ITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR

Fixed Poit Theory, 14(2013), No. 1, 39-52 http://www.ath.ubbcluj.ro/ odeacj/sfptcj.htl ITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN,,, Departet of Matheatics ad Coputer Sciece Uiversity of Oradea, Uiversitatii 1, 410087, Oradea, Roaia E-ail: balaj@uoradea.ro E-ail: lcoroiau@uoradea.ro E-ail: galso@uoradea.ro E-ail: suresa@uoradea.ro Abstract. I this paper we study the sequece of successive approxiatios, the fixed poits ad the Ishikawa iterates for the Berstei ax-product operator. Key Words ad Phrases: Berstei ax-product operator, oexpasive operator, sequece of successive approxiatios, fixed poits, Ishikawa iteratios. 2010 Matheatics Subject Classificatio: 41A36, 47H09, 47H10, 47H12. 1. Itroductio For a fuctio f : 0, 1] R + (here x R + eas x 0), the Berstei axproduct approxiatio operator was for the first tie defied (ad forally studied) i 7], pp. 325-326, by the forula (f)(x) = p,k (x)f ( ) k, p,k (x) where p,k (x) = ( ) k x k (1 x) k ad p,k(x) = ax k={0,...,} {p,k (x)}. Notice that the Berstei ax-product operator is obtaied fro the liear Berstei polyoial writte i the for B (f)(x) = p,k(x)f(k/) p,k(x) ad replacig here the su operator by the axiu operator. Surprisigly, with respect to the classical Berstei polyoials, i the whole class of cotiuous fuctios o 0, 1], the ax-product Berstei operators do ot loose the approxiatio properties. Moreover, they preset the advatage that for large classes of fuctios iprove the order of approxiatio to the Jackso-type order. 39

40 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I ore details, it was proved i 3], 4] that B is a oliear (ore exactly subliear o the space of positive fuctios) operator, well-defied for all x R, ad a piecewise ratioal fuctio o R. Also, i 3] it was proved that B possesses soe iterestig direct approxiatio results ad shape preservig properties. For exaple, while i geeral the order of uifor approxiatio was foud to be ω 1 (f; 1/ ) 0,1], however, for soe subclasses of fuctios icludig for exaple the class of cocave fuctios ad also a subclass of the covex fuctios, the order of approxiatio is essetially better, aely is ω 1 (f; 1/) 0,1]. I additio, i 3] it was proved that B (f) is cotiuous for ay positive fuctio f, preserves the ootoicity ad the quasicovexity of f. For strictly positive fuctios, iproved direct approxiatio results by the Berstei ax-product operator we have obtaied i 5]. For the classical Berstei polyoials B (f)(x), i the paper of Rus 11] the wellkow Kelisky Rivli s result i 10] statig that for all f C0, 1], x 0, 1] ad N it holds li B (f)(x) = f(0) + f(1) f(0)]x = B 1 (f)(x) (here B (f) deotes the th iterate of the sequece of successive approxiatios), is proved i a very siple ad elegat aer, by usig the Baach fixed poit theore. Note here that B 1 (f)(x) = f(0) + f(1) f(0)]x is a fixed poit for the operator B. Also, if = depeds o ad if li = 0, the it is kow that (see e.g. 10]) li B (f)(x) = f(x) uiforly i 0, 1]. Siilar studies for the iterates of other kids of Berstei-type operators were obtaied via fixed poit theory i e.g. Agratii 1], Rus 12] ad Agratii-Rus 2]. The ai ai of the preset paper is to ake a siilar study for the iterates of. It is worth otig that due to the fact that B is ot a cotractio (is oly a o-expasive operator), the ethods used i the case of Berstei polyoials caot be used for the Berstei ax-product operators, so that ew ethods are required. The pla of the paper goes as follows. Although the Berstei ax-product operator is ot a cotractio, as a aalogue of the above etioed Kelisky Rivli s results for the Berstei polyoial, i Sectio 2 of the preset paper firstly we prove by a direct ethod that for ay fixed N ad f : 0, 1] 0, + ), the sequece of successive approxiatios of the the Berstei ax-product operator ] (f)(x), still uiforly coverges for to a fixed poit of B. Also, the liits of the double sequece (a (f)), N for other iterdepedeces betwee ad are calculated. I the sae sectio, iportat subsets of the set of fixed poits of the operator B are cocretely deteried. Fially, i Sectio 3 we study the covergece of so-called Ishikawa iterates for the operator B. oliear operator, deoted by a (f)(x) = 2. The sequece of successive approxiatios ad fixed poits for For the proof of the covergece of the sequece of successive approxiatios of, we eed the followig three auxiliary results.

