FIXED POINT THEOREMS FOR KANNAN TYPE MAPPINGS  IN CONE METRIC SPACES

Đoàn Trọng Hiếu; Trịnh Văn Hà; Hoàng Văn Linh

FIXED POINT THEOREMS FOR KANNAN TYPE MAPPINGS IN CONE METRIC SPACES

About this article

Received: 10/03/20 Revised: 06/05/20 Published: 22/05/20

Authors

1. Doan Trong Hieu , Quang Ninh University of Industry.
2. Trinh Van Ha, TNU - University of Information Technology and Communications
3. Hoang Van Linh, Lang Son College of Education

Abstract

Many important problems in mathematics in particular and in science and technology in general
led to the study of the existence of a fixed point of mapping. That is why fixed point theory
is concerned by many mathematicians in the world. Let X be a nonempty. A point x 0 ∈ X
is called the fixed point of the mapping T : X → X if T x 0 = x 0. In 1922 Banach proved:
"Every contractive mappings of T from a complete metric space X into itself has a unique fixed
point." To generalize the Banach contraction priciple, in this paper, we consider the concepts
of boundedly compact cone metric space and orbitally compact cone metric space. Moreover,
we prove the results of Kannan-type fixed point in these spaces. By expanding the space, our
results solve the problems: In the compact cone metric space and compact cone metric space in orbit every Kannan-type contraction map has a unique fixed point.

Keywords

Fixed point, Kannan-type, cone metric space, contractive mapping, regular cone.

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References

[1]. S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fund. M ath., vol. 3, pp. 133-181, 1922.

[2]. R. Kannan, “Some results on fixed points,” Bul l. Calcutta Math. Soc., vol. 60, pp.
71–76, 1968.

[3]. J. Górnicki, “Fixed point theorems for Kannan type mappings,” J. Fixed Point
Theory Appl, vol. 19(3), pp. 2145–2152, 2017.

[4]. L.-G. Huang, and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” J. Math. Anal. Appl, vol. 332, pp. 1468-1476, 2007.

[5]. Sh. Rezapour, and R. Hamlbarani, “Some notes on the paper: Cone metric spaces and fixed point theorems of contractive mappings,” J. Math. Anal. Appl, vol. 345, pp. 719-724, 2008.

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