Surveys in Mathematics and its Applications

ISSN1842-6298 (electronic), 1843-7265 (print)

Volume 14 (2019), 159 -- 171

This work is licensed under a Creative Commons Attribution 4.0 International License.## GENERALIZED BANACH CONTRACTION MAPPING PRINCIPLE IN GENERALIZED METRIC SPACES WITH A TERNARY RELATION

## Y. U. Gaba, C. A. Agyingi, Binayak S. Choudhury and P. Maity

Abstract. In the present work the use of ternary relations is introduced in fixed point theory to obtain some fixed point results in G-metric spaces. Amongst several generalizations of metric spaces suggested in recent times, G-metric spaces are the ones in which the metric is replaced by a function through which sets of three elements are assigned to non-negative real numbers. A ternary relation is assumed on the space and a generalized contractive condition is assumed for the triplets of elements related by the ternary relation. Fixed point and related results are established for such contractions as generalization of contractive mapping principle. The case without the assumption of ternary relation on the space is also discussed. There are some corollaries and illustrative examples. The illustrations establish the actuality of the generalization. The methodology of the proofs are new in the context of G-metric spaces.2010 Mathematics Subject Classification:Primary 54H25, 47H05; Secondary 47H09, 47H10.

Keywords:G-metric spaces, Ternary relation, Fixed point, Orbital point, Contraction.

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Y. U. Gaba

Institut de Mathématiques et de Sciences Physiques (IMSP)/UAC, 01 BP 613 Porto-Novo, Bénin.

and

Department of Mathematical Sciences, North West University, Private Bag X2046,

Mmabatho 2735, South Africa.

and

African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon.

E-mail: gabayae2@gmail.com

C. A. Agyingi

Department of Mathematics and Applied Mathematics, Nelson Mandela University,

P.O. Box 77000, Port Elizabeth 6031, South Africa.

and

African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon.

E-mail: Collins.agyingi@mandela.ac.za

Binayak S. Choudhury

Department of Mathematics, Indian Institute of Engineering Science and Technology,

Shibpur, Howrah - 711103, West Bengal, India

E-mail: binayak12@yahoo.co.in

P. Maity

Department of Mathematics, National Institute of Technology, Rourkela,

Rourkela-769008, Odisha, India.

E-mail: pranati.math@gmail.com