- Open Access
A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces
© Khan et al.; licensee Springer 2012
- Received: 30 June 2012
- Accepted: 29 August 2012
- Published: 12 September 2012
In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.
- three-step iterative scheme
- ϕ-strongly accretive operator
- ϕ-hemicontractive operator
for all . If , then T is called nonexpansive, and if , T is called contraction.
- (i)T is said to be strongly pseudocontractive if there exists a such that for each , there exists satisfying
- (ii)T is said to be strictly hemicontractive if is nonempty and if there exists a such that for each and , there exists satisfying
- (iii)T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function with such that for each , there exists satisfying
- (iv)T is said to be ϕ-hemicontractive if is nonempty and if there exists a strictly increasing function with such that for each and , there exists satisfying
Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.
- (i)T is called accretive if the inequality
- (ii)T is called strongly accretive if, for all , there exists a constant and such that
- (iii)T is called ϕ-strongly accretive if there exists and a strictly increasing function with such that for each ,
Remark 4 It has been shown in [11, 12] that the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators. If I denotes the identity operator, then T is called strongly pseudocontractive (respectively, ϕ-strongly pseudocontractive) if and only if is strongly accretive (respectively, ϕ-strongly accretive).
Chidume  showed that the Mann iterative method can be used to approximate fixed points of Lipschitz strongly pseudocontractive operators in (or ) spaces for . Chidume and Osilike  proved that each strongly pseudocontractive operator with a fixed point is strictly hemicontractive, but the converse does not hold in general. They also proved that the class of strongly pseudocontractive operators is a proper subclass of the class of ϕ-strongly pseudocontractive operators and pointed out that the class of ϕ-strongly pseudocontractive operators with a fixed point is a proper subclass of the class of ϕ-hemicontractive operators. These classes of nonlinear operators have been studied by various researchers (see, for example, [7–25]). Liu et al.  proved that, under certain conditions, a three-step iteration scheme with error terms converges strongly to the unique fixed point of ϕ-hemicontractive mappings.
In this paper, we study a three-step iterative scheme with error terms for nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.
We need the following results.
Lemma 5 
then the limit exists.
Lemma 6 
Let . Then for every if and only if there is such that .
Lemma 7 
Suppose that X is an arbitrary Banach space and is a continuous ϕ-strongly accretive operator. Then the equation has a unique solution for any .
3 Strong convergence of a three-step iterative scheme to a solution of the system of nonlinear operator equations
For the rest of this section, L denotes the Lipschitz constant of , and , and denote the ranges of , and respectively. We now prove our main results.
where , and are bounded sequences in X and , , , , , , , are sequences in and in satisfying the following conditions: (i) , (ii) , (iii) , , (iv) , and . Then the sequence converges strongly to the solution of the system ; .
for all and for all ; .
yields , contradicting the fact that . Hence, . □
Corollary 9 Let X be an arbitrary real Banach space and be three Lipschitz ϕ-strongly accretive operators, where ϕ is in addition continuous. Suppose or as ; . Let , , , , , , , , , , , , and be as in Theorem 8. Then, for any given , the sequence converges strongly to the solution of the system ; .
Proof The existence of a unique solution to the system ; follows from  and the result follows from Theorem 8. □
Then converges strongly to the common fixed point of , , .
The rest of the argument now follows as in the proof of Theorem 8. □
Remark 11 The example in  shows that the class of ϕ-strongly pseudocontractive operators with nonempty fixed point sets is a proper subclass of the class of ϕ-hemicontractive operators. It is easy to see that Theorem 8 easily extends to the class of ϕ-hemicontractive operators.
Gurudwan and Sharma  studied a strong convergence of multi-step iterative scheme to a common solution for a finite family of ϕ-strongly accretive operator equations in a reflexive Banach space with weakly continuous duality mapping. Some remarks on their work can be seen in .
All the above results can be extended to a finite family of ϕ-strongly accretive operators.
The last author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.
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