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 41 The first result obtaied oe refers to the fact that ulike the classical Berstei (liear) operator B (f) which is a cotractio, the ax-product Berstei (oliear) operator B (f) is oly a oexpasive operator. This eas that the Baach fixed poit theore caot be applied i this case. Theore 2.1. For ay N, the ax-product Berstei operator B : C + 0, 1] C + 0, 1] is oexpasive, that is B (f) B (g) f g, for all f, g C + 0, 1], where C + 0, 1] = {f : 0, 1] R + ; f is cotiuous o 0, 1]}, R + = {x R; x 0} ad deote the uifor or i C + 0, 1]. Proof. We easily get (f)(x) B (g)(x) p,k(x)f(k/) p,k (x)g(k/) p,k(x) f g, which proves the theore. Rearks. 1) I geeral, the iequality i Theore 2.1 is ot strict, that is there exists f, g C + 0, 1], such that B (f) B (g) = f g. Ideed, let us choose, for exaple, f oicreasig o 0, 1] ad g = 0 o 0, 1]. By Corollary 5.6 i 3], it follows that B (f) is also oicreasig o 0, 1], which iplies that f = f(0), B (f) = B (f)(0) ad by the obvious relatioship B (f)(0) = f(0), it iplies B (f) B (g) = B (f) = f(0) = f = f g. 2) Note that Lea 2.5 i 6] shows that for ay bouded f : 0, 1] R + ad N, B (f) Lip L 1, with L = C 2 f, C > 0 beig a costat idepedet of f ad, where Lip L 1 = {f : 0, 1] R; f(x) f(y) L x y, for all x, y 0, 1]}. I the ext result we obtai a explicit value for C i the above Reark 2. Theore 2.2. For all f C + 0, 1] ad h 0 we have ω 1 (B (f); h) 6πe 2 2 f h, where ω 1 (f; h) = sup{ f(x) f(y) ; x, y 0, 1], x y h} deotes the odulus of cotiuity. Proof. Aalysig the proof of Lea 2.5 i 6], we get ω 1 (B (f); h) 1 2 f h, c 2 1 where it is easy to observe that the costat c 1 > 0 (idepedet of x ad ) coes fro Lea 2.4 i 6] as satisfyig the iequality p,k(x) c1, for all x 0, 1] ad N. Aalysig ow the proof of Lea 2.4 i 6], it easily follows that c 1 = c 2 1 e, where c 2 > 0 is ow the costat that appear i the stateet of Lea 2.3 i 6] as satisfyig { j i p,j ( + 1 ), p,j( j + 1 } + 1 ) c 2, for all N, ad j {0, 1,..., }, where c 2 > 0 is a absolute costat idepedet of ad j.

42 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I cotiuatio, aalysig the proof of Lea 2.3 i 6] ad deotig A = (2!) 2 1 (2)! 2+1, sice li A = π 2 ad because it is easy to prove that (A ) is icreasig, we get 2 π < A <, for all N. 3 2 This iediately iplies (2)! 4 (!) 2 > 2 3π 1, for all N. Therefore, followig the lies i the proof of Lea 2.3 i 6], case (i), we iediately obtai ( ) j p > 1 2 + 1 e 3π 1 2 1 =. 3πe Siilarly, followig the lies i the proof of Lea 2.3 i 6], case (ii), we get ( ) 1 + 1 p,1 = (2 1)! + 1 4 1 ( 1 ) 2 2 1 + 1 2 2 1 + 2 > 3π 1 1 2 = 1 1. 6π Cobiig the cases (i) ad (ii) i the proof of Lea 2.3 i 6], sice 2 3πe > 1 6π, it follows that the costat c 2 i the stateet of Lea 2.3 i 6] ca be chose as c 2 = 1 6π. I coclusio, goig back with the values of the costats, we obtai c 1 = 1 6π 1 e ad 1 = 6πe 2, which fiish the proof. c 2 1 Also, we preset: Lea 2.3. For ay f C + 0, 1] ad N we have B (f)](x) B (f)(x), for all x 0, 1]. Proof. Let us choose arbitrary j {0, 1,..., }. By relatio (4.17) i 3], oe has B (f)(x) = f k,,j (x), x j/( + 1), (j + 1)/( + 1)], (1) where ( k) f k,,j (x) = ( j) ( ) k j x f(k/) 1 x for all k {0, 1,..., }. Relatio (1) iplies B (f)(x) f k,,j (x) for all x j/( + 1), (j + 1)/( + 1)] ad k {0, 1,..., }. I particular, for x = j/ j/( + 1), (j + 1)/( + 1)] ad k = j, we get B (f)(j/) f j,,j (j/) = f(j/), j {0, 1,..., }. Therefore, takig ito accout the relatioship of defiitio for B (f)(x) i Itroductio, we iediately get the stateet of the lea. We are ow i positio to prove the first ai result of this sectio.

