Abstract and Applied Analysis

Volume 2013 (2013), Article ID 560258, 9 pages

http://dx.doi.org/10.1155/2013/560258

## Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces

^{1}Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey^{2}Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Received 2 September 2013; Accepted 14 October 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 Vatan Karakaya 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 introduce a new three-step iteration scheme and prove that this new iteration scheme is convergent to fixed points of contractive-like operators. Also, by providing an example, we show that our new iteration method is faster than another iteration method due to Suantai (2005). Furthermore, it is shown that this new iteration method is equivalent to some other iteration methods in the sense of convergence. Finally, it is proved that this new iteration method is -stable.

#### 1. Introduction and Preliminaries

Most of nonlinear equations appearing in physical formulations can similarly be transformed into a fixed point equation of the form . To obtain results on existence and uniqueness of such equations' solution, an approximate fixed point theorem is applied. That is, this application will bring us to the solution of the original equation via help of a particular fixed point iteration method. For this reason, it is crucial to define a new iteration method. To decide whether an iteration method is useful for application, it is of paramount importance to answer the following questions.(i)Does it converge to fixed point of an operator?(ii)Is it faster than the iterations defined in the existing literature?(iii)Is its convergence equivalent to the convergence of the other iteration methods?(iv)Is it -stable? and so forth.

Throughout this paper we examine four essential concepts based on the above questions for a new three-step iteration method when applied to contractive-like mapping.

As a background to our exposition, we now give some information about literature of those concepts.

The first concept of this work is about convergence of fixed point iteration methods. Fixed point iteration methods may exhibit radically different behaviors for various classes of mappings. While a particular fixed point iteration method is convergent for an appropriate class of mappings, it could not be convergent for others. Therefore, it is important to determine whether an iteration method converges to fixed point of a mapping. In this field, there are numerous works regarding convergence of various iterative methods, as one can see in [1–14].

In this work, the second concept is the rate of convergence of iteration methods. After examining convergence of an iteration method, it is important to check whether this iteration method is faster than some well-known iteration methods or not. If it is faster than some current iteration methods, then it could be more useful than the others. More details about the rate of convergence can be found in [10, 15–17].

The third concept for this work is equivalence among convergences of iteration methods. Rhoades and Soltuz [13, 18–20] showed that the convergence of Mann iteration is equivalent to Ishikawa iteration for different classes of operators. They also showed in [21] that the convergence of modified Mann iteration is equivalent to modified Ishikawa iteration under certain mappings. Afterward, Rhoades and Soltuz [14] studied that Mann and Ishikawa iteration sequences are equivalent to a multistep iteration scheme for various classes of the operators. In addition, Soltuz [22] proved that the convergence of Ishikawa iteration is equivalent to that of Mann and Picard iterations for quasicontractive operators. One can find detailed literature concerning this topic in the following list [3–5, 23–25].

The final concept in this work is stability of fixed point iteration methods. The topic of stability, as an application of the theory of fixed point, has been studied by many authors including Harder and Hicks [26, 27], Rhoades [28, 29], Osilike [30, 31], Ostrowski [32], Berinde [33], Olantiwo [9] and Singh and Prasad [34]. First stability result in metric spaces is due to Ostrowski [32], where he established the stability of Picard iteration by employing Banach’s contraction condition. Afterward, several authors studied this concept in different ways.

Throughout this paper, we denote the set of natural numbers by . Let be a normed space, a nonempty convex subset of a normed space , and a self map of . Let , , , , be real sequences satisfying certain conditions. Let be a sequence generated by a particular iteration process including the operator . That is, where is suitable function and is the initial approximation. Suppose that converges to a fixed point of . Let be an arbitrary sequence and set for all . Then, the iteration algorithm (1) is said to be -stable or stable with respect to if and only if implies that . If, in (1), then it is called the Picard iteration process [35].

The Mann iteration procedure given in [7] is defined by The sequence defined by is known as the Ishikawa iteration process [6].

