About Some Linear Operators

Volume 2007, Article ID 91781,13pages doi:10.1155/2007/91781

Research Article

About Some Linear Operators

Ovidiu T. Pop

Received 22 March 2007; Revised 24 May 2007; Accepted 15 July 2007

Recommended by Brigitte Forster-Heinlein

Using the method of Jakimovski and Leviatan from their work in 1969, we construct a general class of linear positive operators. We study the convergence, the evaluation for the rate of convergence in terms of the first modulus of smoothness and we give a Voronovskaja-type theorem for these operators.

Copyright © 2007 Ovidiu T. Pop. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The aim of this paper is to construct a class of linear operators in more general condi-tions. The method was inspired by Jakimovski and Leviatan (see [1]). We do not study the convergence of these operators with the well-known theorem of Bohman-Korovkin. The evaluation theorems for the rate of convergence are different from the well-known theorem of Shisha-Mond. We prove the Voronovskaja-type theorem for these operators. In the end, we give particularizations of these operators.

We recall some notions and results which we will use in this paper.

LetNbe the set of positive integer numbers andN0=N∪ {0}. For a given interval I, we will use the following function sets:B(I)= {f | f :I→R,f bounded onI},C(I)= {f |f :I→R,f continuous onI}, andCB(I)=B(I)∩C(I).

For anyx∈I, consider the functionsψx:I→Rdefined byψx(t)=t−xandei:I→R, ei(t)=tifor anyt∈I,i∈ {0, 1, 2, 3, 4}.

For f ∈CB(I), by the first-order modulus of smoothness of f is meant the function

ω(f;·) : [0,∞)→Rdefined for anyδ≥0 by

In the following, we take into account the properties of the first-order modulus of smoothness and the properties of the linear positive functional.

Lemma 1.1. Let f ∈CB(I). Then,ω(f;·) has the following properties:

(a)ω(f; 0)=0,

(b)ω(f;·) is an increasing function,

(c)ω(f;·) is a uniform continuous function onI,

(d) for anyδ >0,x,t∈I, one has|f(t)−f(x)| ≤[1 +δ−2_ψ2

x(t)]ω(f;δ).

Lemma 1.2. LetA:E(I)→Rbe a linear positive functional. Then,

(a) for f,g∈E(I) with f(x)≤g(x) for anyx∈I, one has

A(f)≤A(g); (1.2)

(b)|A(f)| ≤A(|f|) for anyf ∈E(I), whereE(I) is a subset of the set of real functions

defined onI.

In [2] we have demonstrated the following theorem.

Theorem 1.3. LetIbe an intervalx∈I, and let the function f :I→Rbestimes diff

eren-tiable inx. According to the Taylor Expansion Theorem, one has

f(t)= s

i=0 (t−x)i

i! f

(i)_{(x) + (t−x)}s_{μ(t−x), (1.3)}

whereμis a bounded function and lim_t

→xμ(t−x)=0. If f

(s) _{is a continuous function onI,}

then for anyδ >0 andx∈Ione has

_μ(t−x)≤ 1 s!

1 +δ−2ψx2(t)ωf(s);δ . (1.4)

2. Preliminaries

In this section, we construct a general class of linear and positive operators and we dem-onstrate for these operators an approximation theorem and a Voronovskaja-type theo-rem.

Let I,J be intervals andI∩J is a nonempty interval. For any m∈Nand k∈N0, consider the functionϕm,k:J→Rwith the propertyϕm,k(x)≥0 for anyx∈J and the

In the following, letE(I) andF(J) be subsets of the set of real functions defined onI,J respectively, such that the series∞k=0ϕm,k(x)Am,k(f) is convergent for any f ∈E(I) and

anyx∈J. We suppose thatψxi∈E(I) for anyx∈I∩Jand anyi∈ {0, 1,...,s+ 2}.

In what followss∈N0,sis even.

Definition 2.1. Form∈N, define the operatorLm:E(I)→F(J) by

Lmf (x)=

∞

k=0

ϕm,k(x)Am,k(f) (2.1)

for any f ∈E(I) andx∈J.

Proposition 2.2. The operators (Lm)m≥1are linear and positive onE(I∩J).

Proof. The proof follows immediately.

Definition 2.3. Form∈Nandi∈N0, defineTiby

TiLm (x)=miLmψxi (x)=mi

∞

k=0

ϕm,k(x)Am,k

ψi

x (2.2)

for anyx∈I∩J.

