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J. Nonlinear Sci. Appl., 14 (2021), 310–323 ISSN: 2008-1898 Journal Homepage: www.isr-publications.com/jnsa Approximation by a new generalization of Szász-Mirakjan operators via (p, q)-calculus Reşat Aslana,∗, Aydin Izgib a Provincial Directorate of Labor and Employment Agency, 63050, Şanliurfa, Turkey. b Department of Mathematics, Faculty of Sciences and Arts, Harran University, 63100, Şanliurfa, Turkey. Abstract In this work, we obtain the approximation properties of a new generalization of Szász-Mirakjan operators based on postquantum calculus. Firstly, for these operators, a recurrence formulation for the moments is obtained, and up to the fourth degree, the central moments are examined. Then, a local approximation result is attained. Furthermore, the degree of approximation in respect of the modulus of continuity on a finite closed set and the class of Lipschitz are computed. Next, the weighted uniform approximation on an unbounded interval is showed, and by the modulus of continuity, the order of convergence is estimated. Lastly, we proved the Voronovskaya type theorem and gave some illustrations to compare the related operators’ convergence to a certain function. Keywords: Weighted approximation, Szász-Mirakjan operators, modulus of continuity, (p, q)-calculus. 2020 MSC: 41A25, 41A35, 41A36. c 2021 All rights reserved. 1. Introduction In [23, 36], the Szász-Mirakjan operators on [0, ∞), which are related to the Poisson distribution, were defined as ∞   X k (nx)k −nx Sn (f; x) = e . (1.1) f n k! k=0 In recent years, many modifications and generalizations of the operators (1.1) were considered by some authors. Aral et al. [6] introduced the new generalization of Szász-Mirakyan operators. Some approximation results for the operators of Szász-Mirakjan-Durrmeyer type are obtained by Krech [19]. Çekim et al. [9] considered the Dunkl generalization of Szász beta-type operators. A modification of the Szász-Mirakjan-Kantorovich operators which preserving linear functions introduced by Duman et al. [12]. In [32], a new generalization of Szász-Mirakjan operators on a closed subintervals of [0, ∞) were defined by Ousman and Izgi as follows: Nn (f; x) = e−nx ∞ X f( k=0 k n + a (nx)k ) , n n + b k! ∗ Corresponding 0 6 x < ∞, author Email addresses: resat63@hotmail.com (Reşat Aslan), aydinizgi@yahoo.com (Aydin Izgi) doi: 10.22436/jnsa.014.05.02 Received: 2020-12-10 Revised: 2020-12-22 Accepted: 2020-12-27 (1.2) R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 311 where a, b ∈ N and 0 6 a 6 b. They estimated for the operators (1.2) the rate of convergence and proved the Voronovskaya type result theorem and also obtained the order of approximation of functions on the class of Lipschitz. We refer to the readers some similar type operators in [7, 16, 37]. In the last three decades, the quantum calculus, briefly q-calculus, which has a lot of application fields in mathematics, has played a serious role on the approximation theory. The first experimentation via the implementation of q-calculus to linear positive operators is done by Lupaş [20]. He investigated the qBernstein