Fixed Points of \(\xi\) - (\(\alpha\), \(\beta\))- Contractive Mappings in b-Metric Spaces
Kapil Jain *
Department of Mathematics, RKSD College, Kaithal, Haryana, India.
Jatinderdeep Kaur
School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab, India.
Satvinder Singh Bhatia
School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab, India.
*Author to whom correspondence should be addressed.
Abstract
In the paper [Some new observations on Geraghty and \(\acute{C}\)iri\(\acute{c}\) type results in b-metric spaces, Mathematics, 7, (2019), doi: 10.3390/math7070643] Mlaiki et al. introduced (\(\alpha\), \(\beta\))-type contraction in order to generalize the contraction mapping defined by Pant and Panicker. Also, in the paper [Some fixed point results in b- metric spaces and b-metric-like spaces with new contractive mappings, Axioms, 10(2), (2021), 15 pages, doi: 10.3390/axioms10020055] Jain and Kaur presented the concepts of \(\xi\) -contractive mappings. Now, the aim of the present article is to introduce \(\xi\) - (\(\alpha\), \(\beta\)) -contractive mappings in b-metric spaces by combining the concepts (\(\alpha\), \(\beta\))-type contraction and \(\xi\)-contractive mappings. Also, we establish some fixed point results for newly defined mappings. Our results generalize various theorems in literature. In support, we provide an example.
Keywords: b-metric space, (\(\alpha\), \(\beta\))-admissible mappings, (\(\alpha\), \(\beta\))-type contraction, \(\xi\)-contractive mappings, Cauchy Sequence, fixed point