D Borwein

MA Noor

Nattawut Pholasa

Prasit Cholamjiak

S Ishikawa

W Phuengrattana

W Suwana-adth

WR Mann

Y Qing

Journal of Inequalities and Applications

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Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214RESEARCH Open AccessApproximating ﬁxed points for continuousfunctions on an arbitrary intervalPrasit Cholamjiak* and Nattawut Pholasa*Correspondence:prasitch2008@yahoo.comSchool of Science, University ofPhayao, Phayao, 56000, ThailandAbstractIn this research article, we introduce a new iterative method for solving a ﬁxed pointproblem of continuous functions on an arbitrary interval. We then prove theconvergence theorem of the proposed algorithm. We ﬁnally give numerical examplesto compare the result with Mann, Ishikawa and Noor iterations. Our main resultsextend the corresponding results in the literature.MSC: 47H09; 47H10Keywords: continuous function; convergence theorem; ﬁxed point; iteration1 IntroductionLet C be a closed interval on the real line and let f : C → C be a continuous function.A point p ∈ C is called a ﬁxed point of f if f (p) = p.One classical way to approximate a ﬁxed point of a nonlinear mapping was introducedin  by Mann [] as follows: a sequence {xn} deﬁned by x ∈ C andxn+ = ( – αn)xn + αnf (xn) (.)for all n ≥ , where {αn} is a sequence in [, ]. Such an iteration process is known asMann iteration. In , Borwein and Borwein [] proved the convergence theorem for acontinuous function on the closed and bounded interval in the real line by using iteration(.).Another classical iteration process was introduced by Ishikawa [] as follows: a sequence{xn} deﬁned by x ∈ C andyn = ( – βn)xn + βnf (xn),xn+ = ( – αn)xn + αnf (yn)(.)for all n≥ , where {αn} and {βn} are sequences in [, ]. Such an iterativemethod is knownas Ishikawa iteration. In , Qing andQihou [] proved the convergence theorem of thesequence generated by iteration (.) for a continuous function on the closed interval inthe real line (see also []).© 2013 Cholamjiak and Pholasa; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com-mons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 2 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214In , Noor [] deﬁned the following iterative scheme by x ∈ C andzn = ( –μn)xn +μnf (xn),yn = ( – βn)xn + βnf (zn),xn+ = ( – αn)xn + αnf (yn)(.)for all n ≥ , where {αn}, {βn} and {μn} are sequences in [, ]. Such an iterative methodis known as Noor iteration. Phuengrattana and Suantai [] considered the convergence ofNoor iteration for continuous functions on an arbitrary interval in the real line.In this paper, motivated by the previous ones, we introduce a new modiﬁed iterationprocess for solving a ﬁxed point problem for continuous functions on an arbitrary intervalin the real line. Numerical examples are also presented to compare the result with Mann,Ishikawa and Noor iterations.2 Main resultsWe begin this section by proving the following crucial lemmas.Lemma. Let C be a closed interval on the real line (can be unbounded) and let f : C → Cbe a continuous function. Let {αn}, {βn}, {μn}, {γn} and {τn} be sequences in [, ] with ≤ τn + βn ≤  and ≤ γn + αn ≤ . Let {xn} be a sequence generated iteratively by x ∈ Candzn = ( –μn)xn +μnf (xn),yn = ( – τn – βn)xn + τnzn + βnf (zn), (.)xn+ = ( – γn – αn)zn + γnyn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.If xn → a, then a is a ﬁxed point of f .Proof Let xn → a and suppose a = f (a). Then {xn} is bounded. So, {f (xn)} is bounded bythe continuity of f . So are {yn}, {zn}, {f (yn)} and {f (zn)}. Moreover, zn → a since xn → aand μn → . We also have yn → a since zn → a and βn → .From (.) we obtainxn+ = ( – γn – αn)zn + γnyn + αnf (yn)= zn + γn(yn – zn) + αn(f (yn) – zn)= ( –μn)xn +μnf (xn) + γn(yn – zn) + αn(f (yn) – zn)= xn +μn(f (xn) – xn)+ γn(yn – zn) + αn(f (yn) – zn)= xn +μn(f (xn) – xn)+ γn(( – τn – βn)(xn – zn) + βn(f (zn) – zn))+ αn(f (yn) – zn)= xn +μn(f (xn) – xn)+ γn(( – τn – βn)μn(xn – f (xn))+ βn(f (zn) – zn))+ αn(f (yn) – zn)Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 3 