In this paper, we study the convergence of modi�ed multistep iteration

and use the scheme to approximate the �xed point of a general class of functions

introduced by Bosede and Rhoades [5] in a complete metrisable locally convex

space. As corollaries, the convergence results for SP and Mann iterations are also

established. Moreover, most convergence results in Banach spaces are generalized to

complete metrisable locally convex spaces. Our convergence results generalize and

extend the results of Berinde [2], Olaleru [11], Phuengrattana and Suantai [13],

Ra�q [14] among others

Akewe, H

Covenant University Repository

Acta Math. Univ. ComenianaeVol. LXXXIII, 1 (2014), pp. 39–4539MODIFIED MULTISTEP ITERATION FOR APPROXIMATINGA GENERAL CLASS OF FUNCTIONSIN LOCALLY CONVEX SPACESH. AKEWEAbstract. In this paper, we study the convergence of modified multistep iterationand use the scheme to approximate the fixed point of a general class of functionsintroduced by Bosede and Rhoades [5] in a complete metrisable locally convexspace. As corollaries, the convergence results for SP and Mann iterations are alsoestablished. Moreover, most convergence results in Banach spaces are generalized tocomplete metrisable locally convex spaces. Our convergence results generalize andextend the results of Berinde [2], Olaleru [11], Phuengrattana and Suantai [13],Rafiq [14] among others.1. Introduction and Preliminary DefinitionsA locally Convex space (X,u) with topology u is a topological vector space whichhas a local base of convex neighborhoods of zero [20, Chap. 7]. It is metrisableif it is Hausdorff and has countable zero basis. Consequently, it is metrisable ifu can be described by a countable family of continuous seminorms [20]. X isHausdorff if and only if for each non-zero x ∈ X, there is p ∈ Q with p(x) > 0[11]. A seminorm p corresponds to each absolutely convex absorbent subset U ofX is called the gauge of U defined by p(x) = inf{λ : λ > 0, x ∈ λU} and with theproperty that {x : p(x) < 1} ⊆ U ⊆ {x : p(x) ≤ 1}, U is a neighborhood of zero ifand only if p is continuous.Theorem 1.1 ([11]). The topology of a metrizable locally convex space canalways be defined by a metric which is invariant under translation.Proof. For details of proof see Olaleru [11]. Let X be a metrisable topological space and C be a nonempty subset of X andT : C → C a self map of C. For x0 ∈ C, the sequence {xn}∞n=1xn+1 = (1− αn)xn + αnTxn,(1.1)where {αn}∞n=0 is a real sequence in [0,1] such that∑∞n=0 αn = ∞ is called theMann iterative process [9].Received November 29, 2012.2010 Mathematics Subject Classification. Primary 47H10.Key words and phrases. Strong convergence; modified multistep iteration; general class of func-tions; fixed point.40 H. AKEWEOlaleru [11] proved the convergence of Mann iterative process using the Zam-firescu operators [22] and generalized several results in literature to completemetrisable locally convex spaces.For x0 ∈ C, the sequence {xn}∞n=0 defined byxn+1 = (1− αn)xn + αnTynyn = (1− βn)xn + βnTxn,(1.2)where {αn}∞n=0, {βn}∞n=0 are real sequences in [0,1] such that∑∞n=0 αn = ∞ iscalled Ishikawa iterative scheme [7].Observe that if βn = 0 for each n, then the Ishikawa iterative scheme (1.2)reduces to the Mann iterative scheme (1.1).For x0 ∈ C, the sequence {xn}∞n=0 defined byxn+1 = (1− αn)xn + αnTyn,yn = (1− βn)xn + βnTzn,zn = (1− γn)xn + γnTxn(1.3)where {αn}∞n=0, {βn}∞n=0, {γn}∞n=0 are real sequences in [0,1] such that∑∞n=0 αn=∞ is called the Noor iterative (or three-step) scheme [10].Also observe that if γn = 0 for each n, then the Noor iteration process (1.3)reduces to the Ishikawa iterative scheme (1.2).For x0 ∈ C, the sequence {xn}∞n=0 defined byxn+1 = (1− αn)yn + αnTyn,yn = (1− βn)zn + βnTzn,zn = (1− γn)xn + γnTxn(1.4)where {αn}∞n=0, {βn}∞n=0, {γn}∞n=0 are real sequences in [0,1] satisfying βn ≤αn, γn ≤ αn,∑∞n=0 αn = ∞ is called the SP-iterative (or modified three-step)process [13].Also observe that if γn = 0 for each n and zn = xn, then the SP-iterationprocess (1.4) reduces to the modified Ishikawa iterative schemexn+1 = (1− αn)yn + αnTyn,yn = (1− βn)xn + βnTxn.(1.4b)If γn = 0 and βn = 0 for each n, then the SP-iteration process (1.4) reduces tothe Mann iterative process (1.1).In 2011, Phuengrattana and Suantai [13] used the SP-iterative process (1.4) toapproximate the fixed point of continuous functions on an arbitrary interval. Theyalso compared the convergence speed of Mann, Ishikawa, Noor and SP-iterativeprocesses and proved that the SP-iterative process converges faster than the others.For x0 ∈ C. The sequence {xn}∞n=0 defined byxn+1 = (1− αn)xn + αnTy1nyin = (1− βin)xn + βinTyi+1n ,yp−1n = (1− βp−1n )xn + βp−1n Txn, i = 1, 2, . . . , p− 2, p ≥ 2(1.5)MODIFIED MULTISTEP ITERATION FOR APPROXIMATING FUNCTIONS 41where {αn}∞n=0, {βin}∞n=0, i = 1, 2, . . . , p − 1 are real sequences in [0,1] such that∑∞n=0 αn =∞ is called a multistep iteration scheme [22].Observe that the multistep iteration is a generalization of the Noor, Ishikawaand the Mann iterations. In fact, if p = 1 in (1.5), we have the Mann iteration(1.2); if p = 2 in (1.5), we have the Ishikawa iteration (1.3) and if p = 3, we havethe Noor iteration (1.4).Several generalizations of the Banach fixed point theorem have been proved todate (for example, see [2], [15] and [22]). One of the most commonly studiedgeneralization hitherto is the one proved by Zamfirescu [21] in 1972, which isstated as follows.Theorem 1.2 ([22]). Let X be a complete metric space and T : X → X aZamfirescu operator satisfyingd(Tx, Ty)≤ hmax{d(x, y), 12[d(x, Tx) + d(y, Ty)],12[d(x, Ty) + d(y, Tx)]}(1.6)where 0 ≤ h < 1. Then, T has a unique fixed point and the Picard iterationconverges to p for any x0 ∈ X.Observe that in a Banach space setting, condition (1.5) implies‖Tx− Ty‖ ≤ δ‖x− y‖+ 2δ‖x− Tx‖(1.7)where 0 ≤ δ < 1 and δ = max{h, h2−h}, for details of proof see [3].Several papers have been written on the Zamfirescu operators (1.6), (for exam-ple, see [2], [15], [18], [22]). The most commonly used methods of approximatingthe fixed points of the Zamfirescu operators are Picard, Mann [8], Ishikawa [6] andNoor [10] iterative processes. Rhoades [16, 17] used the Zamfirescu contractioncondition (1.7) to obtan some convergence results for Mann and Ishikawa iterativeprocesses in a uniformly Banach space. Berinde [1] extended the results of theauthor [16, 17] to arbitrary Banach space for the same fixed point iteration pro-cedures. Rafiq [15] proved the convergence of Noor iterative process (1.3) usingthe Zamfirescu operators defined by (1.7). Osilike [12] proved several stabilityresults and generalized several results in literature using the following contractivedefinition. For each x, y ∈ X, there exist δ ∈ [0, 1) and L ≥ 0 such thatd(Tx, Ty) ≤ δd(x, y) + Ld(x, Tx).