jack; Jr., L. F. Guseman. Extensions of Contractive Mappings and Edelstein's Iterative Test. Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 185-192. doi: 10.4153/CMB-1973-033-9
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     title = {Extensions of {Contractive} {Mappings} and {Edelstein's} {Iterative} {Test}},
     journal = {Canadian mathematical bulletin},
     pages = {185--192},
     year = {1973},
     volume = {16},
     number = {2},
     doi = {10.4153/CMB-1973-033-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1973-033-9/}
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