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 43 Theore 2.4. For a fixed f C + 0, 1], let us cosider the iterative sequece of successive approxiatios a () (f)(x) = B ] (f)(x),, N, x 0, 1]. Here B ] 2 (f)(x) = B B (f)](x) ad so o. (i) For ay fixed N, there exists f : 0, 1] R +, such that f C + 0, 1], f Lip L 1 with L = 6πe 2 2 f, f (0) = f(0), f (1) = f(1), B B li + a() (f) = f, uiforly i 0, 1], (f )(x) = f (x) for all x 0, 1] (that is f is a fixed poit for the operator ) ad (f)(x) = a () 1 (f)(x) a() (f)(x) a () +1 (f)(x) f (x) f, for all x 0, 1], N; (ii) For all, N ad x 0, 1], we have the estiate ] (f)(x) f(x) 12 ω 1 (f; ), + 1 where ω 1 (f; δ) = sup{ f(x) f(y) ; x y δ}; (iii) For ay fixed N we have li a () (f)(x) = f(x), uiforly i 0, 1]; (iv) Let = depedig o such that li = 0. The we have li a () (f)(x) = f(x), uiforly i 0, 1]; (v) Suppose, i additio, that f Lip L 1 ad that it is strictly positive o 0, 1]. The, for all, N we have the estiate B ] (f) f ( ) L L + 4, f where f = if{f(x); x 0, 1]} > 0; (vi) Suppose that f Lip L 1 ad that it is strictly positive o 0, 1]. Let = depedig o such that li = 0. The uiforly o 0, 1] we have li a () (f)(x) = f(x). (vii) Suppose that f C + 0, 1] is such that for ay N, the fuctio B (f) is a fixed poit for the operator B. The, for ay sequece of atural ubers, ( ) N, the sequece of iterates a () (f) = ] (f) coverges uiforly o 0, 1] to f, as. Proof. (i) Fro the above Lea 2.3, easily follow the iequalities 0 (f)(x) = a () 1 (f)(x)... a() (f)(x) a () +1 (f)(x)... f, for all, N. The last iequality follows fro the obvious iequality 0 B (f)(x) f. Fixig N ad x 0, 1], the sequece of positive ubers (a () (f)(x)) N is bouded ad ootoically odecreasig, which iplies, for +, its covergece to a liit, deote it by f (x). Sice B (f)(x) f, we easily obtai a () (f)(x) f, for all, that is the sequece (a () (f)) N is uiforly bouded. Passig to liit with + we get f (x) f for all x 0, 1], N.

