On the convergence of a modified superquadratic method for generalized equations

Year 2018, Volume: 1 Issue: 3, 202 - 214, 30.09.2018

Mohammed Harunor Rashid , Md. Zulfiker Ali

https://doi.org/10.32323/ujma.415225

Abstract

Let $X$ and $Y$ be Banach spaces. Let $\Omega$ be an open subset of $X$. Suppose that $f:X\to{Y}$ is Fr\'{e}chet differentiable in $\Omega$ and $\mathcal F:X\rightrightarrows2^Y$ is a set-valued mapping with closed graph. In the present paper, a modified superquadratic method (MSQM) is introduced for solving the generalized equations $0\in{f(x)+\mathcal F(x)}$, and studied its convergence analysis under the assumption that the second Fr\'{e}chet derivative of $f$ is H\"{o}lder continuous. Indeed, we show that the sequence, generated by MSQM, converges super-quadratically in both semi-locally and locally to the solution of the above generalized equation whenever the second Fr\'{e}chet derivative of $f$ satisfies a H\"{o}lder-type condition.

Keywords

Generalized equations, Lipschitz--like mappings, Semi-local convergence, Set-valued mapping

References

• [1] A. L. Dontchev, Local convergence of the Newton method for generalized equation, C. R. A. S Paris Ser.I 322 (1996), 327–331.
• [2] A. L. Dontchev, Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J. 22 (1996), 385–398.
• [3] A. L. Dontchev, Local analysis of a Newton-type method based on partial linearization, Lectures in Applied Mathematics 32 (1996), 295–306.
• [4] A. L. Dontchev and W.W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481–489.
• [5] A. Pi´etrus, Generalized equations under mild differentiability conditions, Rev. R. Acad. Cienc. Exact. Fis. Nat. 94(1) (2000), 15–18.
• [6] A. Pi´etrus, Does Newton’s method for set-valued maps converges uniformly in mild differentiability context?, Rev. Colombiana Mat. 34 (2000), 49–56.
• [7] C. Li, W. H. Zhang and X. Q. Jin, Convergence and uniqueness properties of Gauss-Newton’s method, Comput. Math. Appl. 47 (2004), 1057–1067.
• [8] J. P. Dedieu and M. H. Kim, Newton’s method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002), 187–209.
• [9] J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.
• [10] J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkh¨auser, Boston, 1990.
• [11] J. P. Dedieu and M. Shub, Newton’s method for overdetermined systems of equations, Math. Comp. 69 (2000), 1099–1115.
• [12] J. S. He, J. H. Wang and C. Li, Newton’s method for underdetemined systems of equations under the modified g-condition, Numer. Funct. Anal. Optim. 28 (2007), 663–679.
• [13] M. Geoffroy and A. Pi´etrus, A superquadratic method for solving generalized equations in the H´oder case, Ricerche di Matematica LII (2003), 231–240.
• [14] M. Geoffroy, S. Hilout and A. Pi´etrus, Acceleration of convergence in Dontchev’s iterative method for solving variational inclusions, Serdica Math. J. 29 (2003), 45–54.
• [15] M. Geoffroy, S. Hilout and A. Pi´etrus, Stability of a cubically convergent method for generalized equations, Set-Valued Analysis 14 (2006), 41–54.
• [16] M. H. Rashid, A Convergence Analysis of Gauss-Newton-type Method for Holder Continuous Maps, Indian Journal of Mathematics 57(2) (2014), 181–198.
• [17] M. H. Rashid, Convergence Analysis of a Variant of Newton-type Method for Generalized Equations, International Journal of Computer Mathematics 95(3) (2018), 584–600.
• [18] M. H. Rashid, On the convergence of extended Newton-type method for solving variational inclusions, Cogent Mathematics, 1(1) (2014), DOI 10.1080/23311835.2014.980600.
• [19] M. H. Rashid, Convergence Analysis of Extended Hummel-Seebeck-type Method for Solving Variational Inclusions, Vietnam Journal of Mathematics 44 (2016), 709–726.
• [20] M. H. Rashid, Extended Newton-type Method and its Convergence Analysis for Nonsmooth Generalized Equations, J. Fixed Point Theory and Appl. 19 (2017), 1295–1313.
• [21] M. H. Rashid, J. H. Wang and C. Li, Convergence analysis of a method for variational inclusions, Applicable Analysis 91(10) (2012), 1943–1956.
• [22] M. H. Rashid, M. Z. Ali and A. Pietrus, Extended Cubic Method and Its Convergence Analysis for Generalized Equations, Journal of Advances and Applied Mathematics, 3(3) (2018), 91-108.
• [23] M. H. Rashid, S. H. Yu, C. Li and S,Y. Wu, Convergence analysis of the Gauss-Newton-type method for Lipschitz–like mappings, J. Optim. Theory Appl. 158(1) (2013), 216–233.
• [24] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Progamming Stud.10 (1979), 128–141.
• [25] S. M. Robinson, Strong regular generalized equations, Math. of Oper. Res. 5 (1980), 43–62.
• [26] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Stud.19 (1982), 200–221.
• [27] X. B. Xu and C. Li, Convergence of Newton’s method for systems of equations with constant rank derivatives, J. Comput. Math. 25 (2007), 705–718.
• [28] X. B. Xu and C. Li, Convergence criterion of Newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl. 345 (2008), 689–701.