(1.8)In 2003, Imoru and Olatinwo [6] proved some stability results and generalizedsome known results in the literature using the follwing general contractive defini-tion.For each x, y ∈ X, there exist δ ∈ [0, 1) and a monotone increasing functionϕ : R+ → R+ with ϕ(0) = 0 such thatd(Tx, Ty) ≤ δd(x, y) + ϕ(d(x, Tx)).(1.9)In 2010, Bosede and Rhoades [4] observed that the process of “generalizing”(1.8) could continue ad infinitum. As a result of this observation, the authors [4],42 H. AKEWEintroduced the following class of functions and proved the stability of Picard andMann iterative schemes. That is, if x = p (is a fixed point), then (1.8) becomesd(p, Ty) ≤ δd(p, y)(1.10)for some 0 ≤ δ < 1 and all x, y ∈ X.The contractive condition (1.10) is more general than those considered by Imoruand Olatinwo [6], Osilike [12] and several others in the sense that if by replac-ing L in (1.8) with more complicated expressions, the process of “generalizing”(1.8) could continue ad infinitum. Also, the condition “ϕ(0) = 0” usually imposedby Imoru and Olatinwo [6] in the contractive definition (1.9) is no longer neces-sary in the contractive condition (1.10). However, Bosede [3] also proved strongconvergence of the Noor iterative process for this general class of functions.Definition 1.3. Let X be a metrisable topological space and C be a nonemptysubset of X and T : C → C a self map of C. For x0 ∈ C, the iteration proceduredefined by (1.4) such that the generated sequence {xn}∞n=0 converges to a fixedpoint p of T . Let {un}∞n=0 be arbitrary sequence in X and set εn = fc(un+1 −g(T, un)), for n ≥ 0. We say the iteration procedure (1.4) is T -stable if and onlyif limn→∞ εn = 0 implies that limn→∞ un = p.Definition 1.4. Let x0 ∈ C. The sequence {xn}∞n=0 defined byxn+1 = (1− αn)y1n + αnTy1nyin = (1− βin)yi+1n + βinTyi+1n ,yp−1n = (1− βp−1n )xn + βp−1n Txn, i = 1, 2, . . . , p− 2, p ≥ 2(1.11)where {αn}∞n=0, {βin}, i = 1, 2, . . . , p − 1 are real sequences in [0,1] such that∑∞n=0 αn =∞ is called the modified multistep iteration scheme.Motivated by the above results, we introduce the following modified multistepiteration scheme (1.11) and use it to approximate the fixed point of a generalclass of functions introduced by Bosede and Rhoades [4] in a complete metrisablelocally convex space. As corollaries, some strong convergence results are obtainedfor SP and Mann iterative schemes for this general class of funtions (1.10). Ourconvergence results generalize and extend the results of Berinde [2], Bosede andRhoades [4], Olaleru [11], Phuengrattana and Suantai [13] and Rafiq [14] amongothers.We shall need the following Lemma which appeared in [1] to prove our results.Lemma 1.5 ([21]). Let δ be a real number satisfying 0 ≤ δ < 1 and {εn}∞n=0a sequence of positive numbers such that limn→∞ εn = 0. Then for any sequenceof positive numbers {un}∞n=0 satisfying un+1 ≤ δun + εn, n = 0, 1, 2, . . . , we havelimn→∞ un = 0.2. Main ResultTheorem 2.1. Let (X, fc) be a complete metrisable locally convex space, Ka closed convex subset of X and T : K → K be an operator with a fixed point pMODIFIED MULTISTEP ITERATION FOR APPROXIMATING FUNCTIONS 43satisfying the