Round-off stability for multi-valued maps
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RESEARCH
Open Access
Round-off stability for multi-valued maps Shyam Lal Singh1*, Swami Nath Mishra2 and Sarika Jain3 * Correspondence: vedicmri@gmail. com 1 Department of Mathematics, Pt. L. M. S. Govt. Postgraduate College (Autonomous), 21 Govind Nagar, Rishikesh 249201, India Full list of author information is available at the end of the article
Abstract An iterative procedure for a map T is said to be stable if the approximate sequence arising in numerical praxis converges to the point anticipated by the theoretical sequence. The study of stability of iterative procedures plays a vital role in computational analysis, game theory, computer programming, and fractal geometry. In generation of fractals, a sequence of approximations produces a stable set attractor only if the corresponding iterative procedure shows a stable behavior. The purpose of this article is to discuss stability of the Picard iterative procedure for a map T satisfying Zamfirescu multi-valued contraction on a metric space. MSC (2010): 47H10; 54H25; 65D15; 65D18. Keywords: fixed point, stability of iterative procedures, Picard iterative procedure, Zamfirescu contraction, fractals
1 Introduction Let (X, d) be a metric space and T: X ® X. The solution of a fixed point equation Tx = x for any x Î X, is usually approximated by a sequence {xn} in X generated by an iterative procedure f(T, xn) that converges to a fixed point of T. However, in actual computations, we obtain an approximate sequence {yn} instead of the actual sequence {xn}. Indeed, the approximate sequence {yn} is calculated in the following manner. First, we choose an initial approximation x0 Î X. Then we compute x1 = f(T, x0). But, due to rounding off or discretization of the function, we get an approximate value y1, say, which is close enough to x1, i.e., y1 ≈ x1. Consequently, when computing x2, we actually compute y2 ≈ x2. In this way, we obtain an approximate sequence {yn} instead of the actual sequence {xn}. The iterative procedure f(T, xn) is considered to be numerically stable if and only if the approximate sequence {yn} still converges to the desired solution of the equation Tx = x. Urabe [1] initiated the study of this problem. The study of stability of iterative procedures plays a significant role in numerical mathematics due to chaotic behavior of functions and discretization of computations in computer programming. For a detailed discussion on the role of stability of iterative procedures, one may refer to Czerwik et al. [2,3], Harder and Hicks [4-6], Lim [7], Matkowski and Singh [8], Ortega and Rheinboldt [9], Osilike [10,11], Ostrowski [12], Rhoades [13,14], Rus et al. [15] and Singh et al. [16]. However, Ostrowski [12] was the first to obtain the following classical stability result on metric spaces. Theorem 1.1. Let (X, d) be a complete metric space and T: X ® X a Banach contraction with contraction constant q, i.e., d(Tx, Ty) ≤ qd(x, y) for all x, y Î X, where 0 © 2012 Singh et al; licensee Springer. This is an Open Access article distributed under the terms of the
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