On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis

Sintunavarat, Wutiphol; Pitea, Ariana

doi:10.22436/jnsa.009.05.53

Cited by 26 publications

(22 citation statements)

References 15 publications

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“…This was in part due to the fact that Picard's iterative sequence for nonexpansive mappings does not necessarily converge. For more recently introduced iterative schemes, one can see Noor [8], Agrawal et al [9], Abbas and Nazir, [10], Sintunavarat and Pitea [11], Thakur et al [12][13][14], etc.…”

Section: Introductionmentioning

confidence: 99%

CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

Houmani

Ţurcanu

2019

Symmetry

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We introduce a new class of non-self mappings by means of a condition which is called the (EP)-condition. This class includes proximal generalized nonexpansive mappings. It is shown that the existence of best proximity points for (EP)-mappings is equivalent to the existence of an approximate best proximity point sequence generated by a three-step iterative process. We also construct a CQ-type algorithm which generates a strongly convergent sequence to the best proximity point for a given (EP)-mapping.

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Section: Introductionmentioning

confidence: 99%

CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

Houmani

Ţurcanu

2019

Symmetry

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“…Thus, we get α n α 6,n → 0 as n → ∞. Hence, { p n } converges faster than { p 6,n } to p. Also, Sintunavarat and Pitea (2016) showed that the Varat algorithm converges faster than Mann, Ishikawa, and S iterative algorithms for the class of weak contractions. Thus, F * iterative algorithm converges faster than all the iterative algorithms as discussed earlier.…”

Section: Resultsmentioning

confidence: 78%

“…(e.g., see Ali et al 2020;Khan 2013;Thakur et al 2016;Katchang and Kumam 2010;Maingè and Mȃruşter 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 :…”

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confidence: 99%

Convergence, stability, and data dependence of a new iterative algorithm with an application

Ali

2020

Comp. Appl. Math.

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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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“…This was motivated in part by the fact that, unlike the case of contraction mappings, the successive application of a non-expansive mapping does not necessarily lead to a fixed point. The earliest results in this direction were obtained by Krasnosel'skii [1], Mann [2], Halpern [3], Berinde [4], for one-step iterations; Ishikawa [5], etc., for two-step iterations; Noor [6], Agrawal et al [7], Abbas and Nazir [8], Gürsoy and Karakaya [9], Sintunavarat and Pitea [10], Thakur et al [11,12], for three-step iterations; and the search for new iteration schemes has remained active ever since.…”

Section: Introductionmentioning

confidence: 99%