A NEW EXTENSION OF PARAMETER CONTINUATION METHOD FOR SOLVING OPERATOR EQUATIONS OF THE SECOND KIND

In this paper, we propose an extension of the parameter continuation method for solving operator equations of the second kind. By splitting of the operator into a sum of two operators: one monotone, Lipschitz-continuous and one contractive, the applicability of the method is broader. The suitability of the proposed approach is presented through an example.


Introductions
Parameter continuation method (PCM) was suggested and developed by Bernstein [1] and Schauder [3] which is the inclusion of the equation (    and study the dependence of the solution from parameter. The PCM is a powerful technique for solving operator equations, see for example [5][6][7]. Gaponenko [2] introduced the PCM for solving operator equations of the second kind ( ) , x A x f  (1) where A is a Lipschitz-continuous and monotone operator, which operates in an arbitrary Banach space .
X The monotone operator in Banach space is defined as follows.
The results obtained by Gaponenko are summarized in Theorem 1 and Theorem 2.
The symbolic notation (3) should be understood as the following iteration processes, which consist of Consider the following subsidiary problems.
We shall carry out a change of variable We have  (9) has a unique solution for any (1) xX  , i.e., the operator determined in the whole space . X By virtue of the monotonicity of the operator 1 , After changing the variable (9), the equation (8) will take the following form For any (1) We shall carry out two changes of variables is a contractive operator with contraction coefficient equal to 1 1 q  . Then the equation (12b) has a unique solution for any (2) xX  , i.e., the operator 1 2 G  is determined in the whole space . X By Lemma 1.1, for any (2) ( , with Lipschitz coefficient equal to 1 . After changing the variables (12a) and (12b), the equation (11) will take the following form (2) ( For any (2)  (2 ) x  for any fX  .
We shall carry out N changes of variables Next, we construct the iterative algorithm to find approximate solution of the Problem 2. The approximate solutions of the integral equation (13) are obtained by using the standard iteration process (2) (2) The iteration processes (19a)-(19d) can be written as the following symbolic notation   Then from above inequality it follows that ( (1) x . The error of an iteration process in the calculation of (1) x equals  (1) x to the variable x again introduces the (1) 1 , 1, 2,..., . ((  . Therefore (2) (2) 11 2 1 2 10 .  (1) x .
The error of an iteration process in the calculation of (1) x equals 1  Then the error of an iteration process in the calculation of (1) x equals () n  . Hence (1)