CHAPTER III - Monotonicity, Inequalities, and Other Topics

Abstract

This chapter discusses the application of the theorems of topological fixed-point theorems, monotonicity, and inequalities, such as Schauder's fixed-point theorem and Brouwer's theorem for various equations. Schauder's fixed-point theorem states that if R be a Banach space; if the continuous operator T map the closed and convex set M (in R) into itself, and if the range TM be relatively compact, the operator T has at least one fixed point in M. Application of Schauder's theorem are discussed in the nonlinear differential equations, real systems of linear equations, extrapolation and error estimates for a monotone sequence of iterations, systems of linear equations, and linear differential equations. The chapter discusses the matrices, boundary value problems of monotone kind, and approximation problem. From approximation problem, the chapter discusses the existence of a minimal solution and then in the number of minimal solutions, particularly whether there is a single solution uniqueness problem. Moreover, rational T approximation, Eigenvalue problems, discrete Chebyshev approximation, and exchange methods.