Vol 2, No 6 (2011)

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CONVERGENCE ANALYSIS OF AN ITERATIVE METHOD FOR NONLINEAR

OPERATOR EQUATIONS USING A MAJORIZING SEQUENCE

Atef. I. Elmahdy*

Department of Mathematical and Computational Sciences,National Institute of Technology Karnataka, Surathkal, India-575025.

E-mail: [email protected]

(Received on: 27-05-11; Accepted on: 08-06-11)

---

ABSTRACT

In this paper we consider the Lavrentiev regularization method for obtaining stable approximate solution to nonlinear

ill-posed operator equations

F

(

x

)

=

y

,

where

F

:

D

(

F

)

⊂

X

→

X

is a nonlinear monotone operator and

X

is

a real Hilbert space. Under the assumption that

F

is Lipschitz continuous, the iteration

(

,

)

δ

α n

x

converges to the

unique solution

x

_αδ of the equation

F

(

x

)

=

y

δ

+

α

(

x

₀

−

x

)

.

MSC: 65J20, 65J15, 47J06, 47J35; 47H05

Keywords: Nonlinear ill-posed operator, Monotone operator, Majorizing sequence, Lavrentiev regularization.

---

1. INTRODUCTION:

In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.

Let

F

:

D

(

F

)

⊂

X

→

X

is a nonlinear monotone operator and

X

is a real Hilbert space with inner product

⋅

,

⋅

and the corresponding norm

.

. Recall that

F

is monotone operator if it satisfies the relation

).

(

,

0,

),

(

)

(

x

F

y

x

y

x

y

D

F

F

−

−

≥

∀

∈

We are interested in obtaining a stable approximate solution for nonlinear ill-posed operator equation

(1.1)

,

)

(

x

y

F

=

when the data

y

is not known exactly. Further we assume that:

• Instead of an exact right-hand side

y

we are given only its perturbation

y

δ

∈

X

, such that

(1.2)

,

δ

δ

≤

−

y

y

where

δ

is the known noise level.

• The equation (1, 1) has a solution

x

† (which need not be unique).

• The operator

F

possesses a locally uniformly bounded Fr chet-derivative

F

′

(

⋅

)

in

B

_r

(

x

₀

)

B

_r

(

x

†

)

⊆

D

(

F

)

⊆

X

0

0 ,

† 0 0

x

x

r

=

−

and

x

₀ is the initial guess for the solution.

---

IJMA- 2(6), June-2011, Page: 824-831

The numerical treatment of nonlinear ill-posed problems equation (1.1) in which the solution does not depend continuously on the data requires the application of special regularization methods. Again equation (1.1) is ill-posed,

and then the problem of recovery of

x

†

⊆

D

(

F

)

from a known noisy equation

F

(

x

)

=

y

δ can cause large deviation in the solution. Since

F

is monotone, one can use the Lavrentiev regularization method for solving (1.1). (See [9]). In this method the regularized approximation

x

_αδ is obtained by solving the operator equation

(1.3)

,

)

(

)

(

x

α

x

x

₀

y

δ

F

+

−

=

It is known (cf. [9], Theorem 1.1) that the equation (1.3) has a unique solution for any

α

≥

0

,

which called the

regularization parameter. In practice one has to deal with some sequence

(

,

)

δ

α

n

x

, converging to the solution

x

_αδ of

(1.3), (see [2], and [6]). In [2] Bakushinsky and Smirnova considered an iteratively regularized Lavrentiev method:

(1.4)

))

(

)

(

(

)

(

0

1

1

x

A

I

F

x

y

x

x

x

k

=

k

−

k

+

k

−

+

k k

−

− +

δ δ

δ δ

δ δ δ

α

α

for

k

=

0

,

1

,

2

,

,

where

A

_kδ

=

F

′

(

x

_kδ

)

and

(

α

_k

)

is a sequence of positive real numbers such that

lim

=

0

→∞ k

k

α

, as an approximate solution for (1.1). A general discrepancy principal has been considered in [2] for choosing the stopping

index

k

_δ and showed that

x

_kδ

→

x

†

δ as

δ

→

0

. However no error estimates for

†

x

x

_kδ

−

δ has been given in

[2]. Later in [7], Mahale and Nair considered the method (1.4) and obtained an error estimate for

x

_kδ

−

x

†

,

under weaker assumptions than the assumptions in [2] (see [7]).

