We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families to $k$-number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

The dynamics of a biparametric family for solving nonlinear equations is studied on quadratic polynomials. This biparametric family includes the $c$-iterative methods and the well-known Chebyshev-Halley family. We find the analytical expressions for the fixed and critical points by solving 6-degree polynomials. We use the free critical points to get the parameter planes and, by observing them, we specify some values of $(\alpha , c)$ with clear stable and unstable behaviors.

We study the existence and approximation of a solution for a system of hierarchical variational inclusion problems in Hilbert spaces. In this study, we use Maingé’s approach for finding the solutions of the system of hierarchical variational inclusion problems. Our result in this paper improves and generalizes some known corresponding results in the literature.

From 1998, the International Astronomical Union (IAU) adopted a new Celestial Reference System: the International Celestial Reference System (ICRS). The first optical materialization was the Hipparcos catalogue, defining the Hipparcos Celestial Reference Frame (HCRF). The compilation of subsequent radio sources catalogues culminated in the current representation of the ICRF, the ICRF2 catalogue that is not sufficiently dense to cover all astrometrical purposes. Linking Hipparcos and ICRF2 is essential to uniformize the reference regardless of whether it is visible (HCRF) or not (ICRF). Many working groups provide their own complementary catalogs, some of whose sources are also in the ICRF2, with different reduction processes for positions. The point is that they provide information in more than one reference for a small number of objects. Some of these projects have been used by us to study the Hipparcos-ICRF2 differences: a certain number of couples of catalogs can be interrelated using a set of parameters. With these couples, we build a closed cycle with the same ending and departure couple. The parameters obtained from each couple affect the next; thus we have an iterative process whose fixed point is the solution that stabilizes it, providing a preliminary link for Hipparcos-ICRF2.

The generalized quasilinearization technique is applied to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of a coupled system of second and fourth order elliptic equations.

This paper presents a new iterative method for computing the approximate inverse of nonsingular matrices. The analytical discussion of the method is included to demonstrate its convergence behavior. As a matter of fact, it is proven that the suggested scheme possesses tenth order of convergence. Finally, its performance is illustrated by numerical examples on different matrices.

It is shown how the mid-point iterative method with cubical rate of convergence can be applied for finding the principal matrix square root. Using an identity between matrix sign function and matrix square root, we construct a variant of mid-point method which is asymptotically stable in the neighborhood of the solution. Finally, application of the presented approach is illustrated in solving a matrix differential equation.

We introduced an implicit and an explicit iteration method based on the hybrid steepest descent method for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem.

Using the relation between a principal matrix square root and its inverse with the geometric mean, we present a fast algorithm for computing the geometric mean of two Hermitian positive definite matrices. The algorithm is stable and possesses a high convergence order. Some experiments are included to support the proposed computational algorithm.

Acceleration of convergence is discussed for some families of iterative methods in order to solve scalar nonlinear equations. In fact, we construct mono- and biparametric methods with memory and study their orders. It is shown that the convergence orders 12 and 14 can be attained using only 4 functional evaluations, which provides high computational efficiency indices. Some illustrations will also be given to reverify the theoretical discussions.

This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

We present a new numerical algorithm for two-point boundary value problems. We first present the exact solution in the form of series and then prove that the n-term numerical solution converges uniformly to the exact solution. Furthermore, we establish the numerical stability and error analysis. The numerical results show the effectiveness of the proposed algorithm.

We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.