polynomials and examined approximation and shape-preserving properties. Later, Phillips [33] achieved several convergence results and proved the Voronovskaya-type results for the generalizations of the q-Bernstein operators. Next [20, 33] investigations, the implementing of q-calculus on the approximation theory become very popular and motivated many authors to introduce this technic to some various famous operators. In [31], Örkcü and Doğru introduced the weighted statistical approximation by kantorovich type q-Szász-Mirakjan operators. Ahasan and Mursaleen [2] obtained some approximation results of the generalized Szász-Mirakjan type operators via q-calculus. Mahmudov and Gupta [22] proposed a q-analogue of Szász Kantorovich operators. Mursaleen and Rahman [30] studied a Dunkl generalization of q-Szász-Mirakjan operators which preserve x2 . Also, we refer to [5, 8, 21]. Recently, Mursaleen et al. [28] by using of the post-quantum calculus briefly, (p, q)-calculus, added a new process to approximation theory. Moreover, by association to (p, q)-calculus the Bernstein-Schurer type operators [26], the Kantorovich variant of Szász-Mirakjan operators [27], the king type Szász-MirakjanKantorovich operators [29] and the Szász-Mirakjan [25] type operators were introduced and examined in detail. Also, Alotaibi et al. [3] obtained some approximation results of a Dunkl type generalization of Szász operators via (p, q)-calculus. Karahan and Izgi [18] investigated some approximation results of the (p, q)-Bernstein operators. Acar [1] defined a new modified operators of Szász-Mirakjan based on (p, q)integers. The Chlodowsky variant of Szász-Mirakjan-Stancu and the Szász-Mirakjan-Baskakov-Stancu type operators on the concept of (p, q)-integers were presented by [24]. The Stancu form of operators of (p, q)-Szász-Mirakjan are explored by [10]. In this research, for the operators given by (1.2), motivated by the many studies given above, we construct a new generalization of the Szász-Mirakjan operators by using of (p, q)-calculus and investigate the approximation attributes for these operators. Now, we recollect some basic notations and definitions about the (p, q)-calculus. Suppose that 0 < q < p 6 1. For all integers n,l such that n > l > 0, the [n]p,q is given as: pn − qn [n]p,q := . p−q The factorial and binomial factors related to the (p, q)-integers are given respectively as: [n]p,q ! := [n]p,q [n − 1]p,q · · · 1, n > 1, 1, n = 0, and  n l  := p,q [n]p,q ! [l]p,q ! [n − l]p,q ! The (p, q)-binomial expansion is gives as: (au + bv)n p,q := n X n−l l p( 2 ) q(2) an−l bl un−l vl , l=0 and 2 2 n−1 (u − v)n u − qn−1 v). p,q = (u − v)(pu − qv)(p u − q v) · · · (p Moreover, the two analogue of (p, q)-exponential function are as follows: ep,q (u) := ∞ X p l=0 l(l−1) 2 ul [l]p,q ! and Ep,q (u) := ∞ X q l=0 l(l−1) 2 ul , [l]p,q ! . R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 312 which verify ep,q (u)Ep,q (−u) = 1. Since p = 1, Ep,q (u) and ep,q (u) turn into functions of q-exponential. More detailed information about the (p, q)-calculus can be found [14, 17, 34, 35]. Now, by utilizing of (p, q)-calculus we will construct our