conditionfc(p− Ty) ≤ δfc(p− y)(2.1)for each y ∈ K and 0 ≤ δ < 1. For x0 ∈ K, let {xn}∞n=0 be the modified multi-step iteration scheme defined by (1.11) converging to p (that is, Tp = p), where{αn}∞n=0, {βin}∞n=0 (for each i) are real sequences in [0, 1]. Then the modified mul-tistep iteration scheme (1.11) converges strongly to p.Proof. In view of (1.11) and (2.1), we havefc(xn+1 − p) ≤ fc((1− αn)y1n + αnTy1n − (1− αn + αn)p)≤ (1− αn)fc(y1n − p) + αnfc(Ty1n − p)≤ (1− αn)fc(y1n − p) + δαnfc(y1n − p)= [1− αn(1− δ)]fc(y1n − p).(2.2)An application of (1.11) and (2.1) with y = y1n givesfc(y1n − p) = fc((1− β1n)y2n + β1nTy2n − (1− β1n + β1n)p‖≤ (1− β1n)fc(y2n − p) + β1nfc(Ty2n − p)≤ (1− β1n)fc(y2n − p) + δβ1nfc(y2n − p)= [1− β1n(1− δ)]fc(y2n − p).(2.3)Also an application of (1.11) and (2.1) with y = y2n givesfc(y2n − p) = fc((1− β2n)y3n + β2nTy3n − (1− β2n + β2n)p≤ (1− β2n)fc(y3n − p) + β2nfc(Ty3n − p)≤ (1− β2n)fc(y3n − p) + δβ2nfc(y3n − p)= [1− β2n(1− δ)]fc(y3n − p).(2.4)Similarly, an application of (1.11) and (2.1) with y = y3n givesfc(y3n − p) ≤ (1− β3n)fc(y4n − p) + β3nfc(Ty4n − p)≤ (1− β3n)fc(y4n − p) + δβ3nfc(y4n − p)= (1− β3n)fc(y4n − p) + δβ3nfc(y4n − p)= [1− β3n(1− δ)](y4n − p).(2.5)Continuing the above process, we havefc(xn+1 − p) ≤ [1− αn(1− δ)][1− β1n(1− δ)][1− β2n(1− δ)][1− β3n(1− δ)]. . . [1− βk−2n (1− δ)]fc(yk−1n − p).(2.6)44 H. AKEWEAn application of (1.11) and (2.1) also givesfc(yk−1n − p) ≤ (1− βk−1n )fc(xn − p) + βk−1n fc(Txn − p)≤ (1− βk−1n )fc(xn − p) + δβk−1n fc(xn − p)= (1− βk−1n )fc(xn − p) + δβk−1n fc(xn − p)= [1− βk−1n (1− δ)]fc(xn − p).(2.7)Substituting (2.7) in (2.6), we havefc(xn+1 − p) ≤ [1− αn(1− δ)][1− β1n(1− δ)][1− β2n(1− δ)][1− β3n(1− δ)]. . . [1− βk−2n (1− δ)][1− βk−1n (1− δ)]fc(xn − p)≤ [1− αn(1− δ)]fc(xn − p)≤n∏j=0[1− αj(1− δ)]fc(x0 − p)≤ e−(1−δ)∑∞j=0 αj fc(x0 − p).(2.8)Since 0 ≤ δ < 1, αj ∈ [0, 1) and∑∞n=0 αn =∞, so e−(1−δ)∑∞j=0 αj → 0 as n→∞.Thus limn→∞ fc(xn+1 − p) = 0.Therefore, {xn}∞n=0 converges strongly to p. Theorem 2.1 leads to the following corollaries:Corollary 2.2. Let (X, fc) be a complete metrisable locally convex space, Ka closed convex subset of X and T : K → K be an operator with a fixed point psatisfying the conditionfc(p− Ty) ≤ δfc(p− y)(2.9)for each y ∈ K and 0 ≤ δ < 1. For x0 ∈ K, let {xn}∞n=0 be the SP-iteration schemedefined by (1.4) converging to p (that is, Tp = p), where {αn}∞n=0, {βn}∞n=0,{γn}∞n=0 are real sequences in [0,1]. Then the SP-iteration scheme converges stronglyto p.Corollary 2.3. Let (X, fc) be a complete metrisable locally convex space, Ka closed convex subset of X and T : K → K be an operator with a fixed point psatisfying the conditionfc(p− Ty) ≤ δfc(p− y)(2.10)for each y ∈ K and 0 ≤ δ < 1. For x0 ∈ K, let {xn}∞n=0 be the Mann iterationscheme defined by (1.1) converging to p (that is, Tp = p), where {αn}∞n=0 is a realsequences in [0, 1]. Then the Mann iteration scheme converges strongly to p.Remark 1. Our Theorem 2.1 improves several known results in literature in-cluding the results of Berinde [2] and Rhoades [18, Theorem 2] by extendingthe space to complete metrisable locally convex spaces. The fact that there arecomplete metrisable spaces including many useful function spaces that are notnormable makes our Corollary 2.2 a useful generalization of Berinde’s theorem [2].MODIFIED MULTISTEP ITERATION FOR