Convergence, stability, and data dependence of a new iterative algorithm with an application
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Convergence, stability, and data dependence of a new iterative algorithm with an application Faeem Ali1 · Javid Ali1 Received: 9 March 2020 / Revised: 6 May 2020 / Accepted: 18 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The purpose of this article is to introduce a new two-step iterative algorithm, called F ∗ algorithm, to approximate the fixed points of weak contractions in Banach spaces. It is also showed that the proposed algorithm converges strongly to the fixed point of weak contractions. Furthermore, it is proved that F ∗ iterative algorithm is almost-stable for weak contractions, and converges to a fixed point faster than Picard, Mann, Ishikawa, S, normal-S, and Varat iterative algorithms. Moreover, a data dependence result is obtained via F ∗ algorithm. Some numerical examples are presented to support the main results. Finally, the solution of the nonlinear quadratic Volterra integral equation is approximated by utilizing our main result. The results of the paper are new and extend several relevant results in the literature. Keywords F ∗ iterative algorithm · Weak contraction · Fixed points · Numerically stable · Data dependence · Nonlinear quadratic Volterra integral equation Mathematics Subject Classification 47H05 · 47H09 · 47H10
1 Introduction and preliminaries Throughout this article, we assume that Z+ is the set of nonnegative integers, Y a nonempty, closed and convex subset of a Banach space X , and F(G), the set of fixed points of the self-mapping G defined on Y . The iterative approximation of fixed points of linear and nonlinear mappings is one of the most significant tools in the fixed point theory that has many applications in different fields like Engineering, Differential equations, Integral equations, etc. Hence, a large number of researchers introduced and studied many iterative algorithms for certain classes of mappings
Communicated by Apala Majumdar.
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Javid Ali [email protected] Faeem Ali [email protected]
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Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India 0123456789().: V,-vol
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F. Ali, J. Ali
(e.g., see Ali et al. 2020; Khan 2013; Thakur et al. 2016; Katchang and Kumam 2010; Maingè and M˘aru¸ster 2011). The following iterative algorithms are called (Picard 1890; Mann 1953; Ishikawa 1974), S (Agrawal et al. 2007), normal-S (Sahu 2011), and Varat (Sintunavarat and Pitea 2016) algorithms, respectively, for the self-mapping G defined on Y : p0 ∈ Y , (1.1) pn+1 = Gpn , n ∈ Z+ p0 ∈ Y , (1.2) pn+1 = (1 − r˜n ) pn + r˜n Gpn , n ∈ Z+ ⎧ ⎨ p0 ∈ Y , pn+1 = (1 − r˜n ) pn + r˜n Gqn , (1.3) ⎩ qn = (1 − s˜n ) pn + s˜n Gpn , n ∈ Z+ ⎧ ⎨ p0 ∈ Y , pn+1 = (1 − r˜n )Gpn + r˜n Gqn , (1.4) ⎩ qn = (1 − s˜n ) pn + s˜n Gpn , n ∈ Z+ p0 ∈ Y , (1.5) pn+1 = G((1 − r˜n ) pn + r˜n Gpn ), n ∈ Z+ ⎧ p0 ∈ Y , ⎪ ⎪ ⎨ pn+1 = (1 − r˜n )Gz n + r˜n Gqn , (1.6) z ⎪ n = (1 − t˜n ) pn + t˜n qn , ⎪ ⎩ qn = (1 − s˜n ) pn + s˜n Gpn , n ∈ Z+ , where {r˜n }, {s˜n }, and {t˜n } are sequences in (0, 1). Motivated by the
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