Abstract and Applied Analysis

Volume 2014 (2014), Article ID 683295, 6 pages

http://dx.doi.org/10.1155/2014/683295

Research Article

Implicit Approximation Scheme for the Solution of -Positive Definite Operator Equation

Naseer Shahzad,¹ Arif Rafiq,² and Habtu Zegeye³

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

²Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan

³Departement of Mathematics, University of Botswana, Private Bag Box 00704, Gaborone, Botswana

Received 5 December 2013; Accepted 9 February 2014; Published 23 March 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 Naseer Shahzad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct an implicit sequence suitable for the approximation of solutions of K-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).

1. Introduction

Let be a real Banach space and let denote the normalized duality mapping from to defined by

where denotes the dual space of and denotes the generalized duality pairing. It is well known that if is strictly convex, then is single valued. We will denote the single-valued duality mapping by .

Let be a Banach space. The modulus of smoothness of is the function.

defined by

The Banach space is called uniformly smooth if

A Banach space is said to be strictly convex if for two elements which are linearly independent we have that .

Let be a dense subspace of a Banach space . An operator with domain is called continuously -invertible if the range of , , with in considered as an operator restricted to , is dense in and has a bounded inverse on .

Let be a Banach space and let be a linear unbounded operator defined on a dense domain, , in . An operator will be called positive definite (pd) [1] if there exist a continuously -invertible closed linear operator with and a constant such that ,

Without loss of generality, we assume that .

In [1], Chidume and Aneke established the extension of pd operators of Martynjuk [2] and Petryshyn [3, 4] from Hilbert spaces to arbitrary real Banach spaces. They proved the following result.

Theorem 1. Let be a real separable Banach space with a strictly convex dual and let be a pd operator with . Suppose

Then, there exists a constant such that for all

Furthermore, the operator is closed, , and the equation has a unique solution for any given .

As the special case of Theorem 1 in which () spaces, , Chidume and Aneke [1] introduced an iteration process which converges strongly to the unique solution of the equation , where and are commuting. Recently, Chidume and Osilike [5] extended the results of Chidume and Aneke [1] to the more general real separable -uniformly smooth Banach spaces, , by removing the commutativity assumption on and . Later on, Chuanzhi [6] proved convergence theorems for the iterative approximation of the solution of the pd operator equation in more general separable uniformly smooth Banach spaces.

In [7], Osilike and Udomene proved the following result.

Theorem 2. Let be a real separable Banach space with a strictly convex dual and let be a pd operator with . Suppose for all . Choose any and define by

Then the Picard iteration scheme generated from an arbitrary by

converges strongly to the solution of the equation . Moreover, if denotes the solution of the equation , then

The most general iterative formula for approximating solutions of nonlinear equation and fixed point of nonlinear mapping is the Mann iterative method [8] which produces a sequence via the recursive approach , for nonlinear mapping , where the initial guess is chosen arbitrarily. For convergence results of this scheme and related iterative schemes, see, for example, [9–15].

In [16], Xu and Ori introduced the implicit iteration process , which is the modification of Mann, generated by , , for , nonexpansive mappings, and and . They proved the weak convergence of this process to a common fixed point of the finite family of nonexpansive mappings in Hilbert spaces. Since then fixed point problems and solving (or approximating) nonlinear equations based on implicit iterative processes have been considered by many authors (see, e.g., [17–21]).

It is our purpose in this paper to introduce implicit scheme which converges strongly to the solution of the pd operator equation in a separable Banach space. Even though our scheme is implicit, the error estimate obtained indicates that the convergence of the implicit scheme is faster in comparison to the explicit scheme obtained by Osilike and Udomene [7].

2. Main Results

We need the following results.

Lemma 3 (see [10]). If is uniformly convex then there exists a continuous nondecreasing function such that , for all and

for all .

Lemma 4 (see [22]). If there exists a positive integer such that for all , (the set of all positive integers),

then

where , and .

Remark 5 (see [6]). Since is continuously invertible, there exists a constant such that

In the continuation , and are the constants appearing in (4), (6), and (12), respectively. Furthermore, is defined by

With these notations, we now prove our main results.

