Indonesian Journal of Electrical Engineering and Computer Science Vol. 25, No. 1, January 2022, pp. 540~549 ISSN: 2502-4752, DOI: 10.11591/ijeecs.v25.i1.pp540-549  540 Journal homepage: http://ijeecs.iaescore.com Fixed point theorem between cone metric space and quasi-cone metric space Abdullah Al-Yaari 1,2 , Hamzah Sakidin 1 , Yousif Alyousifi 3 , Qasem Al-Tashi 4,5 1 Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Seri Iskandar, Malaysia 2 Department of Mathematics, Faculty of Education, Thamar University, Dhamar, Yemen 3 Department of Mathematics, Faculty of Applied Science, Thamar University, Dhamar, Yemen 4 Department of Computer and Information Sciences, Universiti Teknologi PETRONAS, Seri Iskandar, Malaysia 5 Faculty of Administrative and Computer Sciences, University of Albaydha, Albaydha, Yemen Article Info ABSTRACT Article history: Received Jul 28 2021 Revised Nov 1, 2021 Accepted Nov 26, 2021 This study involves new notions of continuity of mapping between quasi-cone metrics spaces (QCMSs), cone metric spaces (CMSs), and vice versa. The relation between all notions of continuity were thoroughly studied and supported with the help of examples. In addition, these new continuities were compared with various types of continuities of mapping between two QCMSs. The continuity types are -continuous, -continuous, -continuous, and -continuous. The results demonstrated that the new notions of continuity could be generalized to the continuity of mapping between two QCMSs. It also showed a fixed point for this continuity map between a complete Hausdorff CMS and QCMS. Overall, this study supports recent research results. Keywords: Cone metric spaces second Fixed point theorem Quasi-cone metric spaces This is an open access article under the CC BY-SA license. Corresponding Author: Abdullah Al-Yaari Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS 32610 Seri Iskandar, Malaysia E-mail: abdullah_20001447@utp.edu.my 1. INTRODUCTION More than two decades ago, Huang and Zhang had proposed the notion of a cone metric space [1]. The Cauchy and convergent sequences and a few fixed-point theorems were used for the contractive form of mappings in cone metric spaces (CMSs). This notion is exciting because of the usual metric space where an ordered Banach space switches to the real numbers. Abbas and Jungck [2] have shown some noncommuting mapping causes CMSs. Furthermore, several researchers have proved different contractive-type maps in CMS and fixed points [3]-[8]. The function of a quasi-metric is to verify the triangle disparity. However, quasi-metric is considered an asymmetric metric. As compared to metric space, quasi-metric space (QMS) is more inclusive. It is also a topic of exhaustive research in computer science and the framework for topology. For instance, a quasi-cone metric space (QCMS) definition expands to the QMS given by Turkoglu and Abuloha [3]. Morales and Rojas have introduced the continuity of mapping between CMS (, ),  a cone along with constant  and :  ⟶  and it self [4]. Yaying et al. introduced the continuity of mapping between QCMS (, ) and itself [5]. In addition, although the concept of normality in CMSs is monumental in developing fixed point theory in CMSs, Rezapour and Hamlbarani [6] have rejected it. Several researchers have simplified fixed points in CMSs in many directions. For instance, Jankovic et al. [7] surveyed the latest outcomes in CMSs. Abdeljawad and Karapinar [8] have verified fixed point theorems in QCMSs. They introduced many Cauchy sequences in QCMSs, which are studied as an extension of the Banach contraction