Elementary Fixed Point Theorems by P. V. Subrahmanyam

Author:P. V. Subrahmanyam
Language: eng
Format: epub
ISBN: 9789811331589
Publisher: Springer Singapore

Lemma 6.5.1

Let be analytic and . Then

where , .

Proof

If g is analytic function of single complex variable in , then

This follows from Cauchy’s integral formula

Applying this for we get the inequalities

where and . The choice leads to the estimate stated in the lemma.

Remark 6.5.2

The Cauchy problem (6.5.1) can be written as the equivalent integral equation.

(6.5.2)

(6.5.3)

Theorem 6.5.3

Suppose(i)the functions A(t, z), , C(t, z) are continuous in G and analytic in z for fixed t, the function being analytic in z;

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