44 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN Also, sice it is easy to check that it is iediate that a () (f)(0) = f(0) ad a () therefore iplies that f (0) = f(0), f (1) = f(1). (f)(0) = f(0) ad B (f)(1) = f(1), (f)(1) = f(1) for all N, which Now, fro B (f) f ad applyig successively Theore 2.2, we easily obtai that a () (f) = B ] (f) Lip L 1, for all N. Therefore, the sequece (of fuctios of successive approxiatio) (a () (f)) N clearly is equicotiuous, which cobied with the fact that the sequece is uiforly bouded, by the Arzela- Ascoli theore iplies that it cotais a subsequece (a () k (f)) k N, uiforly coverget. Because the whole sequece is poitwise coverget to f (x), we get that li k a () k (f) = f uiforly i 0, 1] ad as a cosequece, it iediately follows that f C + 0, 1], i fact oreover, that f Lip L 1 with L = 6πe 2 2 f. Applyig ow the well-kow Dii s theore to the poitwise coverget ootoe sequece of cotiuous fuctios (a () (f)) N, it follows that i fact we have li a () (f) = f uiforly i 0, 1]. Also, the ootoicity of the sequece (a () ) N iplies a () all x 0, 1],, N. Fially, sice a () +1 (f) = B a () (f)] ad li a () 0, 1], takig also ito accout that by Theore 2.1, B fixed it follows that for all N we have (f)(x) f (x) for +1 (f) = f uiforly i is oexpasive, for ay (f ) f B (f ) a () +1 (f) + a() +1 (f) f f a () (f) + a () +1 (f) f. Passig here with, we get B (f ) f = 0, that is B (f )(x) f (x) = 0, for all x 0, 1]. (ii) For ay fixed N ad N variable, it is easy to see that the sequece ] (f)) N satisfies the Corollary 2.4 i 3], that is for all δ > 0 we get B ] (f)(x) f(x) 1 + 1 ] δ B ] (ϕ x )(x) ω 1 (f; δ), x 0, 1], ( where ϕ x (t) = t x, for all t 0, 1]. I what follows we prove by atheatical iductio that B ] (ϕ x )(x) 6 +1, for all, N, x 0, 1], which replaced i the above estiate ad by choosig the δ = 6 +1, will iediately iply ) B ] (f)(x) f(x) 12 ω 1 (f;. + 1 Ideed, deotig ( k) k,,j (x) = ( j) ( ) k j x, 1 x

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 45 by 3], relatioship (4.17), we ca write ( ) k B (f)(x) = k,,j (x)f, for all x j/( + 1), (j + 1)/( + 1)]. This iediately iplies ] 2 (f)(x) = = k,,j (x)b (f)(k/) ] k,,j (x) i,,k (k/)f(i/). i=0 Replacig here f(t) = t x = ϕ x (t) with x fixed, ad takig ito accout the iequality i x i k + k x, for all x j/( + 1), (j + 1)/( + 1)] we get ] B ] 2 (ϕ x )(x) = k,,j (x) i,,k (k/) i x i=0 k,,j (x) i,,k (k/) k i ] i=0 ] + k,,j (x) i,,k (k/) k x i=0 = k,,j (x) i,,k (k/) k i ] i=0 + k,,j (x) k x ] i,,k (k/) 6 1 + 1 + 6 i=0 1 + 1 = 6 2 + 1. For the last estiate we used the iequalities which follow fro the relatioship (4.6) i the proof of Theore 4.1 i 3] k,,j (x) k x 6, i,,k (k/) k + 1 i 6 + 1 ad the iequalities obtaied fro Lea 3.2 i 3] k,,j (x) 1, i,,k (k/) 1. Siilarly, takig ito accout that for all x j/( + 1), (j + 1)/( + 1)] we ca write B ] 3 (f)(x)

46 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN = ]] k,,j (x) i,,k (k/) l,,i (i/)f(l/), i=0 replacig here f(t) = t x = ϕ x (t), takig ito accout the iequality l x l i + i k + k x, ad reasoig exactly as i the case of B ] 2, we easily obtai B ] 3 3 (ϕ x )(x) 6, x j/( + 1), (j + 1)/( + 1)], + 1 valid for all j = 0, 1,...,. Therefore, the above iequality is i fact valid for all x 0, 1]. Reasoig ow by atheatical iductio, we get the desired estiate i the stateet for arbitrary N. (iii) It is iediate by passig to liit with i the iequality fro the above poit (ii). (iv) It is iediate by replacig with i the estiate i (ii), by passig to liit with ad takig ito accout that li +1 = 0. (v) We obviously ca write B ] (f) f B ] j (f) B ] j 1 (f), where by covetio B ] 0 (f)(x) = f(x). But by applyig successively Theore 2.1, we easily get that