J. Appl. Math. & Informatics Vol. 35(2017), No. 3 - 4, pp.313 - 321 https://doi.org/10.14317/jami.2017.313 NON-CONVEX HYBRID ALGORITHMS FOR A FAMILY OF COUNTABLE QUASI-LIPSCHITZ MAPPINGS CORRESPONDING TO KHAN ITERATIVE PROCESS AND APPLICATIONS † WAQAS NAZEER, MOBEEN MUNIR, ABDUL RAUF NIZAMI, SAMINA KAUSAR, SHIN MIN KANG * Abstract. In this note we establish a new non-convex hybrid iteration al- gorithm corresponding to Khan iterative process [4] and prove strong con- vergence theorems of common fixed points for a uniformly closed asymp- totically family of countable quasi-Lipschitz mappings in Hilbert spaces. Moreover, the main results are applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. The results presented in this article are interesting extensions of some current results. AMS Mathematics Subject Classification : 47H05; 47H09; 47H10. Key words and phrases : Hybrid algorithm, quasi-Lipschitz mapping, non- expansive mapping, quasi-nonexpansive mapping, asymptotically quasi- nonexpansive mapping 1. Introduction Fixed points of special mappings like nonexpansive, asymptotically nonex- pansive, contractive and other mappings has become a field of interest on its on and has a various applications in related fields like image recovery, signal processing and geometry of objects [13]. Almost in all branches of mathematics we see some versions of theorems relating to fixed points of functions of special nature. As a result we apply them in industry, toy making, finance, aircrafts and manufacturing of new model cars. A fixed-point iteration scheme has been applied in intensity modulated radiation therapy optimization to pre-compute Received August 3, 2016. Revised February 22, 2017. Accepted March 8, 2017. * Corresponding author. † This work was supported by the Higher Education Commission, Pakistan and University of Education, Township, Lahore 54000, Pakistan. c ⃝ 2017 Korean SIGCAM and KSCAM. 313