1 Introduction and Construction of the Operators

Let{ϕn}(n=1,2,···),be a sequence of functions,having the following properties:

i)ϕn is analytic on a domain D containing the disk for each positive integer n,

ii)ϕn(0)=1 for n∈N,

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iii)ϕn(x)>0 and for every positive integer n,x ∈[0,∞)and and for every nonnegative integer k.

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Thus the desired result is obtained.

Furthermore,if we take ϕn(x)=(1+x)n and replace x by anx in the operator(1.1),we have

This operator is known as Bernstein type rational function which was studied in[4,16].

In[8],O.introduced and investigated approximation properties of certain linear operators defined by

where for 0n(x)satisfies the above conditions given by(i)-(iii)and also following condition:

Recently,linear positive operators and their Chlodowsky type generalizations have been widely studied by several authors[1,3–22],because this generalization allows us to investigate approximation properties of functions defined on the infinite interval[0,∞)by using the similar techniques and methods on the classical operators.

Letbe positive increasing sequence of real numbers with the properties

and γ be real number in the intervalAssume that the sequence of functions{ϕn}satisfies the conditions(i)-(iv).For a function f defined on[0,∞)and bounded on every finite interval[0,γ],we define the following sequence of linear positive operators:

Motivated by this work,we give Chlodowsky type generalization of Ln(f;x)operators given by(1.2)as follows:

The aim of this paper is to study some convergence properties of the operators Ln(f;βn,x)defined by(1.4)and modify the operators for differentiable functions,in order to improve the rate of convergence on the interval[0,βn]extending infinity as n→∞.Also we give an application to functional differential equation by using these operators.

2 Approximation properties of Ln(f;βn,x)

In this section we study conditions of Korovkin theorem[2]and the rate of convergence,an asymptotic formula for the operators(1.4)for f ∈C[0,γ].

Now we use the test functions ei(t)=ti,i=0,1,2.Then,we obtain the following result.

Theorem 2.1.Let{βn}be a positive increasing sequence satisfying(1.3)and the operators Ln(f;βn,x)be defined by(1.4)with the conditions(i)-(iv).For every finite intervaland for each we have

Proof.Firstly,from condition(i),we have

so we get

By the definition of the operators(1.4)and using e1(t)=t

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by(iv),we have

Sincewe can fnid a positive constant c such that for any k.Therefore,one gets

or

On the other hand,observe that

combining(2.2)with(2.3),we have

and hence,

on the other hand,it is clear that Ln(e2(t);βn,x)−e2(x)≥0.Using(2.4),we get

Finally,for e2(t)=t2,we obtain

by(iv),we get

using this equalities,we obtain

which implies that

uniformly in[0,γ].

Hence we have

uniformly in[0,γ].Thus,the proof of the theorem is completed.

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Remark 2.1.Let Simple calculations,one can easily obtain ψn=1,αn,k=0.Then Ln(f;βn,x)has the following form:

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Let be a fixed number.For f ∈Cr(I)and n∈N,we consider the operators:

Example 2.1.For n=10,50 andthe convergence of Ln(f;βn,x)to f(x)will be illustrated in Fig.1 and Fig.2.

Example 2.2.Let n=10.For βn=n4/5and βn=n2/3,the convergence of Ln(f;βn,x)to f(x)will be illustrated in Fig.3.

Now we give the approximation order of operators(1.4)with help of asymptotic inequality.

where B(α,r)is a beta function.Therefore we write

Let I=[0,γ]and C(I)be the space of all continuous functions f.For a fixed r∈N we denote by

Theorem 2.2.If the operators Ln are defined by(1.4),then for sufficiently large n and for every f ∈C2(I)

Proof.By the Taylor formula,we write

where λ(t)→0 as t→x.If we Apply the operators(1.4)to(2.5),we get

From(2.1),(2.2)and(2.4),we have

using(2.7a)-(2.7c)in(2.6),one obtains

and hence

which implies

In[5],V.A.Baskakov introduced following the sequence of linear operators{Ln},

3 A generalization of the Ln(f;βn,x)

In recent years several authors[7,8,17,21,22]investigated approximation properties of certain linear operators for differentiable functions.In this section we will modify the operator(1.4)for differentiable functions,in order to improve the rate of convergence of the sequence{Ln f}to f(see Example 3.1)

Chlodowsky type MKZ operators(see[21]).

Note that for r=0,we have

Theorem 3.1.If then

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where B(α,r)is a beta function and Ln is defined by(1.4).

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Proof.From(2.1),we can write

Consider the term in parentheses.Using modified Taylor’s formula we have

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since f(r)∈LipMα,we obtain

On the other hand,we have

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from(3.1)and(3.2),we have

Thus the proof of theorem is completed.

Remark 3.1.If we choose we obtain Chlodowsky type generalized MKZ operators defined by

(see[21]).

Example 3.1.For n=10,βn=n2/3 and r=2,the convergence comparison of and Ln(f;βn,x)to f(x)will be illustrated in Fig.4 and Fig.5.

Example 3.2.Let n=10.For βn=n4/5and βn=n2/3,the convergence of to f(x)=x6 will be illustrated in Fig.6

4 Application an differential equation

Many authors obtained some differential equations by using the linear positive operators which are solution of these equations,we refer the readers to[1,3,8,9,19].In this section,using the same idea and method,as an application to the differential equation,we obtain a functional differential equation so that the linear positive operator Ln(f;βn,x)is a particular solution of it.

Theorem 4.1.Let For each x ∈[0,γ]and f ∈C[0,γ],the operators Ln(f;βn,x)defined by(1.4)satisfy the following differential equation:

where

Proof.By the Theorem 2.1,if f ∈C(I),then Ln(f;βn,x)converges uniformly to f(x)on[0,γ].So we can differentiate both sides of(1.4)term by term to obtain

Thus we can see that:

Since we get

by using and simple calculations,we have

and

hence we get

This gives the desired result.

Corollary 4.1.The operators Ln(f;βn,x)given by(1.4)are a particular solution of the following differential equation:

where βn is given by(4.2).

Proof.Selecting f(t)=βn−t in(4.1),we get

from(2.1),we have

which gives the proof.

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