j=1 B ] j (f) B ] j 1 (f) B ] j 1 (f) B ] j 2 (f) (... B ](f) (f) ω 1 f; 1 ) ] ω1 (f; 1/) + 4, f where for the last estiate above we used Theore 4.6 i 5], valid for strictly positive fuctios oly. Now, takig ito accout that f Lip L 1, fro the above estiate we get l=0 ] j (f) B ] j 1 (f) 1 for all j = 1,...,, which fially iplies B ] (f) f L ( L f + 4 ( )] L L + 4. f )], (vi) It is iediate by takig = ad passig to liit i the estiate fro the above poit (v). (vii) By hypothesis, we have B B (f)] = B (f), for all N, ad therefore it easily follows that B ] (f) = B (f), for all N. Cosequetly, by Theore 4.1 i 3], we obtai B ] (f)(x) f(x) = B (f)(x) f(x) 12 ω 1 (f; 1/ + 1),

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 47 ad passig to liit with, we iediately get the desired coclusio. Rearks. 1) I the class of Lipschitz, strictly positive fuctios, Theore 2.4, (vi), is ore geeral tha Theore 2.4, (iv). Ideed, while li = 0 iplies li = 0, the coverse is ot true. Note that the case of Theore 2.4, (vi), is siilar with what happes i the case of the iterates of Berstei polyoials. 2) As a cosequece of the well-kow Trotter s approxiatio result i the theory of the seigroups of liear operators (see e.g. 9]), it is kow that i the case of Berstei polyoials B (f)(x), if f is twice differetiable ad li = t > 0, the li B (f)(x) = e ta(x), where A(x) = x(1 x)f (x) 2, for all x 0, 1]. It reais as a iterestig ope questio what happes with the iterates ] (f), whe li = t > 0. Let us first observe that by Theore 2.4, (vii), if f satisfies the hypothesis there, the B ] (f) uiforly coverges to f o 0, 1]. It is worth etioig that by the ext Theores 2.5 ad 2.6, we put i evidece large classes of fuctios f satisfyig the hypothesis i Theore 2.4, (vii). Therefore, the above etioed ope proble for the Berstei ax-product operator, gets a sese oly if f does ot satisfy the hypothesis i Theore 2.4, (vii). Also, otice here that the Berstei ax-product operator B ] is ot liear. 3) If f is a fixed poit of B (f)(x) = B, i.e. f(x) = B (f)(x) for all x 0, 1], we easily (f)(x), for all N, x 0, 1], therefore i this case it is get a () trivial i Theore 2.4, (i), that f (x) = B (f)(x), for all x 0, 1]. 4) Accordig to Theore 2.4, (i), for each fixed N it is iportat to deterie the set of the fixed poits for B. I this sese, we preset the followig results. Theore 2.5. (i) If f : 0, 1] 0, ) is odecreasig ad such that the fuctio g : (0, 1] 0, ), g(x) = f(x) x is oicreasig, the for ay N, B (f) is a fixed poit for the operator B, that is B B (f)](x) = B (f)(x), for all x 0, 1]; (ii) If f : 0, 1] 0, ) is oicreasig ad such that the fuctio h : 0, 1) 0, ), h(x) = f(x) 1 x is odecreasig, the for ay N, B (f) is a fixed poit for the operator, that is (f)](x) = (f)(x), for all x 0, 1]. Proof. (i) Fro the relatios (4.46) ad (4.47) i the proof of Corollary 4.7 i 3], for all x j/( + 1), (j + 1)/( + 1)] ad j {0, 1,..., 1} we ca write ad where B (f)(x) = ax{f j,,j (x), f j+1,,j (x)} (f)(x) = f(1), for x /( + 1), 1], ( k) f k,,j (x) = ( j) ( ) k j x f(k/). 1 x Takig above x = j/, by siple calculatio we obtai (f)(j/) = ax{f(j/), f(j + 1)/] j/(j + 1)},

48 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN which by the property of the auxiliary fuctio g i hypothesis, iplies f(j/) f(j + 1)/], which replaced i the above equality gives B (f)(j/) = f(j/). j j+1 But it is clear that if for f C + 0, 1] we have B (f)(j/) = f(j/) for all j {0, 1,..., }, the g = B (f) is a fixed poit for B, which iplies the desired coclusio. (ii) Fro the relatios (4.49) ad (4.50) i the proof of Corollary 4.7 i 3], for all x j/( + 1), (j + 1)/( + 1)] ad j {1,..., } we ca write B (f)(x) = ax{f j 1,,j (x), f j,,j (x)}, ad B (f)(x) = f(0), for x 0, 1/( + 1)]. Takig above x = j/, by siple calculatio we obtai B (f)(j/) = ax{f(j 1)/] ( j)/( j + 1), f(j/)}, which by the property of the auxiliary fuctio g i hypothesis, iplies f(j/) j j+1f(j 1)/], which replaced i the above equality gives B (f)(j/) = f(j/). Therefore, we agai get the desired coclusio. Rearks. 