Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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Local convergence of two Newton-like methods under Holder continuity condition in Banach spaces

International Journal of Mathematics in Operational Research ◽

10.1504/ijmor.2020.10040267 ◽

2020 ◽

Vol 1 (1) ◽

pp. 1

Author(s):

S.K. Parhi ◽

Debasis Sharma

Keyword(s):

Banach Spaces ◽

Local Convergence ◽

Hölder Continuity ◽

Continuity Condition ◽

Holder Continuity

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Convergence of derivative free iterative methods

Creative Mathematics and Informatics ◽

10.37193/cmi.2019.01.03 ◽

2019 ◽

Vol 28 (1) ◽

pp. 19-26

Author(s):

IOANNIS K. ARGYROS ◽

◽

SANTHOSH GEORGE ◽

Keyword(s):

Banach Space ◽

Iterative Methods ◽

Local Convergence ◽

Practical Interest ◽

Systems Of Equations ◽

Hölder Continuous ◽

Numerical Examples ◽

Lipschitz Continuous ◽

Derivative Free ◽

Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

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Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

Abstract and Applied Analysis ◽

10.1155/2013/682167 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Yonghui Ling ◽

Xiubin Xu ◽

Shaohua Yu

Keyword(s):

Banach Spaces ◽

Local Convergence ◽

Nonlinear Operator ◽

Convergence Criterion ◽

Second Derivative ◽

Operator Equations ◽

Convergence Ball ◽

Nonlinear Operator Equations ◽

Steffensen’S Method ◽

Convergence Problems

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

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Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

Nagoya Mathematical Journal ◽

10.1017/s002776300001610x ◽

1974 ◽

Vol 53 ◽

pp. 141-155 ◽

Cited By ~ 1

Author(s):

Mitsuru Nakai

Keyword(s):

Banach Space ◽

Riemann Surface ◽

Banach Spaces ◽

Riemann Surfaces ◽

Space Structure ◽

Bounded Solutions ◽

Linear Isometry ◽

Hölder Continuous ◽

Hyperbolic Riemann Surfaces ◽

Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

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