NOTE ON THE ITERATES OF q AND (p, q) BERNSTEIN OPERATORS

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1 Vasile Alecsadri Uiversity of Bacău Faculty of Scieces Scietific Studies ad Research Series Mathematics ad Iformatics Vol ), No. 2, NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS VOICHIŢA ADRIANA RADU Abstract.I this ote we are cocered with the limit behavior of the iterates of q ad p,q)-berstei operators. The covergece of the iterates of q-berstei operators was proved by H. Oruç ad N. Tucer i 2002 ad by X. Xiag, Q. He ad W. Yag i 2007, usig the q-differeces, Stirlig polyomials ad matrix techiques ad by S. Ostrovsa i 2003, with the aids of eigevalues. We try to prove the covergece from some differet poits of view: usig cotractio priciple Wealy Picard Operators Theory). The theory of wealy) Picard operator is very useful to study some properties of the solutios of those equatios for which the method of successive approximatios wors. Wor preseted at Gheorghe Vrăceau Coferece o Mathematics ad Iformatics, December 6, 2016, Vasile Alecsadri Uiversity of Bacău 1. Itroductio I 1912, usig the theory of probability, Berstei defied the polyomials called owadays Berstei polyomials, i eed to proof the famous Weierstrass Approximatio Theorem. Later, it was foud that this polyomials possess may remarable properties. Due to this fact, these polyomials ad their geeralizatios have bee itesively studied. Keywords ad phrases: Iterates, q - Berstei operator, p, q) Berstei operator, cotractio priciples, wealy Picard operators. 2010) Mathematics Subject Classificatio: 41A10, 41A36, 47H10 83

2 84 V.A.RADU I 1987, A. Lupaş [12] itroduced a q-aalogue of the Berstei operators ad i 1997 aother geeralizatio of these operators based o q-itegers was itroduced by G.M. Phillips [17]. For q = 1, these polyomials are just the classical oes, but for q 1, there are cosiderable differeces betwee them. May properties of q-berstei operators were studied by umerous mathematicias, startig with [17]. Covergece properties for iterates of both q-berstei polyomials ad their Boolea sum have bee studied i 2002 by H. Oruç ad N. Tucer [15] ad i 2007 by X. Xiag, Q. He ad W. Yag [21]. The covergece of B f, q; x) as q ad the covergece of the iterates B j f, q; x) as both ad j have bee ivestigated by S. Ostrovsa i 2003, i [16]. Recetly, i 2015, the poitwise covergece ad properties of the q-derivatives of q-berstei operators have bee studied by T. Acar ad A. Aral [2]. The defiitio ad elemetary properties of q-berstei polyomials are give as follows, see, e.g., [12], [3], [13] ad [17]. For ay fixed real umber q > 0, the q-iteger [] q, for N is defied as [] q = { ) 1 q /1 q), q 1, q = 1. [ Set [0] q = 0. The q-factorial [] q! ad q-biomial coefficiets are defied as follows ] q ad [ [] q! = ] = { []q [ 1] q...[1] q, = 1, 2,... 1, = 0 [] q!, for {0, 1,..., }. [] q![ ] q! The q-aalogue of x a) is the polyomial { 1, = 0 x a) q = x a)x qa)...x q 1 a), 1. I 1997, G.M. Phillips [17] defied the q-berstei operators by: ) []q 1) B f, q; x) = b, x; q)f, x [0, 1], [] q =0