1) Accordig to Reark 4.8 i 3], if f : 0, 1] 0, ) is a covex, odecreasig fuctio satisfyig f(x) x f(1) for all x 0, 1], or if f : 0, 1] 0, ) is a covex, oicreasig fuctio satisfyig f(x) 1 x f(0), the agai f satisfies the hypothesis i Theore 2.5, (i) ad (ii), respectively, ad cosequetly we get (f)](x) = B (f)(x), for all x 0, 1]. 2) Deote by S0, 1] the class of all fuctios f which satisfy the hypothesis i the stateet of Theore 2.5 (i), or of Theore 2.5 (ii), or i the above Reark 1. Also, for ay fixed arbitrary N, let us deote T 0, 1] = B (S0, 1]) = {F C + 0, 1]; f S0, 1] such that F (x) = (f)(x), x 0, 1]}. The if we deote by F 0, 1] = {F : 0, 1] 0, + ); B (F )(x) = F (x), for all x 0, 1]} : C + 0, 1] C + 0, 1], the stateet 0, 1] F 0, 1]. 3) By Lea 4.6 i 3], ay odecreasig cocave fuctio satisfies the hypothesis of Theore 2.5, (i), ad ay oicreasig cocave fuctio satisfies the hypothesis the set of all fixed poits of the operator of Theore 2.5 together with the above Reark 1 eas that we have T of Theore 2.5, (ii). Therefore, the class of all positive, ootoe ad cocave fuctios o 0, 1] deoted by MK + 0, 1], has the property MK + 0, 1] S0, 1], that is the fuctio H = B (f) satisfies B (H)(x) = H(x), for all x 0, 1]. 4) It is easy to cosider cocrete exaples of fuctios i S0, 1] (others tha the costat fuctios which obviously are fixed poits for ), like x, e x, 1 + x 2, si(x), cos(x), l(1 + x), e x, 1 + x 3.

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 49 Ideed, it is easy to check that x, e x ad 1 + x 2 satisfy the first type of hypothesis i the above Reark 1, si(x), cos(x) ad l(1 + x) belog to the class MK + 0, 1] defied i the above Reark 3, while e x satisfy the secod type of hypothesis i the above Reark 1. Therefore, for ay f i this reark we have B B (f)](x) = B (f)(x), for all x 0, 1] ad N. The results expressed by the above Reark 3 ca be geeralized to the whole class of cocave fuctios, as follows. Theore 2.6. If f : 0, 1] 0, ) is a cotiuous cocave fuctio the we have ] (f) = (f) for all N. Proof. By the proof of Corollary 4.6. i 3] we get B (f)(x) = ax{f j 1,,j (x), f j,,j (x), f j+1,,j (x)} for all x j/( + 1), (j + 1)/( + 1)] ad j {1,..., 1}, ad Here recall that (f)(x) = ax{f 0,,0 (x), f 0,,1 (x)} for all x 0, 1/( + 1)] (f)(x) = ax{f,, 1 (x), f,, (x)}, for all x /( + 1), 1]. ( k) f k,,j (x) = ( j) ( ) k j x f(k/). 1 x Sice j/ j/( + 1), (j + 1)/( + 1)], replacig x = j/ i the above forulas for B (f)(x), we easily obtai (see the reasoigs i the proof of Theore 2.5, (i) ad (ii)) that B (f)(j/) = f(j/) for all j {0, 1,..., }, which for the forula of defiitio of B (f)(x) easily iplies the desired coclusio. Rearks. 1) Theores 2.5 ad 2.6 put i evidece large classes of fuctios f C + 0, 1], with the property that B (f) is a fixed poit for the operator B, for all N. The followig exaple of f is that of a fuctio for which there exists N (i fact a ifiity of such of ) such that B (f) is ot ayore fixed poit for the operator B. Ideed, let f : 0, 1] 0, ) be defied by f(x) = 1/2 x if x 0, 1/2] ad f(x) = x 1/2 if x (1/2, 1]. For = 5, by the forula of defiitio of B (f)(x) i Itroductio, we easily get ad 5 (f)(0) = 5 (f)(1) = 1/2, 5 (f)(1/5) = 5 (f)(4/5) = 2/5, 5 (f)(2/5) = 5 (f)(3/5) = 9/40, 5 ( 5 (f))(2/5) = 3/10. Therefore, it follows 5 ( 5 (f))(2/5) 5 (f)(2/5), which clearly iplies that 5 (f) is ot a fixed poit for the operator 5.