The Noor iteration method [8] is defined by Suantai [11] proposed an iterative scheme as follows: Agarwal et al. established an S-iteration method in [1] as follows: Thianwan [12] introduced a two-step Mann iteration by Recently, Phuengrattana and Suantai [10] defined an SP iteration process as follows: Inspired by the above iteration process, we will introduce the following new iterative algorithm: where , , , , and are real sequences in satisfying Some special cases of the new iteration process given in (11) are as follows.(i)If and for all , then (11) reduces to Picard iteration (3).(ii)If for all , then (11) reduces to Mann iteration (4).(iii)If for all , then (11) reduces to Ishikawa iteration (5).(iv)If and for all , then (11) reduces to S-iteration (8).(v)If for all , then (11) reduces to two-step Mann iteration (9).(vi)If for all , then (11) reduces to SP iteration (10).

Quite recently, Imoru and Olatinwo [36] introduced a class of operators called contractive-like mappings by where and is a monotone increasing function with .

In inequality (13), if we take , then it is reduced to the contractive definition due to Osilike and Udomene [31]. Also, by putting in (13), the class of quasicontractive operators reduces to class of operators due to Berinde [2].

In [2] it was shown that the class of these operators is wider than class of Zamfirescu operators given in [37], where , , and , and are real numbers satisfying , , and . Besides, it is easy to see that special case of Zamfirescu operator gives Kannans' and Chatterjeas' results given in [38] and [39], respectively.

In this paper, we will prove that the new iteration method (11) is convergent to fixed point of contractive-like mappings satisfying (13). Also, by using a counterexample given in [17], we compare the rates of convergence between the new iteration method (11) and the iteration method (7) for the same class of mappings satisfying (13). Moreover, we establish an equivalence among convergences of some iteration methods including the new iteration method (11). Finally, we prove that the new iteration method (11) is -stable.

We end this section with the following definition and lemma which will be useful in proving our main results.

*Definition 1 (see [17]). *Assume that and are two real convergent sequences with limits and , respectively. Then is said to converge faster than if

Lemma 2 (see [33]). *If is a real number satisfying and is a sequence of positive numbers such that , then for any sequence of positive numbers satisfying
**
one has
*

#### 2. Main Results

Theorem 3. *Let be a nonempty closed convex subset of an arbitrary Banach space and be a mapping satisfying (13) with . Let a sequence defined by (11) with real sequences , , , , satisfying , and . Then the iterative sequence converges strongly to the fixed point of .*

* Proof. *Let be the fixed point of . It can be seen easily from (13) that is the unique fixed point of . To show that converges to the fixed point , we use condition (13). Hence, we have
By combining (17)–(19), we derive
Since , (20) becomes
By continuing the above processes, we obtain the following estimates
for all . Since , and , we have
Therefore ; that is, for all .

Theorem 3 allows us to give the following example which compares the rates of convergence between the new iteration method (11) and the iteration method (7) for contractive-like operators. In the following example, for convenience, we use sequences and associated with the iterative methods (11) and (7), respectively.

*Example 4 (see [17]). *Define a mapping as . Let , for , and , , for all .

It can be seen easily that the mapping satisfies condition (13) with the unique fixed point . Furthermore, it is easy to see that Example 4 satisfies all the conditions of Theorem 3.

Indeed, let be initial point for iterative methods (11) and (7). By using iterative methods (11) and (7), we have Now, let us compare these results as follows: Since finally we get Thus, from Definition 1, we conclude that the iteration method (11) is faster than the iteration method (7).

Theorem 5. *Let be a nonempty closed convex subset of an arbitrary Banach space and a mapping satisfying (13) with . If and for all , then the following statements are equivalent.*(i)*Mann iteration (4) converges to fixed point .*(ii)*The new iteration (11) converges to fixed point .*

*Proof. *: Suppose that Mann iteration (4) converges to fixed point ; that is, as . We will show that the new iteration (11) converges to the fixed point ; that is, as . Using (4), (11), and (13), we have
Substituting (32) in (31), we get
By combining (30), (32), and (33) and using the assumption , we have
Denote that
Since and , it follows from (13) that . Hence by Lemma 2 we see that
Also, from triangle inequality we have
and this leads to as .

Now, suppose that as . We will show that as . Using (4), (11), and (13), the following estimates can be obtained:
By substituting (40) in (39), we obtain
Again by substituting (40) and (41) in (38) and using the assumption , we have
Now define
Since as and , it follows from (13) that
Since the function is continuous, we get
Thus Lemma 2 and (42) give as .

