Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 5803–5812 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Approximation with modified Phillips operators Danyal Soybas ¸ Department of Mathematics Education, Faculty of Education, Erciyes University, Kayseri 38039, Turkey. Communicated by D. Baleanu Abstract In the present paper, we study modified Phillips operators in simultaneous approximation. The operators discussed here are important as they have link with the well-known Sz´ asz operators. We estimate some direct results for the operators. c 2017 All rights reserved. Keywords: Sz´ asz operators, Phillips operators, simultaneous approximation, modulus of continuity, moment generating function, asymptotic formula, error estimation. 2010 MSC: 41A25, 41A30. 1. Introduction In order to generalize the well-known Bernstein polynomials to the positive real axis Sz´ asz [16] intro- duced the following operators S α (f, x)= ∞  k=0 e -αx (αx) k k! f  k α  , x ∈ [0, ∞). These operators are linear positive operators and play an important role in the theory of approximation. Recently, Gupta [6] discussed some approximation properties of the operators S α (f, x). Four years later Phillips [15] proposed a generalization of the Sz´ asz operator in the following form: L α (f, x)= e -αx f(0)+ α ∞  k=1 s α,k (x)  ∞ 0 s α,k-1 (t)f(t)dt, x ∈ [0, ∞), where s α,k (x)= e -αx (αx) k k! . Later Mazhar and Totik [12], Finta and Gupta [1], Gupta and Srivastava [8], Govil et al. [4], Heilmann and Tachev [10], Gupta [5], Tachev [17], etc. discussed several approximation properties of the operators L α (f, x). Recently, in order to generalize the Phillips operators, based on the parameter ρ> 0, Pˇ altˇ anea in [13] proposed the following operators L ρ α (f, x)=  ∞ 0 k ρ α (x, t)f(t)dt = e -αx f(0)+ ∞  k=1 s α,k (x)  ∞ 0 θ ρ α,k (t)f(t)dt, x ∈ [0, ∞), (1.1) Email address: danyal@erciyes.edu.tr (Danyal Soybas ¸) doi:10.22436/jnsa.010.11.18 Received 2017-05-31