operators as follows: ! ∞ k X [k]p,q [n + a]p,q k(k−1) ([n]p,q x) p,q 2 (1.3) q Rn,a,b (f; x) = Ep,q (− [n]p,q x) f [k]p,q ! qk−2 [n]p,q [n + b]p,q k=0 where a, b, n ∈ N, 0 6 a 6 b, 0 < q < p 6 1, x ∈ [0, ∞). It is clear to see Ep,q (− [n]p,q x) ∞ P q k(k−1) 2 k=0 ([n]p,q x)k [k]p,q ! = 1. The operators given by (1.3) are positive and linear. In special case, for a = b the operators (1.3) reduce to operators given by [1] and also for p = 1 and a = b the operators (1.3) reduce to operators given by [21]. 2. Main results Lemma 2.1. Let Rp,q n,a,b (f; x) operators are given by (1.3). Then, we attain the following relation: s+1 Rp,q ; x) n,a,b (t = s   X s u=0 u xpu [n + a]s+1−u p,q s+1−u q2u−s−1 [n]s−u p,q [n + b]p,q u Rp,q n,a,b (t ; x). Proof. Using the equation below [n]p,q = qn−1 + p [n − 1]p,q , then, we may write p,q Rn,a,b (ts+1 ; x) = Ep,q (− [n]p,q x) ∞ X s+1 [k]s+1 p,q [n + a]p,q q (k−2)(s+1) [n]s+1 [n + b]s+1 p,q p,q k=0 q = Ep,q (− [n]p,q x) ∞ X [k]sp,q [n + a]s+1 p,q q s+1 k(k−1) 2 k(k−1) −k+2 2 (k−2)s [n]s [n + b] p,q p,q k=1 q s  k−1 ∞ p [k − 1] [n + a]s+1 X p,q p,q + q = Ep,q (− [n]p,q x) q k=1 s  ∞ X X = Ep,q (− [n]p,q x) × [n + a]s+1 p,q s+1 [n + b]p,q ([n]p,q x)k (k−2)s [n]s [n + b]s+1 p,q p,q q [k]p,q ! ([n]p,q x)k−1 x [k − 1]p,q ! k(k−1) −k+2 2 ([n]p,q x)k−1 x [k − 1]p,q ! (k−1)(k−2)  +1 2 s (k−1)(s−u) u q q p [k − 1]u ! p,q (k−2)s u [n]sp,q q k=1u=0 ([n]p,q x)k−1 x [k − 1]p,q ! s+1−u u ∞ s   X X [k − 1]u xpu [n + a]p,q s p,q [n + a]p,q = Ep,q (− [n]p,q x) u s−u u q2u−s [n]p,q q(k−2)u [n]p,q [n + b]u [n + b]s+1−u p,q p,q u=0 k=1 ×q (k−1)(k−2) +1 2 ([n]p,q x)k−1 [k − 1]p,q ! u s+1−u ∞ s   X X [k]u xpu [n + a]p,q s p,q [n + a]p,q = Ep,q (− [n]p,q x) u s−u s+1−u u q2u−s−1 [n]p,q q(k−2)u [n]u [n + b]p,q p,q [n + b]p,q u=0 ×q = k(k−1) 2 s  X u=0 which ends the proof.  ([n]p,q x)k [k]p,q ! xpu [n + a]s+1−u s p,q Rp,q (tu ; x), s+1−u n,a,b u q2u−s−1 [n]s−u p,q [n + b]p,q k=0 (2.1) R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 313 Lemma 2.2. Let the Rp,q n,a,b (f; x) operators are given by (1.3). Then, the following identities Rp,q n,a,b (1; x) = 1, Rp,q n,a,b (t; x) = qx 2 Rp,q n,a,b (t ; x) p,q Rn,a,b (t3 ; x) p,q Rn,a,b (t4 ; x) [n + a]p,q [n + b]p,q , (2.2) = ! 2 [n + a]2p,q q 2 x pqx + , [n]p,q [n + b]2p,q = q3 2pq2 + p2 q 2 x + p3 x3 + x [n]p,q [n]2p,q = q4 p6 4 p5 q + 2p3 q2 + 3p3 q3 3 p3 q + 3p2 q2 + 3pq3 2 x + x x + x + 2 2 2 q q [n]p,q [n]p,q [n]3p,q (2.3) ! [n + a]3p,q [n + b]3p,q , ! [n + a]4p,q [n + b]4p,q , are satisfies. Proof. In view of (2.1), it is obvious that Rp,q n,a,b (1; x) = 1. Then, Rp,q n,a,b (t; x) = qx 2 Rp,q n,a,b (t ; x) = px = px [n + a]p,q [n + b]p,q [n + a]p,q [n + b]p,q [n + a]p,q [n + b]p,q Rp,q n,a,b (1; x) = qx [n + a]p,q [n + b]p,q , [n + a]2p,q p,q q2 R (1; x) x [n]p,q [n + b]2p,q n,a,b ! ! [n + a]2p,q [n + a]2p,q [n + a]p,q q2 q2 2 = pqx + x x qx + [n + b]p,q [n]p,q [n + b]2p,q [n]p,q [n + b]2p,q Rp,q n,a,b (t; x) + and 2 [n + a]3p,q p,q p2 [n + a]p,q p,q q3 2pq [n + a]p,q p,q x R (t; x) + x Rn,a,b (1; x) Rn,a,b (t2 ; x) + x [n]p,q [n + b]2p,q n,a,b q [n + b]p,q [n]2p,q [n + b]3p,q ! ! [n + a]2p,q q2 p2 [n + a]p,q 2 pqx + x = x [n]p,q q [n + b]p,q [n + b]2p,q ! 