Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Argyros, Ioannis K.

doi:10.3390/math9161942

Cited by 52 publications

(53 citation statements)

References 25 publications

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“…In this sense, the convergence analysis of iterative techniques should include a measure of closeness of the initial estimate to the solution. In fact, many authors (see [6][7][8][9][10][11][12][13][14][15][16]) have adopted an appropriate methodology to establish the local or semilocal convergence analysis by considering the hypothesis of Lipschitz continuity on first-order derivatives only. Furthermore, the bounds of the error estimates and the convergence radius can be computed by defining some real functions and parameters.…”

Section: Introductionmentioning

confidence: 99%

On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations

et al. 2022

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The local convergence of a generalized (p+1)-step iterative method of order 2p+1 is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for the given iterative method was carried out while assuming the existence of certain higher-order derivatives. In contrast to this approach, the convergence analysis is carried out in the present study by considering the hypothesis only on the first-order Fréchet derivatives. This study further provides an estimate of convergence radius and bounds of the error for the considered method. Such estimates were not provided in earlier studies. In view of this, the applicability of the given method clearly seems to be extended over the wider class of functions or problems. Moreover, the numerical applications are presented to verify the theoretical deductions.

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Section: Introductionmentioning

confidence: 99%

On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations

et al. 2022

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“…A plethora of problems from diverse disciplines are solved using high order iterative methods [1][2][3][4][5][6][7][8][9][10][11][12][13]. But there are some common problems with the applications of these methods.…”

Section: Introductionmentioning

confidence: 99%

“…So, the convergence of the Noor-Waseem method is not guaranteed by the analysis in [9]. The same problem exists with other methods [1][2][3][4][5][6][7][8][10][11][12][13] although the method may converge. Other problems have already been reported in the abstract of this study.…”

Section: Introductionmentioning

confidence: 99%

On a Noor-Waseem-Type Method for Solving Nonlinear Equations

2022

Electronic Journal of Mathematics

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A generalized method due to Noor and Waseem is studied for solving nonlinear equations in Banach space. The Noor-Waseem method is of order three. But, the convergence of this method was shown assuming that the fourth derivative, not on the method, exists. This constraint is limiting its applicability. Moreover, neither computable error bounds nor results about the uniqueness of the solution were given. We address all these problems using only the first derivative which only appears on the method. Hence, we extend the applicability of the method under consideration. Our techniques can be used to obtain the convergence of other similar higher order methods using assumptions only on the first derivative of the operator involved.

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“…This forces researchers and practitioners to resort to iterative approximations to χ * . A plethora of such approximations can be found in the literature [1][2][3][4][5]. Among those the most useful are the high convergence order ones.…”

Section: Introductionmentioning

confidence: 99%

“…The remainder of this paper is organized as follows: Majorizing sequences for method (2) are introduced and studied in Section 2. The semi-local convergence is given in Section 3 for method (2). Numerical applications appear in Section 4.…”

Section: Introductionmentioning

confidence: 99%

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

et al. 2022

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The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a more interesting case. A process is developed based on majorizing sequences and the notion of restricted Lipschitz condition to provide a semi-local convergence analysis for the third convergent order Noor–Waseem method. Due to the generality of our technique, it can be used on other high order methods. The convergence analysis is enhanced. Numerical applications complete are used to test the convergence criteria.

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Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Cited by 52 publications

References 25 publications

On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations

On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations

On a Noor-Waseem-Type Method for Solving Nonlinear Equations

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

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