Theorem 6. Let be a real separable Banach space with a strictly convex dual and let be a pd operator with . Suppose for all . Let denote a solution of the equation . For arbitrary , define the sequence in by

Then, converges strongly to with

where . Thus, the choice yields . Moreover, is unique.

Proof. The existence of the unique solution to the equation comes from Theorem 1. From (4) we have

and from Lemma 1.1 of Kato [23], we obtain that

for all and . Now, from (14), linearity of and the fact that we obtain that

which implies that

so that

With the help of (14) and Theorem 1, we have the following estimate:

which gives

Furthermore, inequality (20) can be rewritten as

In addition, from (17) and (22), we get that

which implies that

where

From (25) and (26), we have that

Hence by Remark 5, we get that

as . Thus, as .

In [6], Chuanzhi provided the following result.

Theorem 7. Let be a real uniformly smooth separable Banach space, and let be a pd operator with . Suppose for all . For arbitrary and , define the sequence by

where is as in , is the constant appearing in inequality (6), is the constant appearing in inequality (4), and

Then, converges strongly to the unique solution of .

However, its implicit version is as follows.

Theorem 8. Let be a real uniformly smooth separable Banach space, and let be a pd operator with . Suppose for all . For arbitrary and , define the sequence by

Then, converges strongly to the unique solution of .

Proof. The existence of the unique solution to the equation comes from Theorem 1. Using (31) and (32) we obtain

Consider

which implies that

Hence, is bounded. Let

Also from (6) it can be easily seen that is also bounded. Let

Denote ; then .

By using (34) and Lemma 3, we have

where

By using (6) and (34) we obtain that

Thus,

Denote

Condition (33) assures the existence of a rank such that , for all . Since is continuous, so (by condition (33)). Now with the help of (33), (42), and Lemma 4, we obtain from (39) that

At last by Remark 5, as ; that is as . Because has bounded inverse, this implies that , the unique solution of . This completes the proof.

Remark 9. According to the estimates (6–8) of Martynjuk [2], we have

where

for or , . Thus,

For , we observe that

Thus, the relation between Martynjuk [2] and our parameter of convergence, that is, between and , respectively, is the following:

Despite the fact that our scheme is implicit, inequality (49) shows that the results of Osilike and Udomene [7] are improved in the sense that our scheme converges faster.

Example 10. Suppose , , , ( is the solution of ); then for the explicit iterative scheme due to Osilike and Udomene [7] we have

which implies that

and hence

Also for the implicit iterative scheme we have that

which implies that

It can be easily seen that for and , (4) and (6) are satisfied. Suppose and ; then , , , , and and so . Take ; then from (52) we have Table 1 and for (54) we get Table 2.

Table 1

1
2
3
4
5
6
7
8

0.0922
0.0851
0.07859
0.07528
0.06947
0.06411
0.05916
0.05459

Table 2

1
2
3
4
5
6
7
8

0.0893
0.0798
0.07136
0.06376
0.05698
0.05092
0.04550
0.04066

Example 11. Let us take , , , ( is the solution of ); then for the explicit iterative scheme due to Osilike and Udomene [7] we have

which implies that

and hence

Also for the implicit iterative scheme we have that

which implies that

It can be easily seen that for and , (4) and (6) are satisfied. Suppose and ; then , , , , and and so . Take ; then from (57) we have Table 3 and for (59) we get Table 4.

Even though our scheme is implicit we observe that it converges strongly to the solution of the pd operator equation with the error estimate which is faster in comparison to the explicit error estimate obtained by Osilike and Udomene [7].

Table 3

1
2
3
4
5
6
7
8

0.0098
0.0096
0.00949
0.00933
0.00917
0.00901
0.00885
0.00870

Table 4

1
2
3
4
5
6
7
8

0.0098
0.0096
0.00941
0.00923
0.00905
0.00887
0.00869
0.00852

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author, therefore, acknowledges his thanks to DSR for the financial support. This paper is dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday.

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