2 [n + a]p,q [n + a]2p,q q3 2pq [n + a]p,q x + qx x + [n]p,q [n + b]2p,q [n + b]p,q [n]2p,q [n + b]2p,q ! [n + a]3p,q q3 2pq2 + p2 q 2 3 3 x + = p x + x , [n]p,q [n]2p,q [n + b]3p,q 3 Rp,q n,a,b (t ; x) = 4 Rp,q n,a,b (t ; x) 2 3p2 [n + a]p,q p,q p3 [n + a]p,q p,q 3 R (t ; x) + x = 2x R (t2 ; x) [n]p,q [n + b]2p,q n,a,b q [n + b]p,q n,a,b 3 [n + a]4p,q p,q q4 3pq2 [n + a]p,q p,q x Rn,a,b (t; x) + x + Rn,a,b (1; x) [n]2p,q [n + b]3p,q [n]3p,q [n + b]4p,q !   [n + a]3p,q q3 2pq2 + p2 q 2 p3 [n + a]p,q 3 3 x x + p x + = 2x [n]p,q q [n + b]p,q [n]2p,q [n + b]3p,q !   2 2 2 [n + a] 3p2 [n + a]p,q q p,q + x x pqx2 + [n]p,q [n + b]2p,q [n]p,q [n + b]2p,q R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 3 3pq2 [n + a]p,q x + [n]2p,q [n + b]3p,q =  qx [n + a]p,q [n + b]p,q  + 314 [n + a]4p,q q4 x [n]3p,q [n + b]4p,q q4 p6 4 p5 q + 2p3 q2 + 3p3 q3 3 p3 q + 3p2 q2 + 3pq3 2 x + x + x x + q2 q2 [n]p,q [n]2p,q [n]3p,q ! [n + a]4p,q [n + b]4p,q , this ends the proof. Corollary 2.3. Taking into account Lemma 2.2, the following central moments Rp,q n,a,b (t − x; x) = {q − 1} 2 Rp,q n,a,b ((t − x) ; x) 4 Rp,q n,a,b ((t − x) ; x) =  pq [n + a]p,q [n + b]p,q [n + a]2p,q − 2q [n + b]2p,q (2.4) x, [n + a]p,q [n + b]p,q  2 +1 x +q 2 [n + a]2p,q [n + b]2p,q x , [n]p,q  4 [n + a]2p,q [n + a]p,q [n + a]3p,q p6 [n + a]p,q 3 = + 6pq − 4q − 4p 3 2 4 2 [n + b]p,q q [n + b]p,q [n + b]p,q [n + b]p,q  4 [n + a]3p,q p5 q + 2p3 q2 + 3p3 q3 [n + a]p,q 2 2 − 4(2pq + p q) × q2 [n + b]3p,q [n + b]4p,q (2.5)  [n + a]2p,q 2 x4 [n + a]4p,q [n + a]3p,q x3 3 2 2 3 3 +6q p q + 3p q + 3pq − 4q [n + b]2p,q [n]p,q [n + b]3p,q [n + b]4p,q +q 4 [n + a]4p,q x [n + b]4p,q [n]3p,q  x2 (2.6) [n]2p,q , are satisfies. Remark 2.4. It can be clearly seen that lim [n]p,q = 0 or n→∞ 1 p−q for 0 < q < p 6 1. For the purpose of to provide the results, we get the sequences q = (qn ) ∈ (0, 1) and p = (pn ) ∈ (qn , 1] so that lim pn = n→∞ n 1, lim qn = 1 and lim pn n = c, lim qn = d, by 0 < c, d 6 1. Then, we obtain lim [n]pn ,qn = ∞. n→∞ n→∞ n→∞ n→∞ Furthermore, following relations hold   [n + a]pn ,qn = α, lim [n]pn ,qn (qn − 1) n→∞ [n + b]pn ,qn   [n + a]pn ,qn [n + a]2pn ,qn − 2qn lim [n]pn ,qn pn qn +1 = β n→∞ [n + b]pn ,qn [n + b]2pn ,qn   3 4 2 [n [n [n + a] + a] + a] p6n [n + a]pn ,qn pn ,qn pn ,qn pn ,qn − 4p3n + 1 = γ. lim [n]pn ,qn + 6pn qn − 4qn 3 2 n→∞ [n + b] q2n [n + b]4p ,q [n [n + b] + b] p ,q n n p ,q p ,q n n n n n n Let us give an example for the sequences (pn ), (qn ) given by Remark 2.4. Taking (pn ) = 1 − 3 1 1 n 1 − n+3 , so it is clear to see that lim pn = 1, lim qn = 1, lim pn n = e2 , lim qn = e3 2 n+2 , (qn ) = 1 lim n→∞ [n]pn ,qn n→∞ n→∞ n→∞ = 0 as n → ∞. Further, we obtain α = c(e−3 − e−2 ), β = e−2 − e−3 , γ = 0. n→∞ Remark 2.5. Let the sequences (pn ), (qn ) are given by Remark 2.4. Then, we obtain pn ,qn (t − x; x) = αx, lim [n]pn ,qn Rn,a,b n→∞ (2.7) pn ,qn ((t − x)2 ; x) = βx2 + x, lim [n]pn ,qn Rn,a,b (2.8) pn ,qn lim [n]pn ,qn Rn,a,b ((t − x)4 ; x) = γx4 . (2.9) n→∞ n→∞ R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 315 3. Local approximation results for Rp,q n,a,b (f; x) operators Suppose that the space CB [0, ∞) indicates for all