A Third Order Newton-Like Method and Its Applications
Abstract
:1. Introduction
- (a)
- Lipschitz condition: for all and for some ;
- (b)
- Hölder Lipschitz condition: for all and for some and ;
- (c)
- ω-continuity condition: for all ,where is a nondecreasing and continuous function.
2. Preliminary
- (1)
- for all .
- (2)
- .
- (1)
- .
- (2)
- The sequence is decreasing, that is for all .
- (3)
- for all .
- (4)
- for all .
- (5)
- for all , where and .
- (1)
- Since the scalar equation defined by Equation (9) has a minimum positive root and is decreasing in with and . Therefore, in the interval and hence
- (2)
- From (1) and Equation (10), we have . This shows that (2) is true for . Let be a fixed positive integer. Assume that (2) is true for Now, using Equation (10), we haveThus holds for . Therefore, by induction, holds for all .
- (3)
- Since for each and for all , it follows that
- (4)
- From (3), one can easily prove that the sequences and are well defined. Using Equations (10) and (11), one can easily observe thatHence holds for and . Let be a fixed integer. Assume that holds for each From Equations (11) and (12), we haveThus holds for . Therefore, by induction, holds for all .
- (5)
- From Equation (11) and (4), one can easily observe that
3. Computation of a Solution of the Operator Equation (1)
- (1)
- for all ;
- (2)
- for all and for some .
- (C1)
- for all ;
- (C2)
- for all and for some ;
- (C3)
- for some ;
- (C4)
- for some ;
- (C5)
- , and ;
- (C6)
- , , and , where .
- (1)
- The sequence generated by Equation (13) is well defined, remains in and satisfies the following estimates:
- (2)
- The sequence converges to the solution of the Equation (1).
- (3)
- The priory error bounds on is given by:
- (4)
- The sequence has R-order of convergence at least .
4. Applications
4.1. Fixed Points of Smooth Operators
- (C7)
- for some ;
- (C8)
- for some ;
- (C9)
- for all ;
- (C10)
- for all and for some .
4.2. Fredholm Integral Equations
- (C12)
- for all ;
- (C13)
- for some ;
- (C14)
- for some ;
- (C15)
- for all , where ;
- (C16)
- and ;
- (C17)
- for all .
- (C18)
- for some
- (C19)
- for some
- (C20)
- for all .
- (a)
- , ;
- (b)
- ;
- (c)
- ;
- (d)
- ;
- (e)
- and .
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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4 | 8.775202786637237 × |
5 | 4.440892098500626 × |
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Sahu, D.R.; Agarwal, R.P.; Singh, V.K. A Third Order Newton-Like Method and Its Applications. Mathematics 2019, 7, 31. https://doi.org/10.3390/math7010031
Sahu DR, Agarwal RP, Singh VK. A Third Order Newton-Like Method and Its Applications. Mathematics. 2019; 7(1):31. https://doi.org/10.3390/math7010031
Chicago/Turabian StyleSahu, D. R., Ravi P. Agarwal, and Vipin Kumar Singh. 2019. "A Third Order Newton-Like Method and Its Applications" Mathematics 7, no. 1: 31. https://doi.org/10.3390/math7010031
APA StyleSahu, D. R., Agarwal, R. P., & Singh, V. K. (2019). A Third Order Newton-Like Method and Its Applications. Mathematics, 7(1), 31. https://doi.org/10.3390/math7010031