Theorem 2.4. If f ∈E(I) is ans-times differentiable function inx∈I∩J, with f(s)

con-tinuous inx, and if there existαs,αs+2∈[0,∞) andm(s)∈Nsuch that

αs+2< αs+ 2 (2.3)

and (TsLm)(x)/mαs_{, (}T_s

+2Lm)(x)/mαs+2are bounded for anym∈N,m≥(s), then

lim

m→∞m

s−αs

Lmf (x)− s

i=0 1 i!mi

TiLm (x)f(i)(x)

=0. (2.4)

Assume thatf is anstimes differentiable function onIwithf(s)_{continuous onIand an}

in-tervalK⊂I∩Jexists such that there existm(s)∈Nand the constantskj(K)∈Rdepending

onK, so that for anym∈N,m≥m(s) andx∈K, one has

TjLm (x)

mαj ≤kj(K), (2.5)

wherej∈ {s,s+ 2}. Then, the convergence given in (2.4) is uniform onKand

ms−αs

Lmf (x)− s

i=0 1 i!mi

TiLm (x)f(i)(x)

≤_s1

!

ks(K) +ks+2(K) ω

f(s)_;√ 1 m2+αs−αs+2

(2.6)

Proof. According to Taylor’s Theorem, we have

f(t)= s

i=0 (t−x)i

i! f

(i)_{(x) + (t−x)}s_{μ(t−x), (2.7)}

whereμis a bounded function and lim

t→xμ(t−x)=0.

Hence, from (2.7), we have

Am,k(f)= s

i=0 f(i)_(x)

i! Am,k

ψi

x +Am,kψxsμx , (2.8)

whereμx:I→R,μx(t)=μ(t−x), for anyt∈I∩J.

Multiplying byϕm,k(x) and summing overk∈N0, we obtain

Lmf (x)= s

i=0 f(i)_(x)

i!

Lmψix (x) +

∞

k=0

ϕm,k(x)Am,kψxsμx . (2.9)

Thus,

ms−αs

Lmf (x)− s

i=0 f(i)_(x)

i!mi

TiLm (x)

=Rmf (x), (2.10)

where

Rmf (x)=ms−αs

∞

k=0

ϕm,k(x)Am,k

ψs

xμx . (2.11)

Then,

_{Rmf (x)≤m}s−αs

∞

k=0

ϕm,k(x)Am,k

ψs

xμx (2.12)

and takingLemma 1.2into account, we obtain

_{Rmf (x)≤m}s−αs

∞

k=0

ϕm,k(x)Am,kψxsμx . (2.13)

According to the relation (1.4), for anyδ >0 andt∈I∩J, we have

_{μx(t)=μ(t−x)≤}1 s!

1 +δ−2_ψ2

x(t)ωf(s);δ , (2.14)

and so

ψs

xμx (t)≤_s1

!

ψs

From (2.13) and (2.15), it results that

_{Rmf (x)≤} 1 s!m

s−αs

_∞

k=0

ϕm,k(x)Am,kψxs +δ−2

∞

k=0

ϕm,k(x)Am,kψxs+2

ωf(s)_{;δ .}

(2.16)

Thus,

_{Rmf (x)≤} 1 s!

TsLm (x)

mαs +δ

−2

Ts+2Lm (x)

mαs+2 m

−2−αs+αs+2

ωf(s)_{;δ . (2.17)}

Consideringδ=1/√m2+α2−αs+2, the inequality above becomes

_{Rmf (x)≤}1 s!

TsLm (x)

mαs +

Ts+2Lm (x) mαs+2

ωf(s)_;√ 1 m2+αs−αs+2

. (2.18)

Taking into account that (TsLm)(x)/mαs_{and (}T_s

+2Lm)(x)/mαs+2 are bounded for anym∈

N,m≥m(s), and considering the fact that

lim

m→∞ω

f(s)_;√ 1 m2+αs−αs+2

=ωf(s)_{; 0 =0, (2.19)}

we have that

lim

m→∞

Rmf (x)=0. (2.20)

From (2.10) and (2.20), (2.4) follows.

If in addition (2.5) takes place then, (2.18) becomes

_{Rmf (x)≤} 1 s!

ks(K) +ks+2(K) ω

f(s)_;√ 1 m2+αs−αs+2

, (2.21)

form≥m(s) andx∈K. Therefore, the convergence from (2.4) is uniform onK. Now,

(2.10) and (2.21) yield (2.6).

In the following, we suppose that for anyk∈N0andm∈N, we have

Am,ke0 =1, (2.22)

and for anyx∈I∩Jandm∈N

∞

k=0

ϕm,k(x)=1. (2.23)

Remark 2.5. Taking (2.22) and (2.23) into account, it results that

T0Lm (x)=1 (2.24)

Remark 2.6. InTheorem 2.4, we choose the smallestαsandαs+2, if they exist.

Remark 2.7. Taking (2.24) into account, we chooseα0=0.

Remark 2.8. Fors=0,s=2, respectively, we state two corollaries which we will use in the section Main results.