In [3], Elmahdy considered an iterative regularization method:

(1.5)

)),

(

)

(

(

, , 0

, ,

1

x

F

x

y

x

x

x

n+

=

n

−

n

−

+

n

−

δ α δ

δ α δ

α δ

α

β

α

for solving the nonlinear equation (1,1) where

β

>

0

and

x

₀

:

=

x

δ_0,αis starting point of the iteration and proved that

)

(

,

δ α n

x

converges to the unique solution

x

_αδ of (1.3) under the following Assumptions.

Assumption: 1.1 There exists

r

₀

>

0

such that

(

₀

)

(

†

)

(

)

0

0

x

B

x

D

F

B

_{r r}

⊆

and

F

is Fr chet differentiable at all

).

(

)

(

₀ †

0 0

x

B

x

B

x

∈

_{r r}

Assumption: 1.2 There exists a constant

L

>

0

such that for every

,

(

₀

)

(

†

).

0 0

x

B

x

B

u

x

∈

_{r r} and

v

∈

X

there exists an element

φ

(

x

,

u

,

v

)

∈

X

satisfying

.

)

,

,

(

),

,

,

(

)

(

)]

(

)

(

[

F

′

x

−

F

′

u

v

=

F

′

u

φ

x

u

v

φ

x

u

v

≤

L

v

Assumption: 1.3 There exists a continuous, strictly monotonically increasing function

ϕ

:

(

0

,

a

]

→

(

0

,

∞

)

with

a

≥

F

′

(

x

†

)

, satisfying

lim

(

)

0

0

=

→

ϕ

λ

λ and there exist

v

∈

X

with

v

≤

1

such that

v

x

F

x

x

₀

−

†

=

ϕ

(

′

(

†

))

and

sup

(

)

(

),

(0,

a].

0

∈

∀

≤

+

≥

λ

α

ϕ

α

λ

λ

αϕ

ϕ λ

c

In this paper we use the following modified form of Assumption 1.2

Assumption: 1.4 There exists a constant

k

₀

>

0

such that for every

,

(

₀

)

(

†

).

0 0

x

B

x

B

u

x

∈

_{r r} and

v

∈

X

there exists an element

φ

(

x

,

u

,

v

)

∈

X

satisfying

.

)

,

,

(

),

,

,

(

)

(

)]

(

)

(

IJMA- 2(6), June-2011, Page: 824-831

Note that from Assumption 1.4 there follows Assumption 1.2 with

L

=

k

₀

x

−

u

. Hence Assumption 1.4 is stronger than Assumption 1.2. We will give a new stopping rule

n

_δfor the iteration different of the stopping rule using in [3]

and provide an optimal order error estimate under a general source condition on †

.

0

x

x

−

Moreover we shall use the adaptive parameter selection procedure suggested by Pereverzev and Schock in [8], for choosing the regularization parameter

α

in

(

,

)

δ α

n

x

.

The plan of this paper is as follows. In Section 2, we introduce the convergence analysis of the method and in Section 3, we give an error bounds under source conditions. Section 4 deals with starting points and algorithm. Finally, we give some concluding remarks in Section 5.

2. CONVERGENCE ANALYSIS:

Here we consider the method (1.5) for solving the nonlinear equation (1.1). The main goal of this section is to provide sufficient conditions for the convergence of method (1.5) to the unique solution

x

_αδ of (1.3) by using a majorizing

sequence and obtain an error estimate for

x

δ_n,α

−

x

_αδ . Recall (see [1], Definition 1.3.11) that a nonnegative sequence

)

(

t

_n is said to be a majorizing sequence of a sequence

(

x

_n

)

in

X

if

.

0

1

1

−

≤

+

−

∀

≥

+

x

t

t

n

x

_{n n n n}

During the convergence analysis we will be using the following Lemma on majorization, which is a reformulation of Lemma 1.3.12 in [1].

Lemma 2.1: Let

(

t

_n

)

be a majorizing sequence for

(

x

_n

)

in

X

. If *

n

lim

t

n

=

t

∞

→ , then

x

lim

n

x

n *

∞ →

=

exists and

_x

*

₋

_x

_≤

_t

*

₋

_t

_∀

_n

_≥

0

.

(2.6)

n n

Proof: Note that

(2.7)

,

)

(

1 n

1 1

1

x

t

t

t

t

x

x

x

_{j j n m}

m n

n j m

n

n j

j j n

m

n

−

≤

−

≤

+

−

=

+

−

− +

= −

+

= + +

so

(

x

_n

)

is a Cauchy sequence in

X

and hence

(

x

_n

)

converges to some

x

*

.

