Available online at www.isr-publications.com/jmcs J. Math. Computer Sci., 18 (2018), 132–145 Research Article Journal Homepage: www.tjmcs.com - www.isr-publications.com/jmcs On (α, p)-convex contraction and asymptotic regularity M. S. Khan a , Y. Mahendra Singh b , Georgeta Maniu c , Mihai Postolache d,e,* a Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, PCode 123 Al-Khod, Muscat, Sultanate of Oman, Oman. b Department of Humanities and Basic Sciences, Manipur Institute of Technology, Takyelpat-795001, India. c Department of Computer Science, Information Technology, Mathematics and Physics, Petroleum-Gas University of Ploies ¸ti, Bucures ¸ti Bvd., No. 39, 100680 Ploies ¸ti, Romania. d China Medical University, Taichung, Taiwan. e University Politehnica of Bucharest, Bucharest, Romania. Abstract In this paper, we present the notions of (α, p)-convex contraction (resp. (α, p)-contraction) and asymptotically T 2 -regular (resp. (T , T 2 )-regular) sequences, and prove fixed point theorems in the setting of metric spaces. Keywords: Approximate fixed point, fixed point, (α, p)-convex contraction, asymptotically regular sequence, asymptotically T (resp. T 2 and (T , T 2 ))-regular sequences. 2010 MSC: 47H10, 54H25. c 2018 All rights reserved. 1. Introduction and preliminaries Let (X, d) be a metric space, and C a nonempty set of X. A mapping T : C → C is called nonexpansive if d(Tx, Ty)  d(x, y) for all x, y ∈ C. In 2007, Goebel and Jap´ on Pineda [8] introduced the class of mean nonexpansive mappings, an extension for the class of nonexpansive mappings. A mapping T : C → C is called mean nonexpansive (or α-nonexpansive) if, for some α =(α 1 , α 2 ,..., α n ) with ∑ n i=1 α i = 1, a i  0 for all i, and α 1 , α n > 0, we have n  i=1 α i d(T i x, T i y)  d(x, y) for all x, y ∈ C. Further, Goebel and Jap´ on Pineda [8] introduced the class of (α, p)-nonexpansive * Corresponding author Email addresses: mohammad@squ.edu.om (M. S. Khan), ymahenmit@rediffmail.com (Y. Mahendra Singh), maniugeorgeta@gmail.com (Georgeta Maniu), emscolar@yahoo.com (Mihai Postolache) doi: 10.22436/jmcs.018.02.01 Received 2017-09-01