Achieving an extended convergence analysis for the secant method under a restricted Hölder continuity condition

SeMA Journal ◽  
2021 ◽  
Author(s):  
H. M. Ren ◽  
I. K. Argyros
Keyword(s):  
Convergence Analysis ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Secant Method ◽  
Holder Continuity

A centred norm inequality for singular integral operators

1992 ◽  
Vol 112 (2) ◽  
pp. 369-383
Author(s):  
Richard F. Bass
Keyword(s):  
Singular Integral ◽  
Hölder Continuity ◽  
Integral Kernel ◽  
Integral Operators ◽  
Continuity Condition ◽  
Norm Inequality ◽  
Singular Integral Operators ◽  
Holder Continuity ◽  
The Mean

AbstractLet K be a standard singular integral kernel on ℝ satisfying the usual Hölder continuity condition of order δ, and define (where c is chosen so that the integral of w is 1), the mean of g with respect to the measure w(x) dx, and ‖·‖p the Lp norm with respect to w(x) dx. Although the inequality is not true in general, the centred norm inequality does hold for 1 < p < ∞ if α < δ.

scholarly journals Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Khedoudj Saibi
Keyword(s):  
Measurable Function ◽  
Lorentz Space ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Lorentz Spaces ◽  
Square Function ◽  
Holder Continuity ◽  
Area Function ◽  
Intrinsic Square Function ◽  
Lusin Area Function

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paley g-function, the intrinsic Lusin area function, and the intrinsic gλ∗-function of the variable Hardy–Lorentz space Hp⋅,qℝn, for p⋅ being a measurable function on ℝn satisfying 0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞ and the globally log-Hölder continuity condition and q∈0,∞, via its atomic and Littlewood–Paley function characterizations.

Some properties of Musielak spaces with only the log-Hölder continuity condition and application

2020 ◽  
Vol 11 (4) ◽  
pp. 1062-1080
Author(s):  
Mustafa Ait Khellou ◽  
Abdelmoujib Benkirane ◽  
Sidi Mohamed Douiri
Keyword(s):  
Hölder Continuity ◽  
Continuity Condition ◽  
Holder Continuity

ON THE R-ORDER CONVERGENCE OF A THIRD ORDER METHOD IN BANACH SPACES UNDER MILD DIFFERENTIABILITY CONDITIONS

2009 ◽  
Vol 06 (02) ◽  
pp. 291-306 ◽  
Author(s):  
P. K. PARIDA ◽  
D. K. GUPTA
Keyword(s):  
Banach Spaces ◽  
Recurrence Relations ◽  
Hölder Continuity ◽  
A Priori ◽  
Continuity Condition ◽  
Order Method ◽  
Holder Continuity ◽  
Third Order ◽  
A Priori Error Bounds ◽  
Two Parameters

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F(x)=0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity condition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails.

Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dharmendra Kumar Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh ◽  
Jose Luis Hueso ◽  
Shwetabh Srivastava ◽  
...  
Keyword(s):  
Recurrence Relations ◽  
Lipschitz Continuity ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Fréchet Derivative ◽  
Second Order ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Dynamical Study ◽  
Frechet Derivative

Abstract The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q, where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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