The error estimate in (2.6) follows from (2.7) as

m

→

∞

.

The next Lemma on majorizing sequence is used to prove the convergence of the method (1.5), proof of which is analogous to the proof of Lemma 2.2 in [3].

Lemma: 2.2 Assume there exist nonnegative numbers

R

,

β

,

η

,

and

r

∈

(

0

,

1

)

such that for all

n

≥

0

,

(2.8)

.

)

1

(

₋

_βα

2

₊

_β

2

_R

2

_η

_r

n

_≤

_r

Then the sequence

(

t

_n

),

n

≥

0

, given by

t

₀

=

0

,

t

₁

=

η

,

(2.9)

)

(

)

1

(

2 2 2 ₁

1 −

+

=

n

+

−

+

n

−

n

n

t

R

t

t

t

βα

β

is increasing, bounded above by

r

t

−

=

1

:

* *

η

, and converges to some

t

* such that

r

t

−

≤

<

1

0

*

η

. Moreover, for

0

≥

n

;

(2.10)

,

)

(

0

_{1 1} n

η

n n n

n

t

r

t

t

r

t

−

≤

−

≤

≤

_{+ −}

and

(2.11)

.

1

*

η

r

r

t

t

n n

IJMA- 2(6), June-2011, Page: 824-831

To prove the convergence of the sequence

(

,

)

δ

α

n

x

defined in (1.5) we introduce the following notations.

• Suppose that the Lipschitz condition is satisfied for the operator

F

, namely there exists a constant

R

>

0

such that

(2.12)

,

)

(

)

(

x

F

y

R

x

y

x,y

D(F)

F

−

≤

−

∀

∈

Let

(2.13)

,

1

)

(

:

0

_β

ω

δ

≤

−

=

F

x

y

with

.

)

(

2

2 2

₊

_R

<

α

α

β

The following Lemma based on the Assumption 1.4 will be used in our proofs.

Lemma: 2.3 ([5] Lemma 2.3) For

,

(

₀

)

0

x

B

v

u

∈

_r

−

−

+

′

=

−

′

−

−

1

0

.

)

,

),

(

(

)

(

)

)(

(

)

(

)

(

u

F

v

F

u

u

v

F

u

v

t

u

v

u

u

v

dt

F

φ

Let

(2.14)

)].

(

)

(

[

:

)

(

x

x

F

x

y

x

x

₀

G

=

−

β

−

δ

+

α

−

Note that with the above notation,

G

(

x

_nδ_,α

)

=

x

δ_n+1,α.

Theorem 2.4: Let

t

*

≤

r

₀ and suppose that the equations (2.12) and (2.13) hold. Let the assumptions in Lemma 2.2 are

satisfied. Then the sequence

(

x

δ_n,α

)

defined in (1.5) is well defined and

x

_n,

∈

B

_t*

(

x

₀

)

δ

α for all

n

≥

0

. Further

)

(

x

_nδ_,α is a Cauchy sequence in

B

_t*

(

x

₀

)

and hence converges to

x

∈

B

_t*

(

x

₀

)

⊂

B

_t**

(

x

₀

)

δ

α and

).

(

)

(

x

_αδ

y

δ

α

x

₀

x

_αδ

F

=

+

−

Moreover, the following estimate hold for all

n

≥

0

,

2.15)

(

,

1 ,

,

1 n n n

n

x

t

t

x

δ_{+ α}

−

δ_α

≤

₊

−

and

2.16)

(

.

1

* ,

r

r

t

t

x

x

n n n

−

≤

−

≤

−

_αδ

η

δ α

Proof: Let G be as in (2.14). Then for

u

,

v

∈

D

(

G

)

,

)]

(

)

(

[

)]

(

)

(

[

)

(

)

(

u

G

v

u

v

F

u

y

u

x

₀

F

v

y

v

x

₀

G

−

=

−

−

β

−

δ

+

α

−

+

β

−

δ

+

α

−

)]

(

)

(

[

)

)(

1

(

=

−

αβ

u

−

v

−

β

F

u

−

F

v

Also we have,

2 2

)]

(

)

(

[

)

)(

1

(

)

(

)

(

u

G

v

u

v

F

u

F

v

G

−

=

−

αβ

−

−

β

−

2 2

2 2

)

(

)

(

1

≤

−

αβ

u

−

v

+

β

F

u

−

F

v

2 2

2

2

_]

)

1

(

[

≤

−

αβ

+

R

β

u

−

v

The last equation by (2.12), so we have

2.17)

(

.

