Stability and convergence of F iterative scheme with an application to the fractional differential equation
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ORIGINAL ARTICLE
Stability and convergence of F iterative scheme with an application to the fractional differential equation Javid Ali1 · Mohd Jubair1 · Faeem Ali1 Received: 15 May 2020 / Accepted: 4 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, we prove that F iterative scheme is almost stable for weak contractions. Further, we prove convergence results for weak contractions as well as for generalized non-expansive mappings due to Hardy and Rogers via F iterative scheme. We also prove that F iterative scheme converges faster than the some known iterative schemes for weak contractions. An illuminative numerical example is formulated to support our assertion. Finally, utilizing our main result the solution of nonlinear fractional differential equation is approximated. Keywords Nonlinear fractional differential equation · Generalized non-expansive mappings · Weak contraction · F iterative scheme · Fixed point · Banach spaces · Almost G-stability Mathematics Subject Classification 47H09 · 47H10 · 54H25
1 Introduction Fixed point theory plays an important role in nonlinear functional analysis, physical sciences, approximation theory, differential equations, etc. The theory itself is a beautiful mixture of analysis, topology and geometry. Over the last six decades or so, the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. It is well known that Banach contraction principle assurances the existence and uniqueness of a fixed point for contraction mapping which can be approximated by Picard iterative scheme. The Banach contraction principle has been generalized in many ways. One of the noted generalizations has given by Berinde [1]. Throughout the paper, ℤ+ denotes the set of nonnegative integers. Let D be a nonempty subset of a Banach space B * Javid Ali [email protected] Mohd Jubair [email protected] Faeem Ali [email protected] 1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
and F(G) = {t ∈ D ∶ Gt = t} . A mapping G ∶ D → D is said to be weak contraction if there exits a constant 𝛿 ∈ (0, 1) and some L ≥ 0 such that for all x, y ∈ D:
‖Gx − Gy‖ ≤ 𝛿‖x − y‖ + L‖x − Gy‖,
(1.1)
and
‖Gx − Gy‖ ≤ 𝛿‖x − y‖ + L‖y − Gx‖.
(1.2)
‖Gx − Gy‖ ≤ 𝛿‖x − y‖ + L‖x − Gx‖, ∀ x, y ∈ B.
(1.3)
Theorem 1.1 [1] Let B be a Banach space and G ∶ B → B a weak contraction with 𝛿 ∈ (0, 1) and some L ≥ 0 , such that
Then, G has a unique fixed point and Picard sequence converges to the fixed point. However, it is well known that Picard sequence need not converge to the fixed points of non-expansive mappings, so we need some other iterative schemes. On the other hand, the concept of generalized non-expansive mappings was coined by Hardy and Rogers [2] which is defined as follows: A mapping G ∶ D → D is called generalized non-expansive if
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‖Gx − Gy‖ ≤ 𝛽1 ‖x − y‖ + 𝛽2 ‖x − Gx‖
+ 𝛽3 ‖y − Gy‖ + 𝛽4 ‖x − Gy‖ + 𝛽5 ‖y − Gx‖
(1.4)
for al
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