#### 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.

*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].

#### 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.