Arif Rafiq

On the Rate of Convergence of Kirk-Type Iterative Schemes

The purpose of this paper is to introduce Kirk-type new iterative schemes called Kirk-SP and Kirk-CR schemes and to study the convergence of these iterative schemes by employing certain quasi-contractive operators. By taking an example, we will compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor iterative schemes for aforementioned class of operators. Also, using computer programs in C

On rate of convergence of various iterative schemes

In this note, by taking an counter example, we prove that the iteration process due to Agarwal et al. (J. Nonlinear Convex. Anal. 8 (1), 61-79, 2007) is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

IMPLICIT MANN FIXED POINT ITERATIONS FOR PSEUDO-CONTRACTIVE MAPPINGS

Abstract Let K be a compact convex subset of a real Hilbert space H and T : K → K a continuous hemi-contractive map. Let { a n } , { b n } and { c n } be real sequences in [0, 1] such that a n + b n + c n = 1 , and { u n } and { v n } be sequences in K . In this paper we prove that, if { b n } , { c n } and { v n } satisfy some appropriate conditions, then for arbitrary x 0 ∈ K , the sequence { x n } defined iteratively by x n = a n x n − 1 + b n T v n + c n u n ; n ≥ 1 , converges strongly to a fixed point of T .

On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings

Abstract For a Lipschitz strongly accretive map considered by Chidume in [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429], it is known that a classical Picard-type iteration process converges strongly to a zero of the operator. He also proved that the rate of convergence, in this case, is at least as fast as a geometric progression. In this paper we study the Mann implicit iteration sequence for strongly accretive and strongly pseudo-contractive mappings. We showed that this implicit scheme gives better convergence rate estimate. Presented results improve the corresponding results of [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429; L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Am. Math. Soc. 125 (2) (1997) 1363–1366; W.R. Sastry, G.V.R. Babu, Approximation of fixed points of strictly pseudo-contractive mappings on arbitrary closed, convex sets in a Banach space, Proc. Amer. Math. Soc. 128 (2000) 2907–2909; Y. Song, R. Chen, Viscosity approximative methods to Cesaro means for non-expansive mappings, Appl. Math. Comput. 186 (2) (2007) 1120–1128].

A New Second-Order Iteration Method for Solving Nonlinear Equations

We establish a new second-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.4142 which is the same as the Newton-Raphson method. By using some examples, the efficiency of the method is also discussed. It is worth to note that (i) our method is performing very well in comparison to the fixed point method and the method discussed in Babolian and Biazar (2002) and (ii) our method is so simple to apply in comparison to the method discussed in Babolian and Biazar (2002) and involves only first-order derivative but showing second-order convergence and this is not the case in Babolian and Biazar (2002), where the method requires the computations of higher-order derivatives of the nonlinear operator involved in the functional equation.

Modified efficient variant of super-Halley method

In this paper, we present a new modified variant of super-Halley method for solving non-linear equations. Numerical results show that the method has definite practical utility.

Solution of nonlinear pull-in behavior in electrostatic micro-actuators by using He's homotopy perturbation method

The work presented here is about the nonlinear pull-in behavior of different electrostatic micro-actuators. Hes homotopy perturbation method (HPM) is applied to solve different types of micro-actuators like Fixed-Fixed beam and Cantilever beam actuators. Simulated results are presented for further analysis. Also the obtained results compare well with the literature.

Some multi-step iterative methods for solving nonlinear equations

In this paper, we develop some new iterative methods for solving nonlinear equations by using the techniques introduced in Golbabai and Javidi (2007) [1] and Rafiq and Rafiullah (2008) [20]. We establish the convergence analysis of the proposed methods and then demonstrate their efficiency by taking some test problems.

Modified Noor iterations for nonlinear equations in Banach spaces

The purpose of this paper is to analyze the modified three-step iterative scheme for solving nonlinear operator equations in real Banach spaces. Our results can be viewed as an extension of three-step and two-step iterative schemes of Glowinski and Le Tallec [R. Glowinski, P. Le Tallec, Augemented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Publishing Co., Philadelphia, 1989] Noor [M.A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math. Anal. Appl. 255 (2001) 589-604; M.A. Noor, Some predictor-corrector algorithms for multivalued variational inequalities, J. Optim. Theory Appl. 108 (3) (2001) 659-670; M.A. Noor, T.M. Rassias, Z. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl. 274 (2002) 59-68] and Ishikawa [S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147-150].

Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces

The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.