COMPLETENESS AND FIXED POINTS IN STRONG b−METRIC SPACE

Phạm Văn Tiến; Hà Trần Phương; Bùi Thế Hùng

doi:10.34238/tnu-jst.4157

COMPLETENESS AND FIXED POINTS IN STRONG b−METRIC SPACE

About this article

Received: 11/03/21 Revised: 30/11/22 Published: 30/11/22

Authors

1. Pham Van Tien , TNU - University of Education
2. Ha Tran Phuong, TNU - University of Education
3. Bui The Hung, TNU - University of Education

Abstract

In 1968, Kannan proves that the following result: Let (X, d) be a
complete metric space and T be a self-mapping on X satisfying
d(Tx, Ty) ≤ r{d(x, Tx) + d(y, Ty)}
for all x, y ∈ X and r ∈ (0, 12). Then, T has a unique fixed point x¯ ∈ X
and for any x ∈ X, the sequence of iterates {T nx} converges to x¯. The
mapping satisfying the above contraction condition is called Kannan
mapping. Another important meaning of the Kannan mapping is being
able to describe the completeness of space in terms of the fixed-point
property of the mapping. This was proved by Subrahmanyam in 1975.
Means, a metric space (X, d) is complete if and only if every Kannan
mapping has a unique fixed point in X. In this paper, we consider the
same problem in the case of strong b-metric space as generalization of
result of Subrahmanyam.

Keywords

Fixed point; Cauchy sequence; Kannan mapping; Complete strong b-metric spaces Strong b-metric spaces;

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References

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DOI: https://doi.org/10.34238/tnu-jst.4157

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