50 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I fact, by usig for exaple MATLAB, oe ca easily show that for ay other values of (sufficietly large), agai we get the sae coclusio. 2) Theore 2.6 is also useful to show that the ethod i the case of Berstei polyoials i 11] caot be use here, because for ay a, b R +, the operator B caot be a cotractio o the subspace U a,b = {f C + 0, 1]; f(0) = a, f(1) = b}. I this sese, we ca prove that for ay atural uber, there exist two cotiuous fuctios f, g : 0, 1] 0, ) satisfyig f(0) = g(0) = a, f(1) = g(1) = b ad such that B (f) B (g) = f g. Ideed, let us defie as y = f(x) the equatio of the straight lie passig through the poits (0, a) ad (1, b) ad let g be the fuctio whose graph is the polygoal lie passig through the poits (0, a), (1/2, c) ad (1, b), where the value c ca be ay real uber which satisfies c > f(1/2). (Note that the graphs of both fuctios f ad g for a triagle.) By siple geoetrical reasoigs we get that f g = g(1/2) f(1/2). Firstly, we suppose that is eve. Sice f ad g are cocave fuctios, by the proof of the above Theore 2.6, we get B (f)(j/) = f(j/) ad siilarly, B (g)(j/) = g(j/) for all j {0, 1,..., }. Therefore, takig j() = /2, we obtai that B (f)(1/2) = f(1/2) ad B (g)(1/2) = g(1/2). I coclusio, we have g(1/2) f(1/2) = f g B (f) B (g) B (f)(1/2) B (g)(1/2) = g(1/2) f(1/2), which iplies B (f) B (g) = f g, for ay eve atural uber. The reasoig is siilar i the case whe is ad odd atural uber, because it suffices to replace the pair (1/2, c) i the defiitio of g with ( 0 /(2 0 + 1), c) where = 2 0 + 1. 3. Ishikawa Iteratios for The results i this sectio are based o the followig two well-kow results. Theore 3.1. (Ishikawa 8]) Let C be a copact covex subset of a Baach space (X, ) ad T : C C be oexpasive. For (λ ) N a sequece i 0, b] with b < 1 ad such that =0 λ = +, let us defie the iterates i X by x +1 := (1 λ )x + λ T (x ). The for ay startig poit x 0 C, the sequece (x ) N coverges to a fixed poit of T. Theore 3.2. (Ishikawa 8]) Let C be a closed bouded covex subset of a Baach space (X, ) ad T : C C be oexpasive. Let (λ ) be as i Theore 3.1. The for ay startig poit x 0 C, the followig sequece, ( x T (x ) ) N, coverges to 0 (i.e. (x ) is a so-called approxiate fixed-poit sequece). Now, i order to ca apply to our case the above Theores 3.1 ad 3.2, firstly we eed to idetify bouded closed covex ad copact covex subsets i C + 0, 1]. For

ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 51 exaple, it is easy to check that the subset C + K 0, 1] = {f C +0, 1]; f K}, is bouded, closed ad covex. Also, it is easy to check that the subset C L,K = C + K 0, 1] Lip L 1 is bouded, closed, covex ad equicotiuous, which by the Arzela- Ascoli theore iplies that C L,K is a covex copact subset i C + 0, 1] edowed with the uifor or. Aother iportat hypothesis i the Theores 3.1 ad 3.2 is the ivariace property of T. I our case, we eed this ivariace property for the Berstei ax-product operator. For this purpose, we will ake use of the Theore 2.2 i Sectio 2. We have: Theore 3.3. (i) If f C + K 0, 1] the for all N we have B (f) C + K 0, 1]; (ii) Let K > 0 ad L 6πe 2 K be fixed costats ad deote C L,K = C + K 0, 1] Lip L 1. The, for all N satisfyig the iequality 2 L 6πe 2 K, the ivariace property B (C L,K ) C L,K holds. Proof. (i) Sice 0 f(k/) f for all N ad k = 0, 1,...,, it is iediate by the forula of defiitio of B (f)(x), because we easily get B (f)(x) f, for all x 0, 1], which iplies B f K, for all N. (ii) Let f C L,K. By (i) it follows that B (f) K for all N ad by (i) it follows that B (f) Lip 6πe 2 2 f 1 Lip 6πe2 2 K 1, for all N. The, by 2 we get B (f) Lip 6πe2 2 K 1 Lip L 1, which leads to the