3 NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS 85 [ ] where b, x; q) = 1 x q j=0 1 q j x) ad a empty product is equal to 1. For ay fuctio f we defie 0 qf i = f i for i = 0, 1,, 1 ad, recursively, +1 q f i = qf i+1 q qf i for 0, where f i deotes ) [i] f. [m] It is established by iductio that [ ] qf i = 1) r q rr 1)/2 f r i+ r. r=0 The q-berstei polyomial may be writte i the q-differece form [ ] 2) B f, q; x) = qf 0 x. =0 We may deduce from 2) that B f, q; x) reproduces liear polyomials, that is, B ax + b, q; x) = ax + b, a, b R. Also, we ca see that these operators verify for the test fuctio e j x) = x j, j = 0, 1,..., that B e 0, q; x) = 1, B e 1, q; x) = x, B e 2, q; x) = x Prelimiaries x1 x) [] q. Fixed poit theory is a fasciatig subject ad became, i the last decades, ot oly a field with a great developmet, but also a strog tool for solvig differet id of problems from various fields of pure ad applied mathematics. A importat result of the metric fixed poit theory is the Baach-Caccioppoli Cotractio Priciple. Today we have may geeralizatios of this result. Oe of the most importat cocepts of fixed poit theory is the Picard operator. I 1993 i [19], Ioa A. Rus itroduced ad developed these theory of Picard ad wealy Picard operators, which is ow oe of the most strog tools of the Fixed poit theory with several applicatios to operatorial equatios ad iclusios. The theory of wealy) Picard operators is very useful to study some properties of the solutios of those equatios for which the method of successive approximatios wors.

4 86 V.A.RADU We ow that a self mappig T of a oempty set X is Picard operator if T has a uique fixed poit ad the sequece of successive approximatio for ay iitial poit coverges to this uique fixed poit. Therefore, a lot of differet types of cotractios are Picard operators. See for more details the article of V. Beride from 2004, [6]). O the other had, i [19] I.A. Rus itroduced the cocept of wealy Picard operator: a self mappig T of a oempty set X is a wealy Picard operator if the set of fixed poits of T is oempty ad the sequece of successive approximatio for ay iitial poit coverges to a fixed poit of T. There exist may classes of wealy Picard operators i the literature, but oe of the most iterestig of them is the class of almost cotractios. I [6], V. Beride showed that the cocept of almost cotractio is more geeral tha several differet types of cotractios i the literature. Recetly, a iterestig geeralizatio i Kasahara spaces was give by A.D. Filip ad A. Petruşel i [9]. I the followig, we recall some basic features from fixed poit theory, see e.g. [5] ad [19]. Defiitio 2.1. Let X, d) be a metric space. The operator A : X X is a wealy Picard operator WPO) if the sequece of iterates A m x)) m 1 coverges for all x X ad the limit is a fixed poit of A. We deote as usually A 0 = I X, A m+1 = A A m, m N. Let F A := {x X Ax) = x} be the set of fixed poits of A. If the operator A is a WPO ad F A has exactly oe elemet, the A is called a Picard operator PO). Moreover we have the followig characterizatio of the WPOs. Theorem 2.1. Let X, d) be a metric space. The operator A : X X is a wealy Picard operator WPO) if ad oly if there exists a partitio {X λ : λ Λ} such that for every λ Λ oe has i) X λ IA), ii) A Xλ : X λ X λ is a PO, where IA) := { Y X AY ) Y } represets the family of all o-empty subsets ivariat uder A. Further o, if A is a WPO we cosider A X X defied by A x) := lim A m x), x X. m Clearly, we have A X) = F A. Also, if A is WPO, the the idetities FA m = F A, m N, hold true.

