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A fixed point theorem for positive k-set contractions

Published online by Cambridge University Press:  20 January 2009

A. J. B. Potter
A. J. B. Potter
Affiliation:
King's College, University of Aberdeen
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The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressive k-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to the k-set contractions (k < 1). It should be remarked that J. D. Hamilton has extended the same result to A-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whether k-set contractions are A-proper or not.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 1 , March 1974 , pp. 93 - 102
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

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