coclusio L 6πe 2 K 6πe 2 K. As iediate cosequeces of the above cosideratios, we get the followig two results. Corollary 3.4. Let K > 0 ad L 6πe 2 K be fixed costats ad C L,K = C + K 0, 1] Lip L 1. Also, let (λ ) N be sequece i 0, b] with b < 1 ad such that =0 λ = +. For ay N ad f,1 C L,K fixed, let us defie the iterated that (f) C L,K for satisfyig 2 L sequece of fuctios f,+1 (x) = (1 λ )f, (x) + λ (f, )(x), N, x 0, 1]. The, for ay fixed N satisfyig the iequality 2 L 6πe 2 K, the sequece of fuctios (f, (x)) N coverges as i the uifor or, to a fixed poit of the operator B. Proof. Firstly, it is clear that C + 0, 1] edowed with the uifor or is a Baach space. By Theore 2.1, by the coets betwee the stateets of the Theores 3.2 ad 3.3 ad by Theore 3.3, (ii), the operator B : C L,K C L,K is oexpasive o the copact covex set C L,K. The the corollary is a iediate cosequece of Theore 3.1. Corollary 3.5. Let K > 0 ad C + K 0, 1] = {f C +0, 1]; f K}. Also, let (λ ) ad the iterated sequece (f,+1 (x)) N be defied as i the stateet of Corollary 3.4. The, for ay N ad f,1 C + K 0, 1] fixed, we have li f, (f, ) = 0,

52 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN where deotes the uifor or. Proof. By Theore 2.1, by the coets betwee the stateets of the Theores 3.2 ad 3.3 ad by Theore 3.3, (i), the operator B : C + K 0, 1] C+ K 0, 1] is oexpasive o the bouded, closed ad covex subset C + K 0, 1]. The the corollary is a iediate cosequece of Theore 3.2. Reark. The ethods i this paper ca be exteded to other ax-product operators of Berstei-type. Ackowledgeet. The work of all authors was supported by a grat of the Roaia Natioal Authority for Scietific Research, CNCS UEFISCDI, project uber PN-II-ID-PCE-2011-3-0861. Also, the work of the secod author was supported by the Sectoral Operatioal Prograe for Hua Resources Developet 2007-2013, co-fiaced by the Europea Social Fud, uder the project uber POSDRU/-/107/1.5/S/76841 with the title Moder Doctoral Studies:Iteratioalizatio ad Iterdiscipliarity. Refereces 1] O. Agratii, O soe Berstei type operators: iterates ad geeralizatios, East J. Approx., 9(2003), o. 4, 415-426. 2] O. Agratii, I.A. Rus, Iterates of a class of discrete liear operators via cotractio priciple, Coet. Math. Uiv. Caroli., 44(2003), o. 3, 555-563. 3] B. Bede, L. Coroiau, S.G. Gal, Approxiatio ad shape preservig properties of the Berstei operator of ax-product kid, Iter. J. Math. ad Math. Sci., volue 2009, Article ID 590589, 26 pages, doi:10.1155/2009/590589. 4] B. Bede, S.G. Gal, Approxiatio by oliear Berstei ad Favard-Szász-Mirakja operators of ax-product kid, Joural of Cocrete ad Applicable Matheatics, 8(2010), o. 2, 193-207. 5] L. Coroiau, S.G. Gal, Classes of fuctios with iproved estiates i approxiatio by the ax-product Berstei operator, Aalysis ad Applicatios, 9(2011), o. 3, 1-26. 6] L. Coroiau, S.G. Gal, Global soothess preservatio by soe oliear ax-product operators, Mateaticki Vesik, (i press), http://v.i.sau.ac.rs 7] S.G. Gal, Shape-Preservig Approxiatio by Real ad Coplex Polyoials, Birkhäuser, Bosto-Basel-Berli, 2008. 8] S. Ishikawa, Fixed poits ad iteratios of a oexpasive appig i a Baach space, Proc. Aer. Math. Soc., 59(1976), 65-71. 9] S. Karli, Z. Ziegler, Iteratio of positive approxiatio operators, J. Approx. Theory, 3(1970), 310-339. 10] R.P. Kelisky, T.J. Rivli, Iterates of Berstei polyoials, Pacific J. Math., 21(1967), 511520. 11] I.A. Rus, Iterates of Berstei operators, via cotractio priciple, J. Math. Aal. Appl., 292(2004), 259-261. 12] I.A. Rus, Iterates of Stacu operators (via fixed poit theory priciples) revisited, Fixed Poit Theory, 11(2010), No. 2, 369-374. Received: Jauary 30, 2012; Accepted: March 15, 2012.