real-valued continuous and bounded functions g. On CB [0, ∞) the norm and K-functional of Peetre’s are given respectively as  kgk = sup |g(x)| and K2 (g, η) = inf kg − hk + η h′′ , h∈C2B x∈[0,∞) where η > 0 and C2B = {h ∈ CB [0, ∞) : h′ , h′′ ∈ CB [0, ∞)} . Taking into account [11], we attain √ K2 (g; η) 6 Cω2 (g; η), η > 0, (3.1) where ω2 (g; η) = sup |g(x + 2h) − 2g(x + h) + g(x)| sup √ 0<h6 η x∈[0,∞) is the second order of modulus of smoothness of function g ∈ CB [0, ∞). Further, ω(g; η) := sup sup |g(x + h) − g(x)| 0<h6η x∈[0,∞) is the ordinary modulus of continuity of g ∈ CB [0, ∞). More details for ω(g, η) can be found by [4]. Theorem 3.1. Suppose that f ∈ CB [0, ∞) and 0 < q < p 6 1. Then, for all x ∈ [0, ∞), we obtain Rp,q n,a,b (f; x) − f(x) 6 Cω2 (f; p ηn (x)) + ω(f; ϑn (x))   2 ; x) + ϑ(x)2 , and ϑ (x) = (q − 1) [n+a]p,q x . where a constant C > 0, ηn (x) = Rp,q ((t − x) n n,a,b [n+b] p,q Proof. Firstly, we give the following auxiliary operators ep,q (f; x) = Rp,q (f; x) − f q R n,a,b n,a,b where x ∈ [0, ∞). From (2.2), [n + a]p,q [n + b]p,q ! x + f(x), (3.2) ep,q (t − x; x) = 0. R n,a,b For g ∈ C2B , making use of Taylor’s expansion, ′ Zt g(t) = g(x) + (t − x)g (x) + (t − u)g′′ (u)du. x ep,q (.; x) operators to (3.3), we have Operating of R n,a,b ep,q (g; x) − g(x) R n,a,b = ep,q ((t − x)g′ (x); x) + R ep,q ( R n,a,b n,a,b Zt (t − u)g′′ (u)du; x) x Zt ep,q (t − x; x) + Rp,q ( (t − u)g′′ (u)du; x) = g′ (x)R n,a,b n,a,b x [n+a] q [n+b]p,q x p,q − Z x (q [n + a]p,q [n + b]p,q x − u)g′′ (u)du (3.3) R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 316 [n+a] Zt p,q = Rn,a,b ( (t − u)g′′ (u)du; x) − q [n+b]p,q x Z p,q x (q x [n + a]p,q [n + b]p,q x − u)g′′ (u)du. In view of Lemma 2.2 and (3.2), [n+a] ep,q (g; x) − g(x) R n,a,b Rp,q n,a,b ( 6 Zt q [n+b]p,q x ′′ (t − u)g (u)du; x) + x Z p,q [n + a]p,q (q [n + b]p,q x x − u)g′′ (u)du [n+a] 6 Rp,q n,a,b ( Zt q [n+b]p,q x ′′ (t − u) g (u) |du|; x) + Z p,q q [n + a]p,q [n + b]p,q x x  !2    [n + a]p,q p,q 6 g′′ (u) Rn,a,b ((t − x)2 ; x) + q x−x .   [n + b]p,q x − u g′′ (u) |du| Further, by (2.2), (2.3), and (3.2), we obtain Then, ep,q (f; x) 6 Rp,q (f; x) + 2 kfk 6 kfk Rp,q (1; x) + 2 kfk 6 3 kfk . R n,a,b n,a,b n,a,b ep,q ep,q Rp,q n,a,b (f; x) − f(x) 6 Rn,a,b (f − g; x) − (f − g)(x) + Rn,a,b (g; x) − g(x) + f(x) − f(q [n + a]p,q [n + b]p,q x)  !2    [n + a]p,q 2 6 4 kf − gk + Rp,q ((t − x) ; x) + (q − 1) g′′ x  n,a,b  [n + b]p,q ! [n + a]p,q x . + ω f; (q − 1) [n + b]p,q Over all g ∈ C2B[0, ∞) utilize the infimum on the right hand side by make use of (3.1) and for a constant [n+a] 2 2 C > 0, ϑn (x) = (q − 1) [n+b]p,q x , ηn (x) = Rp,q n,a,b ((t − x) ; x) + ϑ(x) , then p,q Rp,q n,a,b (f; x) − f(x) 6 4K2 (f; ηn (x)(x)) + ω(f; ϑn (x)) 6 Cω2 (f; which ends the proof. p ηn (x)) + ω(f; ϑn (x)), 4. Order of convergence of Rp,q n,a,b (f; x) operators In this section, by utilizing the ordinary modulus of continuity, we computed the order of convergence. Also, to see the smoothness of approximation for a function g on Lipschitz class LipM (ζ), where M > 0 and 0 < ζ 6 1, we established the degree of convergence of the operators (1.3). Since |g(t) − g(x)| 6 M |t − x|ζ , (t, x ∈ R), holds, then a function g belongs to LipM (ζ). Let Cx2 [0, ∞) := {h : |h(x)| 6 Mh (1 + x2 ), h is continuous, |h(x)| Mh > 0 } and C∗x2 [0, ∞) := {h : h ∈ Cx2 [0, ∞), lim 1+x2 < ∞ }. On C∗x2 [0, ∞), the norm and ordinary x→∞ modulus of continuity of h on [0, δ] are given respectively as follows: |h(x)| 2 x∈[0,∞) 1 + x khkx2 = sup and ωδ (h; η) = sup sup |t−x|6η x,t∈[0,δ] |h(t) − h(x)| δ > 0. R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 317 Theorem 4.1. Assume that f ∈ CB [0, ∞), the sequences (pn ), (qn ) are given as in Remark 2.4 and ωδ+1 (f; η) be its modulus of continuity on [0, δ + 1] ⊂ [0, ∞). Then, for all x ∈ [0, ∞) the following relation p pn ,qn 6 6Mf (1 + δ2 )µn (x) + 2ωδ+1 (f; µn (x)) (f; x) − f(x) Rn,a,b C[0,δ] holds, where µn (x) = (1 − pn qn ) δ2 + δ [n]pn ,qn . Proof. Suppose that x ∈ [0, δ] and t > δ + 1, so t − x > 1, then |f(t) − f(x)| 6 Mf (x2 + t2 + 2) 6 Mf (3x2 + 2(t − x)2 + 2) 6 6Mf (1 + δ2 )(t − x)2 . (4.1) Also, by x ∈ [0, δ] and t 6 δ + 1,   |t − x| |f(t) − f(x)| 6 1 + ωδ+1 (f; η), η Combining (4.1), (4.2), and for t > 0, x ∈ [0, δ], we obtain  |t − x| |f(t) − f(x)| 6 6Mf (1 + δ )(t − x) + 1 + η 2 2 η > 0. (4.2)  (4.3) ωδ+1 (f; η). pn ,qn (.; x) operators to both sides of (4.3), Operating Rn,a,b pn ,qn pn ,qn (|f(t) − f(x)| ; x) (f; x) − f(x) 6 Rn,a,b Rn,a,b 6 6Mf (1 + δ 2 pn ,qn )Rn,a,b ((t − x)2 ; x) + For x ∈ [0, δ] and by (2.5), since a, b, n ∈ N, 0 6 a 6 b, then pn ,qn Rn,a,b ((t − x)2 ; x) = pn qn [n + a]2pn ,qn [n + b]2pn ,qn − 2qn 6 (pn qn − 2qn + 1) δ2 + q2n 6 (1 − pn qn ) δ2 +   21 1 pn ,qn 2 1 + 2 Rn,a,b ((t − x) ; x) ωδ+1 (f; η). η [n+a]pn ,qn [n+b]pn ,qn [n + a]pn ,qn [n + b]pn ,qn 6 1, thus we obtain ! + 1 x2 + q2n [n + a]2pn ,qn [n + b]2pn ,qn x [n]pn ,qn δ [n]pn ,qn δ [n]pn ,qn = µn (x). Choosing η = p µn (x), gives the proof. Theorem 4.2. Assume that the sequences (pn ) and (qn ) are given as in Remark 2.4. Then, for all f ∈ LipM (ζ), M > 0, 0 < ζ 6 1, the following inequality ζ pn ,qn Rn,a,b (f; x) − f(x) 6 M(µn (x)) 2 , holds, where µn (x) is given as Theorem 4.1. pn ,qn (f; x), utilizing the linearity and monotononicity, then Proof. Let f ∈ LipM (ζ) and 0 < ζ 6 1. Since Rn,a,b we get pn ,qn (f; x) − f(x) Rn,a,b 6 Epn ,qn (− [n]pn ,qn x) ∞ X ([n]p k=0 n ,qn x)k [k]pn ,qn ! k(k−1) 2 qn R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 318 [k]pn ,qn [n + a]pn ,qn × f [n]pn ,qn [n + b]pn ,qn qk−2 n 6 MEpn ,qn (− [n]pn ,qn x) ∞ X ([n]p − f(x) n ,qn k=0 x)k [k]pn ,qn ! ! [k]pn ,qn [n + a]pn ,qn × f ! [n]pn ,qn [n + b]pn ,qn qk−2 n k(k−1) 2 qn − f(x) ζ . Utilizing Hölder’s inequality, then pn ,qn (f; x) − f(x) Rn,a,b 6 MEpn ,qn (− [n]pn ,qn x)  ∞  X ([n]p k=0 ×  ([n]pn ,qn x)k k(k−1) qn 2 [k]pn ,qn !   2−ζ 2 x)k n ,qn [k]pn ,qn ! [k]pn ,qn [n + a]pn ,qn k(k−1) 2 qn [n]pn ,qn [n + b]pn ,qn qk−2 n −x ζ pn ,qn ((t − x)2 ; x)) 2 . 