Related papers: Solutions for Underdetermined Generalized Absolute…
This paper investigates the conditions that guarantee unique solvability and unsolvability for the generalized absolute value equations (GAVE) given by $Ax - B \vert x \vert = b$. Further, these conditions are also valid to determine the…
In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $Ax-B|x|=b$ with $A, B\in \mathbb{R}^{n\times n}$ from the optimization field are first presented,…
This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several…
Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted…
We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-|x|=b$. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is…
An inverse-free dynamical system is proposed to solve the generalized absolute value equation (GAVE) with a fixed time convergence, where the time of convergence is finite and is uniformly bounded for all initial points. Moreover, an…
The system of generalized absolute value equations (GAVE) has attracted more and more attention in the optimization community. In this paper, by introducing a smoothing function, we develop a smoothing Newton algorithm with non-monotone…
A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well…
Motivated by the framework constructed by Brugnano and Casulli $[$SIAM J. Sci. Comput. 30: 463--472, 2008$]$, we analyze the finite termination property of the generalized Netwon method (GNM) for solving the absolute value equation (AVE).…
In this paper, we reconsider two new iterative methods for solving absolute value equations (AVE), which is proposed by Ali and Pan (Jpn. J. Ind. Appl. Math. 40: 303--314, 2023). Convergence results of the two iterative schemes and new…
The absolute value equations (AVE) problem is an algebraic problem of solving Ax+|x|=b. So far, most of the research focused on methods for solving AVEs, but we address the problem itself by analysing properties of AVE and the corresponding…
In this article we study a class of generalised linear systems of difference equations with given boundary conditions and assume that the boundary value problem is non-consistent, i.e. it has infinite many or no solutions. We take into…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
The general solutions with free variable to the second-kind Abel equation, a nonlinear ordinary differential equation that has remained unsolved for nearly two centuries, are presented for the first time by using elementary quadrature…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
In this paper, we discussed the unique solvability of the two absolute value matrix equations. The unique solvability condition $\rho (\vert A^{-1} B \vert)<1$ is provided for the generalized absolute value matrix equation (GAVME) $AX + B…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
A generalization of the Newton-based matrix splitting iteration method (GNMS) for solving the generalized absolute value equations (GAVEs) is proposed. Under mild conditions, the GNMS method converges to the unique solution of the GAVEs.…
In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and global error bounds is established, in the…
We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the…