Local Convergence of the Chord Method for Generalized Equations

[1]

**M. H. Rashid**, Department of Mathematics, Faculty of Science, University of Rajshahi, Rajshahi, Bangladesh.

[2]

**A. Basak**, Department of Mathematics, Faculty of Science, University of Rajshahi, Rajshahi, Bangladesh.

[3]

**M. Z. Khaton**, Department of Mathematics, Sapahar Government Degree College, Naogaon, Bangladesh.

Let X be a real or complex Banach space and Y be a normed linear space. Suppose that f : X ⟼ Y is a Frechet differentiable function and F : X→ 2Y is a set-valued mapping with closed graph. In the present paper, we study the Chord method for solving generalized equation 0 ∈ f(x)+ F(x). We prove the existence of the sequence generated by the Chord method and establish local convergence of the sequence generated by this method for generalized equation.

Chord Method, Generalized Equation, Local Convergence, Pseudo-Lipschitz Mapping, Set-Valued Mapping

[1]

K.A. Atkinson, An introduction to Numerical Analysis (second edition), American Mathematics Society, 1988.

[2]

A. L. Dontchev, Local convergence of the Newton method for generalized equation, C. R. Acad. Sci. Paris, Ser. I 322(1996), 327–331.

[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, Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J. 22(1996), 385–398.

[5]

A.L. Dontchev and W.W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121(1994), 481–489.

[6]

M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarily problems, SIAM Rev. 39(1997), 669–713.

[7]

M.H. Geoﬀroy, S. Hilout and A. Pietrus,. Acceleration of convergence in Dontchev’s itera-tive methods for solving variational inclusions, Serdica Math. J. 29(2003), 45–54.

[8]

M.H. Geoffroy and A. Pietrus, A general iterative procedure for solving nonsmooth generalized equations, Comput. Optim. Appl. 31(1)(2005), 57–67.

[9]

C.E. Lawrence, Partial Differential Equations (second edition), American Mathematics Society, 1998.

[10]

R.T. Marinov, Convergence of the method of chords for solving generalized equations, Rendiconti del Circolo Matematico di Palermo 58(2009), 11–27.

[11]

A. Pietrus,Generalized equations under mild differentiability conditions, Rev. S. A. Acad. Cienc. Exact. Fis. Nat., 94(2000)., 15–18.

[12]

A. Pietrus,. Does Newton’s method for set-valued maps converge uniformly in mild diﬀerentiability context? Rev. Columbiana Mat. 34(2000), 49–56.

[13]

A. Pietrus, and S. Hilout, A semilocal convergence of the secant-type method for solving a generalized equation, Positivity 10(2006), 693–700.

[14]

M.H. Rashid, S.H. Yu, C. Li, and S.Y. Wu, Convergence Analysis of the Gauss-Newton Method for Lipschitz--like Mappings, J. Optim. Theory Appl. 158(1) (2013), 216–233.

[15]

S.M. Robinson, Generalized equations and their solutions, Part I, basic theory, Math. Program. Stud. 10(1979), 128–141.

[16]

S.M. Robinson, Generalized equations and their solutions, part II: application to nonlinear programming, Math. Program. Stud. 19(1982), 200–221.

[17]

M. Schatzman, Numerical analysis: a mathematical introduction, Clarendon Press, Oxford, 2002.