)

1

(

)

(

)

(

u

G

v

2

R

2 2

u

v

G

−

≤

−

αβ

+

β

−

Take

)

(

2

2 2

₊

_R

<

α

α

IJMA- 2(6), June-2011, Page: 824-831

On the other hand we shall prove that the sequence

(

t

_n

),

n

≥

0

defined in Lemma 2.2 is a majorizing sequence of the

sequence

(

x

δ_n,α

)

and

x

_n,

∈

B

_t*

(

x

₀

)

δ

α for all

n

≥

0

.

Note that

x

₁δ_,

−

x

₀

=

β

(

F

(

x

₀

)

−

y

δ

)

≤

1

=

η

=

t

₁

−

t

₀

α , assume that

2.18)

(

,

1 , ,

1

x

t

t

i

k

x

_iδ_{+ α}

−

_iδ_α

≤

_i+

−

_i

∀

≤

for some k. Then

x

δ_k+1,α

−

x

₀

≤

x

_kδ_+1,α

−

x

δ_k,α

+

x

_kδ_,α

−

x

δ_k−1,α

+

+

x

₁δ_,α

−

x

₀

≤

t

_k+1

−

t

_k

+

t

_k

−

t

_k−1

+

+

t

₁

−

t

₀

≤

t

_k+1

≤

t

*.

So

x

_i+1,

∈

B

_t*

(

x

₀

)

for

all

i

≤

k

,

and

hence,

by

(2.17)

and

(2.18),

δ α

)

(

)

1

(

)

1

(

2 2 2 _{1, ,} 2 2 2 _{1 2 1}

, 1 ,

2 + + + + +

+

−

k

≤

−

+

k

−

k

≤

−

+

k

−

k

=

k

−

k

k

x

R

x

x

R

t

t

t

t

x

αβ

β

δ

αβ

β

α δ

α δ

α δ

α .

Thus by induction

x

_n+1,

−

x

δ_n,

≤

t

_n+1

−

t

_n

,

α

δ

α for all

n

≥

0

and hence

(

t

n

),

n

≥

0

is a majorizing sequence of the

sequence

(

x

δ_n,α

)

. In particular

x

δ_n,α

−

x

₀

≤

t

_n

≤

t

*

,

i.e.,

x

n_,

∈

B

_t*

(

x

₀

)

δ

α for all

n

≥

0

. So by Lemma 2.1, the

sequence

(

,

)

δ

α n

x

,

n

≥

0

is a Cauchy sequence and converges to some

x

∈

B

_t*

(

x

0

)

⊂

B

_t**

(

x

0

)

δ

α and

.

1

*

,

η

δ α δ

α

r

r

t

t

x

x

n n n

−

≤

−

≤

−

Now by letting n in (1.5) we obtain

F

(

x

_αδ

)

=

y

δ

+

α

(

x

₀

−

x

_αδ

).

3. ERROR BOUNDS UNDER SOURCE CONDITIONS:

To obtain an error estimate for

x

_nδ_,α

−

x

† it is enough to obtain an error estimate for

x

_αδ

−

x

†

.

To obtain an error

estimate for

x

_αδ

−

x

† we use the error estimate for

x

_αδ

−

x

_α and

x

_α

−

x

† where

x

_αis the unique solution of

the equation

F

(

x

)

+

α

(

x

−

x

₀

)

=

y

.

It is known (cf. [9] Proposition 3.1) that

(3.19)

α

δ

α δ

α

−

x

≤

x

and (cf. [5] Theorem 3.1) that

(3.20)

).

(

)

1

(

_{0 0}

†

α

ϕ

ϕ

α

x

k

r

c

x

−

≤

+

Theorem: 3.1 Let

x

δ_α be the unique solution of (1.3) and

x

δ_n,α be as in (1.5). Let the assumptions in Theorem 2.4 and (3.19), (3.20) be satisfied. Then we have the following:

x

_nδ_,α

−

x

†

≤

x

δ_n,α

−

x

_αδ

+

x

_αδ

−

x

_α

+

x

_α

−

x

†

(

1

)

(

).

(3.21)

1

α

00

ϕ

α

δ

η

ϕ

c

r

k

r

r

n

+

+

+

−

≤

Theorem: 3.2 Let

x

_nδ_,α be as in (1.5). Let the assumptions in Theorem 2.4 and Theorem 3.1 be satisfied.

