The Jungck–Khan iterative scheme for a pair of nonself operators contains as a special case Jungck–Ishikawa and Jungck–Mann iterative schemes. In this paper, we establish improved results about convergence, stability, and data dependence for the Jungck–Khan iterative scheme. | Turk J Math (2016) 40: 631 – 640 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Stability and data dependence results for the Jungck–Khan iterative scheme 1 2,∗ ¨ Abdul Rahim KHAN1 , Faik GURSOY , Vivek KUMAR3 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia 2 Department of Mathematics, Adiyaman University, Adiyaman, Turkey 3 Department of Mathematics, KLP College, Rewari, India Received: • Accepted/Published Online: • Final Version: Abstract: The Jungck–Khan iterative scheme for a pair of nonself operators contains as a special case Jungck–Ishikawa and Jungck–Mann iterative schemes. In this paper, we establish improved results about convergence, stability, and data dependence for the Jungck–Khan iterative scheme. Key words: Jungck–Khan iterative scheme, convergence, stability, weak w2 − stability, data dependency 1. Introduction The case of nonself mappings is much more complicated than that of self ones and therefore it is not considered in many situations. Inspired by the work of Khan [7], here we tackle this problem in the context of two nonself operators. Definition 1 [5] Let X be a set and S , T : X → X be mappings. 1. A point x in X is called: (i) coincidence point of S and T if Sx = T x , (ii) common fixed point of S and T if x = Sx = T x . 2. If w = Sx = T x for some x in X , then w is called a point of coincidence of S and T . 3. A pair (S, T ) is said to be: (i) commuting if T Sx = ST x for all x ∈ X , (ii) weakly compatible if they commute at their coincidence points, . ST x = T Sx whenever Sx = T x . Let X be a Banach space, Y be an arbitrary set, and S ,T : Y → X be two nonself operators such that T (Y ) ⊆ S (Y ). ∞ ∞ Definition 2 ([15]) We say that the sequences {Sxn }n=0 and {Syn }n=0 in X are S− equivalent if lim ∥Sxn − Syn ∥ = .