6 M(Rn,a,b !2  ζ2   pn ,qn ((t − x)2 ; x), gives the proof. Taking µn (x) = Rn,a,b 5. Approximation in weighted spaces In this section, we determined the approximation features of the operators (1.3) on the weighted spaces of continuous functions on [0, ∞). Theorem 5.1. Assume that the sequences (pn ) and (qn ) are given as in Remark 2.4. For all x ∈ [0, ∞) and f ∈ C∗x2 [0, ∞), we have the following relation: lim sup n→∞ pn ,qn (f; x) − f(x) Rn,a,b 1 + x2 x∈[0,∞) = 0. Proof. Considering to the Korovkin’s theorem which is given by [13], we have to show that operators (1.3) fulfill the following three conditions: n ,qn s s Rp n,a,b (t ; x) − x lim sup n→∞ x∈[0,∞) 1 + x2 = 0, s = 0, 1, 2. (5.1) By (2.1), for s = 0 the first condition in (5.1) is trivial. Also, from (2.3), sup x∈[0,∞) pn ,qn (t; x) − x Rn,a,b 1 + x2 6 qn [n + a]pn ,qn [n + b]pn ,qn −1 [n + a]pn ,qn x − 1 6 |qn − 1| . 6 qn 2 [n + b]pn ,qn x∈[0,∞) 1 + x sup Then, lim sup n→∞ x∈[0,∞) pn ,qn Rn,a,b (t; x) − x 1 + x2 = 0. Analogously, from (2.4), we get sup x∈[0,∞) pn ,qn 2 Rn,a,b (t ; x) − x2 1 + x2 6 pn qn [n + a]2pn ,qn [n + b]2pn ,qn [n + a]2pn ,qn x2 2 − 1 sup + qn 2 [n]pn ,qn [n + b]2pn ,qn x∈[0,∞) 1 + x x 2 x∈[0,∞) 1 + x sup R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 6 |1 − pn qn | + 319 1 . [n]qn Hence, lim sup n→∞ pn ,qn 2 (t ; x) − x2 Rn,a,b 1 + x2 x∈[0,∞) = 0, which ends the proof. Now, we take the modulus of continuity Ω(g; η) on weighted spaces for every g ∈ C∗x2 [0, ∞) as below Ω(g; η) = |g(x + h) − g(x)| . 2 0<h6η,x>0 1 + (x + h) sup Lemma 5.2 ([15]). Let g ∈ C∗x2 [0, ∞). The following relations are fulfilled: i) Ω(g; η) is a monotone increasing function of g; ii) lim+ Ω(g; η) = 0; η→0 iii) for λ > 0, Ω(g; λη) 6 2(1 + λ)(1 + η2 )Ω(g; η). Theorem 5.3. Suppose that the sequences (pn ) and (qn ) are given as in Remark 2.4. Then, for all x ∈ [0, ∞), f ∈ C∗x2 [0, ∞), we get sup pn ,qn Rn,a,b (f; x) − f(x) (1 + x2 ) x∈[0,∞) 5 2 1 ) 6 KΩ(f; p ϑpn ,qn (n) where K > 0 is a constant and ϑpn ,qn (n) = max 1 − pn qn , 1 [n]pn ,qn . Proof. Let x ∈ [0, ∞), η > 0, making use of Ω(f; η) and by Lemma 5.2, we attain
|f(t) − f(x)| 6 (1 + (t − x)2 ) 1 + x2 Ω(f; |t − x|)  
|t − x| (1 + η2 )Ω(f; η). 6 2(1 + (t − x)2 ) 1 + x2 1 + η pn ,qn pn ,qn Consider Rn,a,b (1; x) = 1 and making use of monotononicity of Rn,a,b operators, we get pn ,qn pn ,qn (|f(t) − f(x); x)|). Rn,a,b (f; x) − f(x) 6 Rn,a,b By (5.2), pn ,qn (f; x) − f(x) Rn,a,b 2 6 2(1 + η )Ω(f; η) 1 + x 2 6 2(1 + η2 )Ω(f; η) 1 + x2

n ,qn Rp n,a,b (   |t − x| (1 + (t − x)2 ); x) 1+ η pn ,qn pn ,qn Rn,a,b (1; x) + Rn,a,b ((t − x)2 ; x) 1 pn ,qn 1 pn ,qn + Rn,a,b (|t − x| ; x) + Rn,a,b (|t − x| (t − x)2 ; x) . η η Next, utilizing the Cauchy-Schwarz inequality,
pn ,qn pn ,qn pn ,qn ((t − x)2 ; x) (1; x) + Rn,a,b (f; x) − f(x) 6 2(1 + η2 )Ω(f; η) 1 + x2 Rn,a,b Rn,a,b q 1 q pn ,qn 1 q pn ,qn pn ,qn + Rn,a,b ((t − x)2 ; x) + Rn,a,b ((t − x)2 ; x) Rn,a,b ((t − x)4 ; x) . η η (5.2) R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 320 Also, from (2.5), pn ,qn ((t − x)2 ; x) Rn,a,b = pn qn [n + a]2pn ,qn [n + b]2pn ,qn 6 (1 − pn qn ) x2 + − 2qn [n + a]pn ,qn [n + b]pn ,qn ! 2 +1 x + q2n [n + a]2pn ,qn x [n + b]2pn ,qn [n]pn ,qn x [n]pn ,qn 6 K1 O(ϑpn ,qn (n))(1 + x2 ), where K1 > 0 and ϑpn ,qn (n) = max 1 − pn qn , [n] 1 pn ,qn hence there consists a constant K2 > 0 such that n→∞ n→∞
2 2 n ,qn Rp n,a,b ((t − x) ; x) 6 K2 1 + x . Moreover, by (2.6), q and pn ,qn ((t − x)4 ; x) 6 K3 1 + x2 Rn,a,b K4 1 q pn ,qn Rn,a,b ((t − x)2 ; x) 6 O η η for K3 > 0 and K4 > 0. Then, we obtain pn ,qn (f; x) − f(x) 6 2(1 + Rn,a,b Taking η = . Since lim pn qn = 1 and lim [n]pn ,qn = ∞, 1
