A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

Minimal and maximal solutions to first-order differential equations with piecewise constant generalized delay

In this paper we employ the method of maximal and minimal solutions coupled with comparison principles and the monotone iterative technique to obtain results of existence and approximation of solutions for differential equations with piecewise constant delay of generalized type (DEPCAG).

Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications

In this paper, we extend the applications of the method of upper and lower solutions for a class of nonlinear nabla fractional difference equations involving Caputo derivative. We obtain the existence of coupled minimal and maximal solutions which constructed by two monotone sequences. In order to illustrate our main results, we present two numerical examples in the end.

Backward doubly SDEs with continuous and stochastic linear growth coefficients

We study backward doubly stochastic differential equations when the coefficients are continuous with stochastic linear growth. Via an approximation and comparison theorem, the existence of minimal and maximal solutions are obtained.

Approximation methods in the theory of hybrid differential equations with linear perturbations of second type

In this paper, some existence theorems for the extremal solutions are proved for an initial value problem of nonlinear hybrid differential equations via constructive methods. The monotone iterative techniques for initial value problems of first order hybrid differential equations are developed and it is shown that the sequences of successive iterations defined in a certain way converge to the minimal and maximal solutions of the hybrid differential equations.

Periodic Boundary Value Problems for First-Order Impulsive Functional Integrodifferential Equations with Integral-Jump Conditions

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.