Page 475 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 32: The Fredholm Theorem

CONTENTS
Objectives

Introduction
32.1 Fredholm Alternate Theorem
32.2 Proof of Fredholm Theorem

32.3 Summary
32.4 Keywords

32.5 Review Question
32.6 Further Readings

Objectives

After studying this unit, you should be able to:

 Learn that Fredholm integral equations are of two types – of first kind and of second kind
 Prove that if is not an eigenvalue then the Fredholm Integral equation has a solution for
the second kind and the solution for the homogeneous equation is zero.

 Show that for an eigenvalue problem the Fredholm integral equation of second kind has
a solution which also contains a set of r-constants in addition to one of its solution.

Introduction

The proof of the Fredholm theorem consists of two parts. In the first part the solution is unique
and is not an eigenvalue.
The second part explains the eigenvalue problem of the homogeneous Fredholm integral equation
and explains the structure of the main integral equation and the conjugate one.

32.1 Fredholm Alternate Theorem

The theorem states that:
Either the integral equation of the second kind

b
t
s
s
s
t
Q
f ( ) Q ( ) K ( , ) ( )dt ...(1)
a
with fixed , admits a unique continuous solutions Q(s) for any continuous function f(s), in
particular Q(s) = 0 for f(s) 0, or the associated homogeneous equation
b
Q
t
s
Q ( ) K ( , ) ( )dt 0 ...(2)
t
s
a
admits a number (r r 1) of linearly independent continuous solutions Q ( ), Q ( )....Q ( ) . In
s
s
s
1 2 n
the first case, the conjugate equation
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