Thai Journal of Mathematics : (2018) 367-378 Special Issue (ACFPTO2018) on : Advances in fixed point theory towards real world optimization problems http://thaijmath.in.cmu.ac.th Online ISSN 1686-0209 On New Fixed Point Results in E b -Metric Spaces P. Sumati Kumari † , C. Boateng Ampadu ‡ ,J. Nantadilok § 1 † Department of Mathematics, Basic Sciences and Humanities, GMR Institute of Technology, Rajam-532127,Andhra Pradesh, India e-mail : mumy143143143@gmail.com ‡ 31 Carrolton Road, Boston, MA 02132-6303, USA. e-mail : DrAmpadu@hotmail.com § Department of Mathematics, Faculty of Science, Lampang Rajabhat University, Lampang, Thailand e-mail : jamnian2010@gmail.com Abstract : In this paper, we introduce the concepts of an extended cyclic Banach contraction and an extended cyclic orbital F -expanding contraction. Thereby, we prove pertinent fixed point theorems in an extended b-metric space (simply E b - metric space). Moreover, we present the characterization of the Hardy and Rogers mapping theorem for (a pair of) non-self maps, which gives a positive answer to the question raised by C. Boateng Ampadu [12] (Fixed Point Theory, 19(2018), No. 2, 449-452 DOI: 10.24193/fpt-ro.2018.2.35). Keywords : Extended b-metric space, an extended cyclic Banach contraction, ex- tended cyclic orbital F -expanding contraction, proximal cyclic Hardy and Rogers type mapping. 2000 Mathematics Subject Classification : 47H10, 54H25. (2000 MSC ) 1 Introduction and Preliminaries 1 Corresponding author email: jamnian2010@gmail.com Copyright c ⃝ 2018 by the Mathematical Association of Thailand. All rights reserved.