ON THE MODIFIED NEWTON'S METHOD FOR THE SOLUTION OF A CLASS OF NONLINEAR SINGULAR INTEGRO-DIFFERENTIAL

In this paper we investigate the solvability of a certain class of nonlinear singular integro-differential equations with Cauchy kernel in the usual Holder space and its generalization by producing the sufficient conditions for the convergence of the modified Newton's method.

In this paper we study the sufficient conditions for the applicability and convergence of the modified Newton's method for the following class of NSIDE with Cauchy kernel in the usual Holder space ( ) ( ) where u(x) is the unknown function and which is taken as a Cauchy principle value .Also , we consider a generalized form of (1.1) which has the form :

BASIC NOTIONS AND AUXILIARY RESULTS
We shall introduce some notations , definitions and assumptions , which will be used in the sequel .Definition 2.1 [5]  ,(k =0,1,2,…,n), we denote the space of all k-times differentiable , we define the following norms: (i) Let the function G in equation (1.1) be defined and continuous on the region : and possess partial derivatives up to the second order and satisfy the following condition: 2) be defined and continuous on the and possess partial derivatives up to the second order and satisfy the following condition: Let the function G in the equation (1.1) satisfies the assumption (i).Then , the following inequality is valid

Proof
It is easy to get the proof from definition 2.1 and condition (2.5).

Lemma 2.2
Let the function G in equation (1.2) satisfies the assumption (ii).Then , the following inequality is valid ) ( ) (

Proof
It is easy to get the proof from definition 2.

APPLICABILITY OF THE MODIFIED NEWTON'S METHOD TO A CERTAIN CLASS OF NSIDE
In this section we shall consider the applicability of the modified Newton's method to the class given by equation (1.1).The following two lemmas are fundamental in our study.

Lemma 3.1
Let the function G in the equation (1.1) satisfies the assumption (i) .Then, the operator P defined in the equation (1.1) is Frechet differentiable with derivative given by : ] , [ b a H δ at every fixed point of the space ( ) ( ) ( ) in the sphere By applying Lagrange's formula , [10], we get : Hence , P(u) is differentiable in the sense of Frechet and its derivative is given by: From equation (3.1) and Lagrange's formula , [10],we get : . with 0 , 0 , and 0 , 0 , , 0 , 0 , Therefore, from the assumption (i) , Lipschitz's condition (3.8) has the form : where: Thus the Lemma is true .

Lemma 3.2
Let the conditions of lemma (3.1) be satisfied .Then, the linear operator that can be rewritten , by using (3.1) , as follows : ( ) and by using the condition (3.11) we can show that : ( ) hence , the Sokhotski formulae , [6] , are : ( ) ( ) and Substituting from equations (3.14) , (3.15) into (3.13)we obtain : which can be rewritten in the following Boundary value problem (B.V.P) : (3.19) where: ( ) ( ) From the theory of linear singular integral equations [6] By using relations (3.23) , we obtain : where The equation (3.21) is a first order linear ordinary differential equation has the following solution: ( ) Thus, the linear operator (3.1) has a bounded inverse and the lemma is true .
Thus, all the conditions of applicability and convergence of the modified Newton's method are satisfied .Hence , the following theorem is valid.
the equation (1.1) has a unique solution v in the sphere of the modified Newton's method converges and the rate of convergence is given by the inequality ( )

ON A GENERALIZATION FORM OF NSIDE
In this section , we can generalize the class of NSIDE represented by (1.1) to a more general class of NSIDE written in the form (1.2).By the same technique have used in section § 2 , we shall study the NSIDE (1.2).Lemma 4. 1 Let the function G in the equation (1.2) satisfies the assumption (ii) .Then, the operator P defined in the equation (1.2) is Frechet differentiable with derivative given by : ] , [ b a H δ at every fixed point of the space ( )  Let the conditions of lemma (4.1) be satisfied , then the linear operator ( )

Proof
Consider the holomorphic function ( ) Then, by using condition (4.6) , we can show that : q n q n n q n q n q n n Equation (4.11) can be rewritten in the form: where (4.12) .) Thus , all the conditions of applicability and convergence of the modified Newton's method are satisfied .Therefore , the following theorem is valid.
functions whose k-derivatives satisfy Holder's condition with exponent δ , : are first order linear ordinary differential equations.