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214= xn +μn( – γn( – τn – βn))(f (xn) – xn)+ γnβn(f (zn) – zn)+ αn(f (yn) – zn). (.)Let pk = f (xk) – xk , qk = f (zk) – zk and rk = f (yk) – zk . Then we observelimk→∞pk = limn→∞(f (xk) – xk)= f (a) – a = ,limk→∞qk = limn→∞(f (zk) – zk)= f (a) – a = ,limk→∞rk = limn→∞(f (yk) – zk)= f (a) – a = .So, from (.) we obtainxn = x +n∑k=μk( – γk( – τk – βk))(f (xk) – xk)+n∑k=γkβk(f (zk) – zk)+n∑k=αk(f (yk) – zk)= x +n∑k=μk( – γk( – τk – βk))pk +n∑k=γkβkqk +n∑k=αkrk .It is easy to see that∑∞k= μk( – γk( – τk –βk))pk <∞ since limk→∞ pk =  and∑∞k= μk <∞. Similarly, we have∑∞k= γkβkqk <∞ since limk→∞ qk =  and∑∞k= βk <∞. This showsthat {xn} is a divergent sequence since limk→∞ rk =  and∑∞k= αk = ∞. This contradictsthe convergence of {xn}. Hence f (a) = a and a is a ﬁxed point of f . Lemma. Let C be a closed interval on the real line (can be unbounded)and let f : C → Cbe a continuous function. Let {αn}, {βn}, {μn}, {γn} and {τn} be sequences in [, ] with ≤ τn + βn ≤  and ≤ γn + αn ≤ . Let {xn} be a sequence generated iteratively by x ∈ Candzn = ( –μn)xn +μnf (xn),yn = ( – τn – βn)xn + τnzn + βnf (zn),xn+ = ( – γn – αn)zn + γnyn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.If {xn} is bounded, then {xn} is convergent.Proof Suppose {xn} is not convergent. Let a = lim infn xn and b = lim supn xn. Then a < b.We ﬁrst show that if a <m < b, then f (m) =m. Suppose f (m) =m. Without loss of gener-ality, we suppose f (m) –m > . Since f is continuous, there exists δ with  < δ < b– a suchthat for |x –m| ≤ δ,f (x) – x > .Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 4 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214Since {xn} is bounded and f is continuous, {f (xn)} is bounded. Hence {zn}, {yn}, {f (zn)} and{f (yn)} are all bounded. Usingxn+ – xn = ( – γn – αn)(zn – xn) + γn(yn – xn) + αn(f (yn) – xn),yn – xn = τn(zn – xn) + βn(f (zn) – xn),zn – xn = μn(f (xn) – xn),we can easily show that |zn – xn| → , |yn – xn| →  and |xn+ – xn| → . Thus there existsa positive integer N such that for all n >N ,|xn+ – xn| < δ , |yn – xn| <δ , |zn – xn| <δ . (.)Since b = lim supn xn >m, there exists k > N such that xnk >m. Let nk = k, then xk >m.For xk , there exist two cases as follows.(i) xk >m + δ , then xk+ > xk –δ ≥m using (.). So, we have xk+ >m.(ii)m < xk <m+ δ , thenm–δ < yk <m+ δ andm–δ < zk <m+ δ by (.). So, we obtain|xk –m| < δ < δ, |yk –m| < δ, |zk –m| < δ. Hencef (xk) – xk > , f (yk) – yk > , f (zk) – zk > . (.)We observe thatyk – zk = ( – τk – βk)(xk – zk) + βk(f (zk) – zk)= μk( – τk – βk)(xk – f (xk))+ βk(f (zk) – zk). (.)From (.), (.) and (.), we havexk+ = xk +μk( – γk( – τk – βk))(f (xk) – xk)+ γkβk(f (zk) – zk)+ αk(f (yk) – zk)= xk +μk( – γk( – τk – βk))(f (xk) – xk)+ γkβk(f (zk) – zk)+ αk(f (yk) – yk)+ αk(yk – zk)= xk +μk( – γk( – τk – βk))(f (xk) – xk)+ γkβk(f (zk) – zk)+ αk(f (yk) – yk)+ αk(μk( – τk – βk)(xk – f (xk))+ βk(f (zk) – zk))= xk +μk( – (γk + αk)( – τk – βk))(f (xk) – xk)+ αk(f (yk) – yk)+ βk(γk + αk)(f (zk) – zk)> xk .Thus xk+ > xk > m. From (i) and (ii), we have xk+ > m. Similarly, we get that xk+ > m,xk+ >m, . . . . Thus we have xn >m for all n > k = nk . So, a = limk→∞ xnk ≥ m, which is acontradiction with a <m. Thus f (m) =m.We next consider the following two cases.Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 5 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214(i) There exists xM such that a < xM < b. Then f (xM) = xM . It follows thatzM = ( –μM)xM +μMf (xM) = xMandyM = ( – τM – βM)xM + τMzM + βMf (zM)= ( – τM – βM)xM + τMxM + βMf (xM)= xM.Hence, we obtainxM+ = ( – γM – αM)zM + γMyM + αMf (yM)= ( – γM – αM)xM + γMxM + αMf (xM)= xM.Similarly, we obtain xM = xM+ = xM+ = · · · . So, we conclude that xn → xM . Since thereexists xnk → a, xM = a. This shows that xn → a, which is a contradiction.