Also, from triangle inequality we have
and this yields as .

With regard to ([5], Corollary 2) and Theorem 5, we can without hesitation give the following corollary.

Corollary 6. *Let be a nonempty closed convex subset of an arbitrary Banach space and a mapping satisfying (13) with . If the initial point is the same for all iterations and , for all , then the following expressions are equivalent.*(1)*The Picard iteration (1) converges to the fixed point of .*(2)*The Krasnoselskij iteration [40] converges to the fixed point of .*(3)*The Mann iteration (4) converges to the fixed point of .*(4)*The Ishikawa iteration (5) converges to the fixed point of .*(5)*The Noor iteration (6) converges to the fixed point of .*(6)*The S-iteration (8) converges to the fixed point of .*(7)*The two-step Mann iteration (9) converges to the fixed point of .*(8)*The SP iteration (10) converges to the fixed point of .*(9)*The multistep iteration [14] converges to the fixed point of .*(10)*The new multistep iteration [41] converges to the fixed point of .*(11)*The new iteration (11) converges to the fixed point of .*

Theorem 7. *Let be an arbitrary Banach space, a self-map of satisfying (13) with , and the unique fixed point of . For , let be the new iteration method defined by (11) with real sequences , , , , satisfying , for all . Then the new iteration method (11) is -stable.*

*Proof. *Let be an arbitrary sequence in . Define
for all , where and . Suppose that as and . Then, we prove that . From condition (13), we have the following estimates:
Substituting (49) and (50) in (48) and using the assumption , for all , we obtain
Thus an application of Lemma 2 to (51) yields .

Conversely, assume that . We now show that . From condition (13) and triangle inequality we have
Since and , for all ,
By taking the limit as of both sides of (52) and using the assumption , we have .

