Third-order iterative methods with applications to Hammerstein equations: a unified approach. (English) Zbl 1216.65064
This article deals with the family of iterations
\[ x_{n+1} = x_n - \left(I + \tfrac12L_F(x_n)(I + b(x_n)F'(x_n)^{-1}f(x_n))^{-1}\right)F'(x_n)^{-1}F(x_n), \quad n = 0,1,2,\dots \]
(\(I\) is the identity operator on \(X\), \(L_F(x) = F'(x)^{-1}F''(x)F'(x)^{-1}F(x)\)) for approximately solving a nonlinear operator equation \(F(x) = 0\) with a nonlinear operator \(F\) between two Banach spaces \(X\) and \(Y\). This family covers a lot of concrete iterations, in particular, Halley’s and Chebyshev’s methods. The authors present two theorems about the convergence to a solutions \(x_*\) of \(F(x) = 0\) and the uniqueness of this solution. The assumptions in these theorems are formulated in terms of some estimates for \(F'(x_1)^{-1}\), \(\|F'(x_1)^{-1}f(x_1)\|\), \(\|F''(x_1)\|\), \(\|B(x_1)\|\) and moduli of smoothness for \(F''(x)\) and \(B(x)\), where \(B(x)\) is an auxiliary function with values in the space of bilinear operators of type \(X \times X \to Y\). As an application, the Hammerstein integral equation
\[ u(s) = \psi(s) + \int_0^1 H(s,t)f(t,u(t)) \, dt \]
is considered.
There are some vague places in this paper; in particular, it is not clear why the assumptions in both theorems are given at the point \(x_1\) instead of \(x_0\); on examination of the Hammerstein integral equation, the authors do not fix the spaces \(X\) and \(Y\), and, as a result, their arguments are not correct and so on.
\[ x_{n+1} = x_n - \left(I + \tfrac12L_F(x_n)(I + b(x_n)F'(x_n)^{-1}f(x_n))^{-1}\right)F'(x_n)^{-1}F(x_n), \quad n = 0,1,2,\dots \]
(\(I\) is the identity operator on \(X\), \(L_F(x) = F'(x)^{-1}F''(x)F'(x)^{-1}F(x)\)) for approximately solving a nonlinear operator equation \(F(x) = 0\) with a nonlinear operator \(F\) between two Banach spaces \(X\) and \(Y\). This family covers a lot of concrete iterations, in particular, Halley’s and Chebyshev’s methods. The authors present two theorems about the convergence to a solutions \(x_*\) of \(F(x) = 0\) and the uniqueness of this solution. The assumptions in these theorems are formulated in terms of some estimates for \(F'(x_1)^{-1}\), \(\|F'(x_1)^{-1}f(x_1)\|\), \(\|F''(x_1)\|\), \(\|B(x_1)\|\) and moduli of smoothness for \(F''(x)\) and \(B(x)\), where \(B(x)\) is an auxiliary function with values in the space of bilinear operators of type \(X \times X \to Y\). As an application, the Hammerstein integral equation
\[ u(s) = \psi(s) + \int_0^1 H(s,t)f(t,u(t)) \, dt \]
is considered.
There are some vague places in this paper; in particular, it is not clear why the assumptions in both theorems are given at the point \(x_1\) instead of \(x_0\); on examination of the Hammerstein integral equation, the authors do not fix the spaces \(X\) and \(Y\), and, as a result, their arguments are not correct and so on.
Reviewer: Peter Zabreiko (Minsk)
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 47J25 | Iterative procedures involving nonlinear operators |
| 65R20 | Numerical methods for integral equations |
| 45G10 | Other nonlinear integral equations |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
Keywords:
iterative methods; semilocal convergence; Hammerstein integral equations; Halley method; Chebyshev method; two-step method; nonlinear operator equation; Banach spacesReferences:
| [1] | Amat, S.; Busquier, S.; Gutiérrez, J. M., Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math., 157, 1, 197-205 (2003) · Zbl 1024.65040 |
| [2] | Amat, S.; Busquier, S., Third-order iterative methods under Kantorovich conditions, J. Math. Anal. Appl., 336, 1, 243-261 (2007) · Zbl 1128.65036 |
| [3] | Ezquerro, J. A.; Hernández, M. A., Halley’s method for operators with unbounded second derivative, Appl. Numer. Math., 57, 3, 354-360 (2007) · Zbl 1252.65098 |
| [4] | Hernández, M. A.; Romero, N., On a characterization of some Newton-like methods of \(R\)-order at least three, J. Comput. Appl. Math., 183, 1, 53-66 (2005) · Zbl 1087.65057 |
| [5] | Hernández, M. A.; Romero, N., Application of iterative processes of \(R\)-order at least three to operators with unbounded second derivative, Appl. Math. Comput., 185, 1, 737-747 (2007) · Zbl 1115.65063 |
| [6] | Safiev, R. A., On some iterative processes, Zh. Vychisl. Mat. Mat. Fiz., 4, 139-143 (1965) |
| [7] | Argyros, I. K., On a two-point Newton method in Banach spaces and the Pták error estimates, Rev. Acad. Cienc. Zaragoza (2), 54, 111-120 (1999) · Zbl 1006.65058 |
| [8] | Chen, D.; Argyros, I. K.; Qian, Q. S., A note on the Halley method in Banach spaces, Appl. Math. Comput., 58, 2-3, 215-224 (1993) · Zbl 0787.65034 |
| [9] | Hernández, M. A.; Salanova, M. A., Indices of convexity and concavity. Application to Halley method, Appl. Math. Comput., 103, 1, 27-49 (1999) · Zbl 0981.65071 |
| [10] | Yamamoto, T., On the method of tangent hyperbolas in Banach spaces, J. Comput. Appl. Math., 21, 1, 75-86 (1988) · Zbl 0632.65070 |
| [11] | Babajee, D. K.R.; Dauhoo, M. Z.; Darvishi, M. T.; Karami, A.; Barati, A., Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, J. Comput. Appl. Math., 233, 8, 2002-2012 (2010) · Zbl 1204.65050 |
| [12] | Kou, J.; Li, Y.; Wang, X., Third-order modification of Newton’s method, J. Comput. Appl. Math., 205, 1, 1-5 (2007) · Zbl 1118.65039 |
| [13] | Koçak, M. Çetin, A class of iterative methods with third-order convergence to solve nonlinear equations, J. Comput. Appl. Math., 218, 2, 290-306 (2008) · Zbl 1143.65041 |
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