Tools and Fundaments to Integral Equations and Their Solution Methods

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Chapter 2

Abstract

Applications

Nawab Hussain and Iram Iqbal

matrix equations, manageable function

bol k:k denotes the spectral norm, that is,

where <sup>λ</sup><sup>þ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> is the largest eigenvalue of <sup>A</sup><sup>∗</sup>

1. Introduction

norm defined by

9

Contraction Mappings and

The aim of the chapter is to find the existence results for the solution of nonhomogeneous Fredholm integral equations of the second kind and non-linear matrix equations by using the fixed point theorems. Here, we derive fixed point theorems for two different type of contractions. Firstly, we utilize the concept of manageable functions to define multivalued α<sup>∗</sup> � η<sup>∗</sup> manageable contractions and prove fixed point theorems for such contractions. After that, we use these fixed point results to find the solution of non-homogeneous Fredholm integral equations of the second kind. Secondly, we introduce weak F contractions named as α-F-weak-contraction to prove fixed point results in the setting of metric spaces and

by using these results we find the solution for non-linear matrix equations.

Let H nð Þ denote the set of all n � n Hermitian matrices, P nð Þ the set of all n � n Hermitian positive definite matrices, S nð Þ the set of all n � n positive semidefinite matrices. Instead of X ∈P nð Þ we will write X . 0. Furthermore, X ≥0 means X ∈S nð Þ. Also we will use X ≥ Y Xð Þ ≤ Y instead of X � Y ≥0ð Þ Y � X ≥0 . The sym-

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup><sup>þ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup>

> > sið Þ A ,

,

A. We denote by k k: <sup>1</sup> the Ky Fan

Keywords: contraction mapping, fixed point, integral equations,

kAk ¼

where sið Þ A , i ¼ 1, …, n, are the singular values of A. Also,

q

k k A <sup>1</sup> ¼ ∑

n i¼1

k k <sup>A</sup> <sup>1</sup> <sup>¼</sup> tr A<sup>∗</sup> ð Þ <sup>A</sup> <sup>1</sup>=<sup>2</sup> � �,

which is tr Að Þ for (Hermitian) nonnegative matrices. Then the set H nð Þ endowed with this norm is a complete metric space. Moreover, H nð Þ is a partially