#### Conflict of Interests

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

#### References

- R. P. Agarwal, O. O'Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically non expansive mappings,”
*Journal of Nonlinear and Convex Analysis*, vol. 8, no. 1, pp. 6179–6189, 2007. View at Google Scholar - V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,”
*Acta Mathematica Universitatis Comenianae*, vol. 73, no. 1, pp. 119–126, 2004. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. Chugh and V. Kumar, “Strong convergence of SP iterative scheme for Quasi-contractive operators,”
*International Journal of Computer Applications*, vol. 31, no. 5, pp. 21–27, 2011. View at Google Scholar - R. Chugh, V. Kumar, and S. Kumar, “Strong convergence of a new three step iterative scheme in Banach space,”
*American Journal of Computational Mathematics*, vol. 2, pp. 345–357, 2012. View at Publisher · View at Google Scholar - F. Gürsoy, V. Karakaya, and B. E. Rhoades, “The equivalence among new multistep iteration, S-iteration and some other iterative schemes,” http://arxiv.org/pdf/1211.5701.pdf.
- S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at Publisher · View at Google Scholar - W. R. Mann, “Mean value methods in iterations,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar - M. A. Noor, “New approximation schemes for general variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Olantiwo, “Some stability and strong convergence results for the Jungck-Ishikawa iteration process,”
*Creative Mathematics and Informatics*, vol. 17, pp. 33–42, 2008. View at Google Scholar - W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 9, pp. 3006–3014, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Suantai, “Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 506–517, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Thianwan, “Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space,”
*Journal of Computational and Applied Mathematics*, vol. 224, no. 2, pp. 688–695, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. E. Rhoades and S. M. Soltuz, “On the equivalence of Mann and Ishikawa iteration methods,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 7, pp. 451–459, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 58, no. 1-2, pp. 219–228, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - V. Berinde, “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,”
*Fixed Point Theory and Applications*, vol. 2004, no. 2, pp. 97–105, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. Hussain, A. Rafiq, B. Damjanović, and R. Lazović, “On rate of convergence of various iterative schemes,”
*Fixed Point Theory and Applications*, vol. 2011, p. 45, 2011. View at Publisher · View at Google Scholar - Y. Qing and B. E. Rhoades, “Comments on the rate of convergence between mann and ishikawa iterations applied to zamfirescu operators,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 387504, 3 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. E. Rhoades and S. M. Soltuz, “The equivalence of mann iteration and ishikawa iteration for non-lipschitzian operators,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 42, pp. 2645–2651, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. E. Rhoades and S. M. Soltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map,”
*Journal of Mathematical Analysis and Applications*, vol. 283, no. 2, pp. 681–688, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. E. Rhoades and S. M. Soltuz, “The equivalence between the Mann and Ishikawa iterations dealing with generalized contractions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2006, Article ID 54653, 5 pages, 2006. View at Publisher · View at Google Scholar - B. E. Rhoades and S. M. Soltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 289, no. 1, pp. 266–278, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. M. Soltuz, “The equivalence of Picard, Mann, and Ishikawa iterations dealing with quasi- contractive operators,”
*Mathematical Communications*, vol. 10, pp. 81–89, 2005. View at Google Scholar - A. Rafiq, “On the equivalence of Mann and Ishikawa iteration methodes with errors,”
*Mathematical Communications*, vol. 11, pp. 143–152, 2006. View at Google Scholar - S. M. Soltuz, “The equivalence between Krasnoselkij, Mann, Ishikawa, Noor and multistep iteration,”
*Mathematical Communications*, vol. 12, pp. 53–61, 2007. View at Google Scholar - X. Zhiqun, “Remarks of equivalence among Picard, Mann, and Ishikawa Iterations in normed spaces,”
*Fixed Point Theory and Applications*, vol. 2007, Article ID 61434, 5 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. M. Harder and T. L. Hicks, “Stability results for fixed point iteration procedures,”
*Mathematica Japonica*, vol. 33, pp. 693–706, 1988. View at Google Scholar - A. M. Harder and T. L. Hicks, “A stable iteration procedure for nonexpansive mappings,”
*Mathematica Japonica*, vol. 33, pp. 687–692, 1988. View at Google Scholar - B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures,”
*Indian Journal of Pure and Applied Mathematics*, vol. 21, no. 1, pp. 1–9, 1990. View at Google Scholar - B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures. II,”
*Indian Journal of Pure and Applied Mathematics*, vol. 24, no. 11, pp. 691–703, 1993. View at Google Scholar - M. O. Osilike, “Stability results for fixed point iteration procedures,”
*Journal of Nigerian Mathematical Society*, vol. 26, no. 10, pp. 937–945, 1995. View at Google Scholar - M. O. Osilike and A. Udomene, “Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings,”
*Indian Journal of Pure and Applied Mathematics*, vol. 30, no. 12, pp. 1229–1234, 1999. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Ostrowski, “The round-off stability of iterations,”
*Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 47, no. 2, pp. 77–81, 1967. View at Publisher · View at Google Scholar - V. Berinde,
*Iterative Approximation of Fixed Points*, Springer, Berlin, Germany, 2007. - S. L. Singh and B. Prasad, “Some coincidence theorems and stability of iterative procedures,”
*Computers and Mathematics with Applications*, vol. 55, no. 11, pp. 2512–2520, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Picard, “Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives,”
*Journal de Mathématiques Pures et Appliquées*, vol. 6, pp. 145–210, 1890. View at Google Scholar - C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and Mann iteration processes,”
*Carpathian Journal of Mathematics*, vol. 19, no. 2, pp. 155–160, 2003. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Zamfirescu, “Fix point theorems in metric spaces,”
*Archiv der Mathematik*, vol. 23, no. 1, pp. 292–298, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. Kannan, “Some results on fixed points,”
*Bulletin of Calcutta Mathematical Society*, vol. 10, pp. 71–76, 1968. View at Google Scholar - S. K. Chatterjea, “Fixed point theorems,”
*Comptes Rendus de l'Academie Bulgare des Sciences*, vol. 25, pp. 727–730, 1972. View at Google Scholar - H. Schaefer, “Uber die methode sukzessiver approximationen,”
*Jahresbericht-Deutsche Mathematiker-Vereinigung*, vol. 59, pp. 131–140, 1957. View at Google Scholar - F. Gürsoy, V. Karakaya, and B. E. Rhoades, “Data dependence results of new multistep and S-iterative schemes for contractive-like operators,”
*Fixed Point Theory and Applications*, vol. 2013, p. 76, 2013. View at Publisher · View at Google Scholar