Hilbert's metric and positive contraction mappings in a Banach space

1973 ◽  
Vol 52 (4) ◽  
pp. 330-338 ◽  
Author(s):  
P. J. Bushell
Keyword(s):  
Banach Space ◽  
Positive Contraction ◽  
Contraction Mappings ◽  
Hilbert’S Metric

scholarly journals Fixed points and their approximation in Banach spaces for certain commuting mappings

1982 ◽  
Vol 23 (1) ◽  
pp. 1-6
Author(s):  
M. S. Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Contraction Mappings ◽  
Continuous Mappings ◽  
Banach Contraction ◽  
The Fixed Point Theorem

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

scholarly journals A fixed point theorem for contraction mappings

1982 ◽  
Vol 5 (2) ◽  
pp. 301-304
Author(s):  
V. M. Sehgal
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Closed Subset ◽  
Contraction Mapping ◽  
Contraction Mappings ◽  
Strict Contraction

LetSbe a closed subset of a Banach spaceEandf:S→Ebe a strict contraction mapping. Suppose there exists a mappingh:S→(0,1]such that(1−h(x))x+h(x)f(x)∈Sfor eachx∈S. Then for anyx0∈S, the sequence{xn}inSdefined byxn+1=(1−h(xn))xn+h(xn)f(xn),n≥0, converges to au∈S. Further, if∑h(xn)=∞, thenf(u)=u.

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