An application of the contraction mapping principle in the theory of operators in an indefinite metric space

1978 ◽  
Vol 12 (1) ◽  
pp. 70-71 ◽  
Author(s):  
V. A. Khatskevich
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Indefinite Metric ◽  
Indefinite Metric Space

scholarly journals Generalizations of the Converse of the Contraction Mapping Principle

1966 ◽  
Vol 18 ◽  
pp. 1095-1104 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contraction Mapping Principle ◽  
Converse Statement ◽  
Natural Formulation

This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).

scholarly journals Banach contraction principle on cone rectangular metric spaces

2009 ◽  
Vol 3 (2) ◽  
pp. 236-241 ◽  
Author(s):  
Akbar Azam ◽  
Muhammad Arshad ◽  
Ismat Beg
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Contraction Mapping Principle ◽  
Banach Contraction Principle ◽  
Space Setting ◽  
Banach Contraction ◽  
Rectangular Metric Space ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We introduce the notion of cone rectangular metric space and prove Banach contraction mapping principle in cone rectangular metric space setting. Our result extends recent known results.

Another Proof of the Contraction Mapping Principle

1968 ◽  
Vol 11 (4) ◽  
pp. 605-606
Author(s):  
D.W. Boyd ◽  
J. S. W. Wong
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Alternative Proof ◽  
Intersection Theorem ◽  
Cantor’S Theorem ◽  
Special Case

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.

scholarly journals On iterative solution of nonlinear functional equations in a metric space

1983 ◽  
Vol 6 (1) ◽  
pp. 161-170
Author(s):  
Rabindranath Sen ◽  
Sulekha Mukherjee
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Iterative Solution ◽  
Contraction Mapping Principle ◽  
Iterative Sequence ◽  
Constructive Proof ◽  
Nonlinear Functional ◽  
Positive Integers

Given thatAandPas nonlinear onto and into self-mappings of a complete metric spaceR, we offer here a constructive proof of the existence of the unique solution of the operator equationAu=Pu, whereu∈R, by considering the iterative sequenceAun+1=Pun(u0prechosen,n=0,1,2,…). We use Kannan's criterion [1] for the existence of a unique fixed point of an operator instead of the contraction mapping principle as employed in [2]. Operator equations of the formAnu=Pmu, whereu∈R,nandmpositive integers, are also treated.

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