Let

}

1

:

min{

:

α

δ

η

δ

≤

−

=

r

r

n

n

n

. Then we have:

(3.22)

).

)

(

}(

)

1

(

,

2

max{

₀₀

† ,

α

δ

α

ϕ

ϕ δ

α

−

x

≤

k

r

+

c

+

IJMA- 2(6), June-2011, Page: 824-831

3.1. A PRIORI CHOICE OF THE PARAMETER:

The error estimate

α

δ

α

ϕ

(

)

+

in (4.33) is of optimal order if

α =

:

α

_δ satisfies,

ϕ

(

α

_δ

)

α

_δ

=

δ

. Now using the

function

_ψ

(

_λ

)

:

₌

_λϕ

−1

(

_λ

)

,

0

<

λ

≤

a

we have

δ

=

α

_δ

ϕ

(

α

_δ

)

=

ψ

(

ϕ

(

α

_δ

))

, so that

_α

_ϕ

1

(

_ψ

1

(

_δ

))

δ

− −

=

.

Theorem: 3.3 Let

_ψ

(

_λ

)

:

_λϕ

−1

(

_λ

)

=

for

0

<

λ

≤

a

and assumptions in Theorem 3.2 holds. For

δ

>

0

, let

))

(

(

1

1

_ψ

_δ

ϕ

α

_δ

₌

− −

and let

n

_δ

}

1

:

min{

:

α

δ

η

δ

≤

−

=

r

r

n

n

n

then

)).

(

(

1

†

,

ψ

δ

δ α

δ

−

=

−

x

O

x

_n

3.2.AN ADAPTIVE CHOICE OF THE PARAMETER:

Now, we will present a parameter choice rule based on the adaptive method studied in [8].

In practice, the regularization parameter

α

is often selected from some finite set

(3.23)

}

,

,

1

,

0

,

{

:

)

(

₀

i

M

D

_M

α

=

α

_i

=

µ

i

α

=

Where

µ

>

1

, M is big enough but not too large and

α

₀

=

δ

.

Let

(3.24)

}.

1

:

min{

:

M n

M

r

r

n

n

α

δ

η

≤

−

=

Then we have

(3.25)

1

0

,

,

x

i

,

,

, M.

x

i

nM i

−

i

≤

_α

∀

=

δ

δ α δ

α

Let δ _α i M

n

i

x

x

:

=

_, . The parameter choice strategy that we are going to consider in this paper, we select

α

=

α

_i from

)

(

α

M

D

and operate only with corresponding

x

_i,

i

=

0

,

1

,

, M.

Theorem 3.4: Assume that there exists

i

∈

{

0

,

1

,

, M

}

such that

i i

α

δ

α

ϕ

(

)

≤

. Let assumptions of Theorem 3.3

and equations (3.24), (3.25) hold and let

M

i

l

i i

≤

<

=

max{

:

(

)

}

:

α

δ

α

ϕ

,

(3.26)

}.

,

,

1

,

0

,

)

)

1

(

2

4

(

:

max{

:

i

x

x

k

0

r

0

c

j

i

k

j j

i

−

≤

+

+

=

=

α

δ

ϕ

and

k

l

Then

≤

†

ψ

−1

(

δ

),

≤

−

x

c

x

_k

where

c

=

3

(

2

+

(

k

₀

r

₀

+

1

)

c

_ϕ

)

µ

.

Proof: To see that

l

≤

k

, it is enough to show that, for each

i

∈

{

1

,

, M

}

,

.

,

,

1

,

0

,

)

)

1

(

2

4

(

)

(

x

x

k

₀

r

₀

c

j

i

j j

i i

i

≤

−

≤

+

+

∀

=

α

δ

α

δ

α

ϕ

_ϕ

IJMA- 2(6), June-2011, Page: 824-831

.

)

)

1

(

2

4

(

2

)

(

)

1

(

2

)

(

)

1

(

0 0 0 0 0 0 † † j j j i i j i j i

c

r

k

c

r

k

c

r

k

x

x

x

x

x

x

α

δ

α

δ

α

ϕ

α

δ

α

ϕ

ϕ ϕ ϕ

+

+

≤

+

+

+

+

+

≤

−

+

−

≤

−

Thus the relation

l

≤

k

is proved. Next we observe that

.