q q (1 + x2 ) ϑpn ,qn (n) )(1 + x2 )Ω(f; p 
) 1 + K2 1 + x2 1 ϑpn ,qn (n) ϑpn ,qn (n) q q q q
K4 2 K4 2 (1 + x ) + K3 1 + x (1 + x2 ) . + O O ϑpn ,qn (n) ϑpn ,qn (n) η η p ϑpn ,qn (n), in above equation, pn ,qn Rn,a,b (f; x) − f(x) 6 2(1 + Thus, for ϑpn ,qn (n) 6 1, we may write sup
1 1 ) + CK4 1 + x2 2 . )(1 + x2 )Ω(f; p ϑpn ,qn (n) ϑpn ,qn (n) 1 pn ,qn Rn,a,b (f; x) − f(x) 1 6 KΩ(f; p ), ϑpn ,qn (n) 5 (1 + x2 ) 2 x∈[0,∞) where K = 4 (1 + K2 + CK4 + K1 K3 K4 ) , which ends the proof. 6. Voronovskaya type theorem Theorem 6.1. Let the sequences (qn ) ∈ (0, 1) and (pn ) ∈ (qn , 1] such that lim pn = 1, lim qn = 1 and n→∞ n→∞ ′ ′′ ∗ n ∗ lim pn n = c, lim qn = d, by 0 < c, d 6 1. For any f ∈ Cx2 [0, ∞) such that f , f ∈ Cx2 [0, ∞) we get n→∞ n→∞ lim [n]pn ,qn n→∞  uniformly on the interval [0, A], A > 0.  pn ,qn (f; x) − f(x) Rn,a,b
βx2 + x ′′ f (x) = αxf (x) + 2 ′ Proof. Suppose that x ∈ [0, ∞) and considering f, f′ , f′′ ∈ C∗x2 [0, ∞) and making use of Taylor’s expansion formula, then 1 f(t) = f(x) + (t − x)f′ (x) + (t − x)2 f′′ (x) + ψ(t; x)(t − x)2 . (6.1) 2 Here, ψ(t; x) is a form of Peano of the rest term. Since ψ(.; x) ∈ C∗x2 [0, ∞), then lim ψ(t; x) = 0. Operating pn ,qn Rn,a,b (.; x) to (6.1), then we get, t→x R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 321 pn ,qn ′ n ,qn [n]pn ,qn (Rp n,a,b (f; x) − f(x)) = [n]pn ,qn Rn,a,b ((t − x); x)f (x) + 1 pn ,qn 2 ′′ 2 n ,qn [n]pn ,qn Rp n,a,b ((t − x) ; x)f (x) + [n]pn ,qn Rn,a,b (ψ(t; x)(t − x) ; x). 2 Utilizing the Cauchy-Schwarz inequality to last part of the above equality, it pursues that q q pn ,qn pn ,qn pn ,qn (ψ2 (t; x); x) Rn,a,b ((t − x)4 ; x). (ψ(t; x)(t − x)2 ; x) 6 Rn,a,b Rn,a,b Considering ψ(t; x) ∈ C∗x2 [0, ∞) and by Theorem 5.1, we get lim ψ(t; x) = 0. Then, (6.2) t→x lim Rpn ,qn (ψ2 (t; x); x) n→∞ n,a,b = ψ2 (t; x) = 0 (6.3) uniformly on x ∈ [0, A]. Combining (6.2), (6.3), and by (2.9), we get pn ,qn (ψ(t; x)(t − x)2 ; x) = 0. lim [n]pn ,qn Rn,a,b n→∞ (6.4) Consequently, in view of (2.7), (2.8), and (6.4), lim [n]pn ,qn n→∞ which gives the desired result.  pn ,qn Rn,a,b (f; x) − f(x) 
βx2 + x ′′ f (x), = αxf (x) + 2 ′ 7. Some plots In this section, we compare the convergence of Rp,q n,a,b (f; x) operators with the different parameters to a particular function. (a) (b) (c) x −3 under the different parameters. Figure 1: The convergence of Rp,q n,a,b (f; x) operators to f(x) = xe R. Aslan, A. Izgi, J. Nonlinear Sci. Appl., 14 (2021), 310–323 322 x −3 by keeping In Figure 1a, we examine the convergence of Rp,q n,a,b (f; x) operators to the f(x) = xe the parameters n, a, b constant and increasing the values of p and q. It is obvious that, in view of 0 < q < p 6 1, as the values of p and q increasing then the convergence of the Rp,q n,a,b (f; x) operators to − x3 the f(x) = xe becomes better. − x3 by keeping the In Figure 1b, we examine the convergence of Rp,q n,a,b (f; x) operators to the f(x) = xe parameters p, q, a, b constant and increasing the values of n. 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