APPROXIMATING FUNCTIONS 45Acknowledgment. The author is thankful to Prof. J. O. Olaleru for givinguseful comments/suggestions leading to the improvement of this paper.References1. Berinde V., Iterative approximation of fixed points, Baia Mare, Editura Efemeride (2002).2. , A convergence theorem for Mann iteration in the class of Zamfirescu operators,Seria Mathematica Informatica XLV(1), 33–41.3. Bosede A. O., Strong convergence of Noor iteration for a general class of functions, Bulletinof Mathematical Analysis and Applications 3(4) (2011), 140–145.4. Bosede A. O. and Rhoades B. E., Stability of Picard and Mann iteration for a general classof functions, Journal of Advanced Mathematical Studies 3(2) (2010), 1–3.5. Chatterjea S. K., Fixed point theorems, R. Acad. Bulgare Sci. 25(6) (1972), 727–730.6. Imoru C. O. and Olatinwo M. O., On the stability of Picard and Mann iteration, CarpathianJ. Math. 19 (2003), 155–160.7. Ishikawa S., Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974),147–150.8. Kannan R., Some results on fixed points, Bulletin Calcutta Mathematical Society 10 (1968),71–76.9. Mann W. R., Mean value methods in iterations, Proc. Amer. Math. Soc. 44 (1953), 506–510.10. Noor M. A., New approximation schemes for general variational inequalities, Math. Anal.Appl. 251 (2000), 217–229.11. Olaleru J. O., On the convergence of the Mann iteration in locally convex spaces, CarpathianJ. Math. 22(1–2) (2006), 115–120.12. Osilike M. O., Stability results for Ishikawa fixed point iteration procedure, Indian J. PureAppl. Math. 26(10) (1995/96), 937–941.13. Phuengrattana W. and Suantai S., On the rate of convergence of Mann, Ishikawa, Noor andSP-iterations for continuous functions on an arbitrary interval, Comput. Appl. Math. 235(2011), 3006–3014.14. Rafiq A., A convergence theorem for Mann fixed point iteration procedure, Applied Mathe-matics E-Notes, 6 (2006), 289–293.15. , On the convergence of the three-step iteration process in the class of quasi-contractive operators, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 22 (2006), 305–309.16. Rhoades B. E., Fixed point iteration using infinite matrices, Trans. Amer. Math. Soc. 196(1974), 161–176.17. , A comparison of various definition of contractive mapping, Trans. Amer. Math.Soc. 226 (1977), 257–290.18. , Fixed point theorems and stability results for fixed point iteration procedure II,Indian J. Pure Appl. Math. 24(11) (1993), 691–703.19. Rhoades B. E. and Soltuz S. M., The equivalence between Mann-Ishikawa iterations andmulti-step iteration, Nonlinear Analysis, 58(2004), 219–228.20. Schaffer H. H., Topological Vector Spaces, Springer-Verlag, 1999.21. Weng X., Fixed point iteration for local strictly pseudo-contractive mapping, Proc. Amer.Math. Soc. 113(3) (1991), 727–731.22. Zamfirescu T., Fixed point theorems in metric spaces, Arch. Math. (Basel) 23(1972),292–298.H. Akewe, Department of Mathematics, University of Lagos, Akoka, Yaba, Lagos, Nigeria,e-mail : hakewe@unilag.edu.ng, hudsonmolas@yahoo.com

MODIFIED MULTISTEP ITERATION FOR APPROXIMATING

A GENERAL CLASS OF FUNCTIONS

IN LOCALLY CONVEX SPACES

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MODIFIED MULTISTEP ITERATION FOR APPROXIMATING A GENERAL CLASS OF FUNCTIONS IN LOCALLY CONVEX SPACES

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