Convergence for a class of improved sixth-order Chebyshev-Halley type methods

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• 作者：Xiuhua Wang^a ; ^b ; Jisheng Kou ; ^a ; ^{kjsfine@aliyun.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chebyshev&ndash ; Halley type methods ; Convergence condition ; Nonlinear equations in Banach space ; Local Ho¨ ; lder continuous ; R-order of convergence
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：273
• 期：Complete
• 页码：513-524
• 全文大小：320 K

文摘

In this paper, we consider the semilocal convergence on a class of improved Chebyshev–Halley type methods for solving le="Click to view the MathML source">F(x)=0,$F\left(x\right)=0,$ where F: Ω ⊆ X → Y is a nonlinear operator, X and Y are two Banach spaces, Ω is a non-empty open convex subset in X. To solve the problems that F ′′′(x) is unbounded in Ω   and it can not satisfy the whole Lipschitz or Hle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315009777&_mathId=si2.gif&_user=111111111&_pii=S0096300315009777&_rdoc=1&_issn=00963003&md5=1bb5d5dcb40a5273538b69cb98a07550">le="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315009777-si2.gif">$\ddot{\text{o}}$lder continuity, ‖F ′′′(x)‖ &le; N   is replaced by le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315009777&_mathId=si3.gif&_user=111111111&_pii=S0096300315009777&_rdoc=1&_issn=00963003&md5=420f04364e440d1902a4c2c8cd6b9149">le="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315009777-si3.gif">$\parallel F^{\prime \prime \prime }\left(x_0\right)\parallel \&<font\; color="red">le</font>;\overline{N},$ for all x ∈ Ω  , where le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315009777&_mathId=si4.gif&_user=111111111&_pii=S0096300315009777&_rdoc=1&_issn=00963003&md5=30abc92afa5e85c76920f836e574bbe3">le="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315009777-si4.gif">$N,\overline{N}\ge 0,$x₀ is an initial point. Moreover, F ′′′(x  ) is assumed to be local Hle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315009777&_mathId=si2.gif&_user=111111111&_pii=S0096300315009777&_rdoc=1&_issn=00963003&md5=1bb5d5dcb40a5273538b69cb98a07550">le="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315009777-si2.gif">$\ddot{\text{o}}$lder continuous. So the convergence conditions are relaxed. We prove an existence-uniqueness theorem for the solution, which shows that the R  -order of these methods is at least le="Click to view the MathML source">5+q,$5+q,$ where q ∈ (0, 1]. Especially, when F ′′′(x) is local Lipschitz continuous, the R-order will become six.