5 NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS Iterates of q-berstei operator I the last decades the iterated of positive ad liear operators i various classes were itesively ivestigated. The study of the covergece of iterates of Berstei operators has coectios, for example, to probability theory, spectral theory, matrix theory, Korovi-type theorems, quatitative results about the approximatio of fuctios by positive liear operators, fixed poit theorems ad the theory of semigroups of operators. We emphasize here the importace of the wor of some mathematicias, such as U. Abel, H.H. Gosa, M. Iva ad I. Raşa [1], [10], [18]. The directio of studyig the covergece of iterates of liear operators usig the fixed poit theory was itroduced by O. Agratii ad I.A. Rus [4], [5] ad [20], based o theory of wealy Picard operators developmet by I.A. Rus, i [19]. For Berstei operators, the study of iterative properties was developed, for example, i [7] ad [8]. We wat to exted the study of the iterate of Berstei operators usig q-calculus. Our aim is to study the covergece for iterates of q-berstei operators, usig cotractio priciple wealy Picard operators theory). I case of classical Berstei polyomials, the iterates coverge to liear ed poit iterpolatio o [0, 1]. R.P. Kelisy ad T.J. Rivli see [11]) cosidered this problem both whe M is depedet of the degree of B f ad whe M is depedet o. We have the followig well ow result: Theorem 3.1. Kelisy & Rivli, [11]) If N is fixed, the for all f C[0, 1], lim M BM f; x) = f0) + f1) f0)) x, x C[0, 1]. I [20] I. A. Rus proved the result from above, usig cotractio priciple. The iterates of the q-berstei polyomial are defied by H. Oruç ad N. Tucer i 2002 i [15] as B M+1 f, q; x) = B B M f, q; x); x ) M = 1, 2,..., ad B 1 f, q; x) = B f, q; x). I the same paper, [15] the authors proved the covergece of the iterates of the q-berstei operators, o the geeral case whe q > 0, usig the q-differeces, Stirlig polyomials ad matrix techiques.

6 88 V.A.RADU I 2003, i paper [16], S. Ostrovsa proved also the covergece of the iterates of the q-berstei operators by usig eigevalues. I the followig, we study the covergece, from a differet poit of view, amely, by usig the cotractio priciple. Theorem 3.2. Let B f, q; x) be the q-berstei operator defied i 1), 0 < q < 1. The the iterates sequece ) B M verifies, for all f C[0, 1] ad M 1 all x [0, 1], 3) lim M BM f, q; x) = f0) + f1) f0)) x. Proof. First we defie X α,β := {f C[0, 1] f0) = α, f1) = β}, α, β R. Clearly every X α,β is a closed subset of C[0, 1] ad {X α,β, α, β) R R} is a partitio of this space. It follows directly from the defiitio that q-berstei polyomials possess the ed-poit iterpolatio property [16], i.e. B f, q; 0) = f0), B f, q; 1) = f1). for all q > 0 ot oly for 0 < q < 1) ad all = 1, 2,... These relatios esure that for all α, β) R R ad N, X α,β is a ivariat subset of f B f, q; ). Further o, we prove that the restrictio of f B f, q; ) to X α,β is a cotractio for ay α R ad β R. Let u := mi x [0,1] b,0 x; q) b, x; q)), i.e. u := mi x [0,1] x + 1 x) q ). The 0 < u 1. Let f, g X α,β. From the defiitio of q-berstei operators we ca write 1 ) []q B f, q; x) B g, q; x) = b, x; q)f g) [] q 1 b, x; q) f g [0,1] = 1 b,0 x; q) b, x; q)) f g [0,1] =1 = 1 x 1 x) q ) f g [0,1] 1 u ) f g [0,1] ad fially =1 B f, q; ) B g, q; ) [0,1] 1 u ) f g [0,1]. The restrictio of f B f, q; ) to X α,β is a cotractio. O the other had, p α,β := αe 0 + β α)e 1 X α,β.

7 NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS 89 Sice B e 0, q; x) = e 0 ad B e 1, q; x) = e 1, it follows that p α,β is a fixed poit of B f, q; ). For ay f C[0, 1]) oe has f X f0),f1) ad, by usig the cotractio priciple, we obtai the desired result 3). I terms of WPOs, usig 2), we ca formulate the above theorem as follows Theorem 3.3. The q-berstei operator f B f, q; ) is WPO ad B f, q; x) = f0) + f1) f0)) x. 4. p,q)-berstei operator Durig the last two decades, the applicatios of q-calculus have emerged as a ew area i the field of approximatio theory. Now, a ew extesio of the q-calculus arises. Recetly, M. Mursalee, K.J. Asari ad A. Kha [14] used p, q) calculus i approximatio theory ad defied the p, q)-aalogue of Berstei operators. They estimated uiform covergece of the operators, as well as the rate of covergece, ad obtaied Voroovsaya theorem as well. Let us recall some defiitios ad otatios regardig p, q)-calculus, see, e.g., [14]. Thep, q)-iteger of the umber N is defied as [] p,q := p q ) /p q), 0 < q < p 1. [ ] The p, q)-factorial [] p,q! ad the p, q)-biomial coefficiets are defied as follows ad [ [] p,q! := [] p,q [ 1] p,q...[1] p,q, = 1, 2,... ] p,q := [] p,q!, for {0, 1,..., }. [] p,q![ ] p,q! Now by some simple calculatio ad iductio o, we obtai the p, q)-biomial expasio as follows x + y) p,q := x + y)px + qy)p 2 x + q 2 y)...p 1 x + q 1 y), ax+by) p,q := 1 x) p,q := 1 x)p qx)p 2 q 2 x)...p 1 q 1 x), ) 1) 1) [ p 2 q 2 =0 ] p,q p,q a b x y.