)

)

1

(

2

(

3

)

)

1

(

2

4

(

2

)

(

)

1

(

0 0 0 0 0 0 † † l l l l k l l k

c

r

k

c

r

k

c

r

k

x

x

x

x

x

x

α

δ

α

δ

α

δ

α

ϕ

ϕ ϕ ϕ

+

+

≤

+

+

+

+

+

≤

−

+

−

≤

−

Now since

α

_δ

≤

α

_l+1

≤

µα

_l, it follows that

µϕ

(

α

)

µψ

1

(

δ

).

α

δ

µ

α

δ

δ δ −

=

=

≤

l

).

(

)

)

1

(

2

(

3

_{0 0} 1

†

_µψ

_δ

ϕ −

+

+

≤

−

x

k

r

c

x

_k

This completes the proof of the theorem.

4. IMPLEMENTATION OF ADAPTIVE CHOICE RULE:

Here we provide an algorithm for the determination of a parameter fulfilling the balancing principle (3.26) and also provide a starting point for the iteration (1.5) approximating the unique solution

x

_αδ of (1.3). The choice of the starting point involves the following steps:

• Choose

0

<

α

₀

<

1

,

0

<

r

<

1

and

µ

>

1

.

• Choose

η

>

0

such that

(

1

−

βα

₀

)

2

+

β

2

R

2

η

≤

r

.

• Choose

x

₀

∈

D

(

F

)

such that ( 0) 1,

β

δ

≤ −y x

F with

.

)

(

2

2 2 0 0

R

+

<

α

α

β

The choice of the stopping index

n

_M involves the following two steps:

• Choose the parameter

α

_M

=

µ

M

α

₀ big enough with

µ

>

1

, not too large.

• Choose

n

_M such that

}.

1

:

min{

:

M n M

r

r

n

n

α

δ

η

≤

−

=

Finally the adaptive algorithm associated with the choice of the parameter specified in Theorem 3.4 involves the following steps:

4.1. ALGORITHM:

• Set

i

←

0

• Solve δ _α i M

n

i

x

x

:

=

_, by using the iteration (1.5).

• If

x

_i

−

x

_j

≥

(

4

+

2

(

k

₀

r

₀

+

1

)

c

)

_j

,

j

≤

i

,

then take

k

=

i

−

1

µ

δ

ϕ .

IJMA- 2(6), June-2011, Page: 824-831

5. CONCLUDING REMARKS:

In this paper we used the adaptive method considered by Pereverzev and Schock in [8] for choosing the regularization parameter

α

in (1.3) and we provided a method for computing the unique solution

x

_αδ of the equation (1.3). Here, an optimal error estimate has been obtained under a general source condition. We also provide a new stopping rule for the iteration index as (3.24). Note that our method always gives the optimal order

(

ψ

−1

(

δ

))

O

under a general source

condition;

x

₀

−

x

†

=

ϕ

(

F

′

(

x

†

))

v

,

v

∈

X

with

v

≤

1

. Here

ϕ

is a monotonically increasing function as in

Assumption 1.3 and

ψ

(

λ

)

:

₌

λϕ

−1

(

λ

)

.

ACKNOWLEDGEMENTS:

I wish to express my sincere gratitude to the anonymous referee for helpful comments and hints. This helped to make the presentation of this paper much clearer.

REFERENCE:

[1] Argyros, I. K., Convergence and applications of Newton-type iterations, (2008) Springer.

[2] Bakushinsky, A. and Smirnova, A., On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal. And Optimiz., 26(1), (2005), 35 - 48.

[3] Elmahdy, A. I., Lavrentiev regularization and majorizing sequence for solving non-linear ill-posed operator equations with monotone operators, (2011) (Communicated).

[4] George, S. and Elmahdy, A. I., An analysis of Lavrentiev regularization for nonlinear ill-posed problems using an iterative regularization method, Int.J.Comput.Appl.Math., 5(3), (2010) ,369 - 381.

[5] George, S. and Elmahdy, A. I., A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations, CMAM, (2011) (To appear).

[6] Jin. Qi-Nian, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems, Mathematics of Computation, 69(232), (2000), 1603 - 1623.

[7] Mahale. P. and Nair. M. T., Iterated Lavrentiev regularization for nonlinear ill-posed problems, ANZIAM Journal, 51, (2009), 191 -217.

[8] Perverzev, S. V. and Schock, E., On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal., 43(5), (2005), 2060 - 2076.

[9] Tautanhahn, U., On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18(1), (2002), 191 - 207.