(ii) For all n,xn ≤ a or xn ≥ b. Since b – a >  and limn→∞ |xn+ – xn| = , there exists N¯such that |xn+ – xn| < (b–a) for n > N¯ . So, it is always that xn ≤ a for n > N¯ ; or it is alwaysthat xn ≥ b for n > N¯ . If xn ≤ a for n > N¯ , then b = liml→∞ xnl ≤ a, which is a contradictionwith a < b. If xn ≥ b for n > N¯ , then a = limk→∞ xnk ≥ b, which is a contradictionwith a < b.Thus we conclude that xn → a. This completes the proof. We are now ready to prove the main results of this paper.Theorem . Let C be a closed interval on the real line (can be unbounded) and let f :C → C be a continuous function. Let {αn}, {βn}, {μn}, {γn} and {τn} be sequences in [, ]with  ≤ τn + βn ≤  and  ≤ γn + αn ≤ . Let {xn} be a sequence generated iteratively byx ∈ C andzn = ( –μn)xn +μnf (xn),yn = ( – τn – βn)xn + τnzn + βnf (zn),xn+ = ( – γn – αn)zn + γnyn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.If {xn} is bounded, then {xn} converges to a ﬁxed point of f .Proof Let {xn} be a bounded sequence. Then, by Lemma (.), {xn} is a convergent se-quence. Hence, by Lemma (.), it converges to a ﬁxed point of f . As a direct consequence of Theorem ., we obtain the following result.Theorem . Let C be a closed interval on the real line (can be unbounded) and let f :C → C be a continuous function. Let {αn}, {βn}, {μn}, {γn} and {τn} be sequences in [, ]Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 6 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214with  ≤ τn + βn ≤  and  ≤ γn + αn ≤ . Let {xn} be a sequence generated iteratively byx ∈ C andzn = ( –μn)xn +μnf (xn),yn = ( – τn – βn)xn + τnzn + βnf (zn),xn+ = ( – γn – αn)zn + γnyn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.Then {xn} converges to a ﬁxed point of f if and only if {xn} is bounded.Corollary . Let f : [a,b]→ [a,b] be a continuous function. Let {αn}, {βn}, {μn}, {γn} and{τn} be sequences in [, ] with ≤ τn + βn ≤  and ≤ γn + αn ≤ . Let {xn} be a sequencegenerated iteratively by x ∈ [a,b] andzn = ( –μn)xn +μnf (xn),yn = ( – τn – βn)xn + τnzn + βnf (zn),xn+ = ( – γn – αn)zn + γnyn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.Then {xn} converges to a ﬁxed point of f .If we take τn = γn = , then we obtain the following result.Corollary . Let C be a closed interval on the real line (can be unbounded) and let f :C → C be a continuous function. Let {αn}, {βn} and {μn} be sequences in [, ]. Let {xn} bea sequence generated iteratively by x ∈ C andzn = ( –μn)xn +μnf (xn),yn = ( – βn)xn + βnf (zn),xn+ = ( – αn)zn + αnf (yn), n≥ ,where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.Then {xn} converges to a ﬁxed point of f if and only if {xn} is bounded.If we take τn + βn =  and γn + αn = , then we obtain the following result.Corollary . Let C be a closed interval on the real line (can be unbounded) and let f :C → C be a continuous function. Let {αn}, {βn} and {μn} be sequences in [, ]. Let {xn} bea sequence generated iteratively by x ∈ C andzn = ( –μn)xn +μnf (xn),yn = ( – βn)zn + βnf (zn),xn+ = ( – αn)yn + αnf (yn), n≥ ,Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 7 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214where∑∞n= αn =∞, limn→∞ αn = ,∑∞n= βn <∞ and∑∞n= μn <∞.Then {xn} converges to a ﬁxed point of f if and only if {xn} is bounded.Remark . Corollary . extends the main result obtained in [] from the modiﬁedIshikawa iteration to the modiﬁed Noor iteration.3 Numerical examplesIn this section, we give numerical examples to demonstrate the convergence of the algo-rithm deﬁned in this paper. For convenience, we call the iteration (.) the CP-iteration.Example . Let f : [,∞) −→ [,∞) be deﬁned by f (x) = √. lnx + . Then f is a con-tinuous function. Use the initial point x =  and the control conditions αn = (n+). ,βn = (n+). , μn =(n+). , τn =n+ and γn = .Table 1 Comparison of the convergence rate of Mann, Ishikawa, Noor and CP iterations forthe function given in Example 3.1n Mann Ishikawa Noor CP-iterationxn xn xn xn |f (xn) – xn|1 3.000000 3.000000 3.000000 3.000000 1.58976910 1.074110 1.071538 1.071437 1.043308 0.02440820 1.015478 1.014946 1.014925 1.008428 0.00465830 1.004821 1.004656 1.004650 1.002531 0.00139440 1.001822 1.001760 1.001757 1.000934 0.00051450 1.000777 1.000750 1.000749 1.000392 0.00021660 1.000360 1.000348 1.000348 1.000180 0.00009970 1.000178 1.000172 1.000172 1.000088 0.00004880 1.000093 1.000089 1.000089 1.000045 0.00002590 1.000050 1.000048 1.000048 1.000024 0.000013100 1.000028 1.000027 1.000027 1.000014 0.000007Figure 1 Convergence behavior of Mann, Ishikawa, Noor and CP iterations for the function given inExample 3.1.Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 8 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214Table 2 Comparison of the convergence rate of Mann, Ishikawa, Noor and CP iterations forthe function given in Example 3.2n Mann Ishikawa Noor CP-iterationxn xn xn xn |f (xn) – xn|1 5.000000 5.000000 5.000000 5.000000 2.36393210 1.529828 1.504824 1.501931 1.359649 0.07366820 1.293319 1.284893 1.283943 1.224654 0.02321830 1.223891 1.219794 1.219335 1.187049 0.01103340 1.193599 1.191253 1.190991 1.171120 0.00620450 1.177752 1.176279 1.176115 1.162994 0.00382760 1.168542 1.167561 1.167452 1.158380 0.00250570 1.162807 1.162126 1.162050 1.155568 0.00171080 1.159055 1.158568 1.158513 1.153766 0.00120590 1.156509 1.156152 1.156112 1.152565 0.000870100 1.154730 1.154463 1.154433 1.151740 0.000641110 1.153458 1.153255 1.153233 1.151160 0.000480120 1.152530 1.152374 1.152357 1.150743 0.000365130 1.151842 1.151721 1.151707 1.150438 0.000281140 1.151325 1.151230 1.151219 1.150212 0.000218150 1.150931 1.150856 1.150847 1.150042 0.000171160 1.150628 1.150568 1.150561 1.149912 0.000136170 1.150393 1.150345 1.150339 1.149813 0.000108180 1.150209 1.150169 1.150165 1.149735 0.000087190 1.150063 1.150031 1.150027 1.149675 0.000070200 1.149947 1.149920 1.149918 1.149627 0.000057Figure 2 Convergence behavior of Mann, Ishikawa, Noor and CP iterations for the function given inExample 3.2.Example . Let f : [,∞) −→ [,∞) be deﬁned by f (x) = .√x –  + √x. Then f is acontinuous function. Use the initial point x =  and the control conditions αn = (n+). ,βn = (n+) , μn =(n+). , τn =n+ and γn = .Remark . From Table , Figure , Table  and Figure , we observe that the sequencegenerated by the CP-iteration converges to a ﬁxed point faster than that ofMann, Ishikawaand Noor iterations.Cholamjiak and Pholasa Journal of Inequalities and Applications 2013, 2013:214 Page 9 of 9http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/214Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsPC and NP contributed equally. All authors read and approved the ﬁnal manuscript.AcknowledgementsThe authors wish to thank Professor Suthep Suantai for the valuable guidance and suggestion. The ﬁrst author wassupported by the Thailand Research Fund, the Commission on Higher Education, and University of Phayao under GrantNo. MRG5580016.Received: 22 November 2012 Accepted: 15 April 2013 Published: 29 April 2013References1. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953)2. Borwein, D, Borwein, J: Fixed point iterations for real functions. J. Math. Anal. Appl. 157, 112-126 (1991)3. Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974)4. Qing, Y, Qihou, L: The necessary and suﬃcient condition for the convergence of Ishikawa iteration on an arbitraryinterval. J. Math. Anal. Appl. 323, 1383-1386 (2006)5. Qing, Y, Cho, SY, Qin, X: Convergence of Ishikawa iteration with error terms on an arbitrary interval. Commun. KoreanMath. Soc. 26, 229-235 (2011)6. Noor, MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217-229 (2000)7. Phuengrattana, W, Suantai, S: On the rate of convergence of Mann Ishikawa, Noor and SP iterations for continuousfunctions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006-3014 (2011)8. Suwana-adth, W: The necessary and suﬃcient condition for the convergence of a new ﬁxed point approximationmethod for continuous functions on an arbitrary interval. Thai J. Math. 8, 627-632 (2010)doi:10.1186/1029-242X-2013-214Cite this article as: Cholamjiak and Pholasa: Approximating ﬁxed points for continuous functions on an arbitraryinterval. Journal of Inequalities and Applications 2013 2013:214.

Approximating fixed points for continuous functions on an arbitrary interval

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Approximating fixed points for continuous functions on an arbitrary interval

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