8 90 V.A.RADU Discoverig that the p, q)-berstei operators itroduced i [14] do ot reproduce the test fuctio e 0, i a recet ote corrigedum), M. Mursalee, K.J. Asari ad A. Kha itroduced a revised form of the p, q)-aalogue of Berstei operators as follows: 4) B f, p, q; x) = where b, x; p, q)f =0 [ ] 1 b, x; p, q) = 1) p 2 p,q []p,q [] p,q ), x [0, 1], 1) 1 p 2 x p j q j x). Note that for p = 1, the p, q)-berstei operators give by 4) tur out to be q-berstei operators 1). We have the followig basic result j=0 B e 0, p, q; x) = 1, B e 1, p, q; x) = x, B e 2, p, q; x) = p 1 [] p,q x+ q[ 1] p,q [] p,q x Iterates of p,q)-berstei operator Now we exted the study of the iterate of Berstei operators i the framewor of p, q)-calculus. The iterates of the p, q)-berstei polyomial are defied as B M+1 f, p, q; x) = B B M f, p, q; x); x ) M = 1, 2,..., ad B 1 f, p, q; x) = B f, p, q; x). Usig the wealy Picard operators techique ad the cotractio priciple, we study the covergece of the iterates of the p, q)- Berstei operators. Theorem 5.1. Let B f, p, q; x) be the p, q)-berstei operator defied i 4), where 0 < q < p 1. the give p, q)-berstei operator is a wealy Picard operator ad its iterates sequece ) B M verifies M 1 5) lim M BM f, p, q; x) = f0) + f1) f0)) x.

9 NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS 91 Proof. The proof follows the same steps as i Theorem 3.2. Let X α,β := {f C[0, 1] f0) = α, f1) = β}, α, β R. p, q)-berstei polyomials possess the ed-poit iterpolatio property B f, p, q; 0) = f0), B f, p, q; 1) = f1). The X α,β is a ivariat subset of f B f, p, q; ), for all α, β) R R ad N. We prove that the restrictio of f B f, p, q; ) to X α,β is a cotractio for ay α R ad β R. Let α, β R. Let f, g X α,β. From defiitio of p, q) -Berstei operator we ca write 1 B f, p, q; x) B g, p, q; x) = b, x; p, q)f g) =1 []p,q [] p,q 1 b, x; p, q) f g [0,1] = 1 b,0 x; p, q) b, x; p, q)) f g [0,1]. =1 Let v := mi x [0,1] b,0 x; p, q) + b, x; p, q)), i.e. The 0 < v 1. Therefore, v := mi x [0,1] x + 1 x) p,q). B f, p, q; x) B g, p, q; x) 1 x 1 x) p,q) f g [0,1] 1 v ) f g [0,1] for ay f, g X α,β, that is the restrictio of f B f, p, q; ) to X α,β is a cotractio. O the other had, p α,β := αe 0 + β α)e 1 X α,β, where e 0 x) = 1 ad e 1 x) = x for all x [0, 1]. Sice B e 0, p, q; x) = e 0 ad B e 1, p, q; x) = e 1, it follows that p α,β is a fixed poit of B f, p, q; ). By Theorem 2.1, the p, q) Berstei operator is a WPO ad usig the cotractio priciple, we obtai the claim 5). ) 6. Acowledgemet The author wish to tha the referee for useful commets ad suggestios.

