Approximation of fixed point and its application to fractional differential equation
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Approximation of fixed point and its application to fractional differential equation Sabiya Khatoon1 · Izhar Uddin1 · Dumitru Baleanu2 Received: 3 August 2020 / Revised: 6 October 2020 / Accepted: 9 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this study, we prove some convergence results for generalized α-Reich–Suzuki nonexpansive mappings via a fast iterative scheme. We validate our result by constructing a numerical example. Also, we compare our results with the other well known iterative schemes. Finally, we calculate the approximate solution of nonlinear fractional differential equation. Keywords Generalized α-Reich–Suzuki non-expansive mappings · Nonlinear fractional differential equation · Fixed point Mathematics Subject Classification 47H10 · 54H25
1 Introduction The theory of non-expansive mappings plays crucial role in nonlinear analysis due to its plenty of applications. In 1965, Browder [9], Göhde [13] and Kirk [22] independently proved the fixed point results for non-expansive mappings. Hereafter, so many researchers came forward with different notions and enhanced this mapping with great improvement, there is a vast literature on the generalizations, extensions of the obtained results and several new concepts on non-expansive mappings. In 1980, Greguš [14] generalized the work of Kannan [20] and joined the ideas of non-expansive and Kannan mappings to obtain a unfamiliar class which is known as the Reich non-expansive
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Sabiya Khatoon [email protected] Izhar Uddin [email protected] Dumitru Baleanu [email protected]
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Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
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Department of Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey
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S. Khatoon et al.
mappings. In 2008, Suzuki [29] proposed a different class of mappings which is known as Suzuki’s generalized non-expansive mapping. Recently, Ali et al. [6] proved some weak and strong convergence results using a three step iterative scheme for Suzuki’s generalized non-expansive mappings in uniformly convex Banach spaces. In 2011, Aoyama and Kohsaka [7] introduced a generalization of non-expansive mappings known as α-non-expansive mapping and obtained some results for this type of mappings. Recently, Pandey et al. [26] proposed a different extension of non-expansive mappings which contains α-non-expansive and Suzuki generalized non-expansive mappings named as generalized α-Reich–Suzuki non-expansive mappings and obtained interesting results containing this kind of mappings. Numerical reckoning of nonlinear operators is very fascinating research problem of nonlinear analysis. However, it is not a easy task to find the fixed points of some operators. To overcome this kind of problem so many iteration procedures has been evolved over the time. Mann [23], Ishikawa [19] and Halpern [18] are the three basic iterative algorithms utilized to approximate the fixed points of non-expansive mappings. After getting motivation by the abov
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