10 92 V.A.RADU Refereces [1] Abel, U., Iva, M., Over-iterates of Berstei operators: a short ad elemetary proof, Amer. Math. Mothly, 116, 2009, [2] Acar, T., Aral, A., O Poitwise Covergece of q-berstei Operators ad Their q-derivatives, Numerical Fuctioal Aalysis ad Optimizatio, 363), 2015, [3] Acu, A., Barbosu, D., Sofoea, D.F., Note o a q-aalogue of Stacu- Katorovich operators, Misolc Mathematical Notes, 161), 2015, [4] Agratii, O., O the iterates of a class of summatio-type liear positive operators, Computers ad mathematics with Applicatios, 55, 2008, [5] Agratii, O., Rus, I.A., Iterates of a class of discrete liear operators via cotractio priciple, Commet. Math. Uiv. Caroli., 443), 2003, [6] Beride, V., Approximatig fixed poits of wea cotractios usig the Picard iteratio, Noliear Aalysis Forum, 91), 2004, [7] Bica, A., Galea, L.F., O Picard Iterative Properties of the Berstei Operators ad a Applicatio to Fuzzy umbers, Commuicatios i mathematical Aalysis, 51), 2008, [8] Cătiaş, T., Otrocol, D., Iterates of Berstei type operators o a square with oe curved side via cotractio priciple, Iteratioal joural o fixed poit theory computatio ad applicatios, 141), 2013, [9] Filip, A.D., Petruşel, A., Fixed poit theorems for operators i geeralized Kasahara spaces, Revista de la Real Academia de Ciecias Exactas, Fisicas y Naturales. Serie A. Mathematicas, 1091), 2015, [10] Gosa, H.H., Raşa, I., The limitig semigroup of the Berstei iterates: degree of covergece, Acta Math. Hugar., 111, 2006, [11] Kelisy, R.P., Rivli, T.J., Iterates of Berstei polyomials, Pacific J. Math., 21, 1967, [12] Lupas, A., A q-aalogue of the Berstei operator, Uiversity of Cluj-Napoca, Semiar o Numerical ad Statistical Calculus, Preprit 9, 1987, [13] Muraru, C.V., Note o the q-stacu-schurer Operator, Misolc Mathematical Notes, 141), 2013, [14] Mursalee, M., Asari, K.J., Kha, A., O p,q)-aalogue of Berstei operators, Appl. Math. Comput., 266, 1 september 2015, [15] Oruç, H., Tucer, N., O the Covergece ad Iterates of q-berstei Polyomials, Joural of Approximatio Theory, 117, 2002, [16] Ostrovsa, S., q-berstei polyomials ad their iterates, Joural of Approximatio Theory, 123, 2003, [17] Phillips, G.M., Berstei Polyomials Based o the q-itegers, A. Numer Math., 4, 1997, [18] Raşa, I., C 0 semigroups ad iterates of positive liear operators: asymptotic behaviour, Red. Circ. Mat. Palermo, 2 Suppl. 82, 2010, [19] Rus, I.A., Wealy Picard mappigs, Commet. Math. Uiv. Caroli., 344), 1993, [20] Rus, I.A., Iterates of Berstei operators, via cotractio priciple, J. Math. Aal. Appl., 292, 2004,

11 NOTE ON THE ITERATES OF q AND p, q) BERNSTEIN OPERATORS 93 [21] Xiag, X., He, Q., Yag, W., Covergece Rate for Iterates of q-berstei Polyomials, Aalysis i Theory ad Applicatios, 233), 2007, Departmet of Statistics-Forecasts-Mathematics Babeş-Bolyai Uiversity, FSEGA, Cluj-Napoca, Romaia voichita.radu@eco.ubbcluj.ro, voichita.radu@gmail.com