Corollary 2.9. If f ∈E(I) is a continuous function inx∈I∩J, and if there existα2and

m(0)∈Nsuch that

0≤α2<2 (2.25)

and (T2Lm)(x)/mα2_{is bounded for any}m∈N_,m≥m_{(0), then}

lim

m→∞

_L

mf (x)=f(x). (2.26)

Assume that f is continuous onI and an intervalK⊂I∩J exists, such that there exist

m(0)∈Nandk2(K) so that for anym∈N,m≥m(0), andx∈K, one has

T2Lm (x)

mα2 ≤k2(K). (2.27)

Then, the convergence given in (2.26) is uniform onKand

_{Lmf (x)−f(x)≤}

1 +k2(K) ω

f;√_m1_2−α

2

(2.28)

for anyx∈Kandm≥m(0).

Corollary 2.10. If f ∈E(I) is a two-times differentiable function inx∈I∩J, with f(2)

continuous inx, and if there existα2,α4andm(2)∈Nsuch that

0≤α2<2,

0≤α4< α2+ 2, (2.29)

(T2Lm)(x)/mα2and ((T4Lm)(x))/mα4are bounded for anym∈N,m≥m(2), then

lim

m→∞m

2−α2L_mf ₍x₎−f₍x₎− 1

m

T1Lm (x)f(1)(x)− 1 2m2

T2Lm (x)f(2)(x)

=0.

(2.30)

Assume that f is a two-times differentiable function onIwith f(2)_{continuous onIand an}

intervalK⊂I∩Jexists, such that there existm(2)∈Nandkj(K), so that for anym≥m(2)

andx∈K, one has

TjLm (x)

mαj ≤kj(K), (2.31)

wherej∈ {2, 4}. Then, the convergence given in (2.30)is uniform onK.

3. Main results

In this section, we construct a general class of linear positive operators. LetR0=[0,∞) andJbe an interval withJ⊂R0. Let the sequence (am)m≥1so thatamx∈Jfor anym∈N

andx∈J. The indefinitely differentiable functionsa,b:J→Rhave the property:

b(x)>0 (3.1)

for anyx∈R0,

a(1)=0 (3.2)

and for any compactK⊂Jthe constantsM1(K),M2(K) depending onKexist, such that

_a(k)_(x)≤M 1(K),

_b(k)_(x)≤M 2(K)

(3.3)

for anyx∈Kandk∈N0. Then, it is known that

a(x)=

∞

n=0 1 n!a

(n)_(0)xn_,

b(x)=

∞

p=0 1 p!b

(p)_(0)xp

(3.4)

for anyx∈J.

Ifu,x,ux∈J, we calculate

a(u)b(ux)= _∞

n=0 1 n!a

(n)_(0)un

_∞

p=0 1 p!b

(p)_(0)(ux)p

(3.5)

and we take it to the form

a(u)b(ux)=

∞

k=0

pk(x)uk, (3.6)

where

pk(x)=

k

i=0 1 i!(k−i)!a

Remark 3.1. Ifu=1, then from (3.6), we obtain

a(1)bamx =

∞

k=0

pkamx (3.8)

for anym∈Nandx∈J.

Remark 3.2. We consider that the conditionsa(i)_(0)b(k−i)_{(0)/a(1)≥0,i∈ {0, 1,...,k}and} k∈N0, hold and then it results thata(1)pk(x)≥0 for anyx∈Jand anyk∈N0.

In the following, let a fixed functionw:R0→(0,∞), called the weight function, and the set functions

E(w)=f |f :R0→Rsuch thatw f is bounded on[0,∞)

. (3.9)

Forf ∈E(w), there exists a positive constantMsuch thatw(x)|f(x)| ≤Mfor anyx∈R0. Form∈Nandx∈J, and taking in the end (3.8) into account, we have

a(1)b1amx

∞

k=0

pkamx f _k

m

≤a(1)b1amx

∞

k=0

pkamx f _k

m

≤_wM

(x) 1 a(1)bamx

∞

k=0

pkamx =_wM

(x),

(3.10)

from where it results that the series (1/a(1)b(amx))∞k=0pk(amx)f(k/m) is convergent.

Definition 3.3. Form∈N, define the operatorLm:E(w)→F(J) by

Lmf (x)=_a 1

(1)bamx

∞

k=0

pkamx f _k

m

(3.11)

for any f ∈E(w) andx∈J, whereF(J) is a subset of the set of real functions defined on J.

In the following, we consider that for anyx∈J, we haveψxi∈E(w),i∈ {1, 2, 3, 4}.

Definition 3.5. Form∈Nandi∈ {1, 2, 3, 4}, defineTiby

TiLm (x)=miLmψxi (x)=mi_a(1)b1_a mx

∞

k=0

pkamx _mk −x i

(3.12)

for anyx∈J.

Lemma 3.6. One has

Lme0 (x)=1, (3.13)

Lme1 (x)=a_mmb (1)_a

mx

bamx x+

1 ma

(1)₍₁₎

a(1) ,

Lme2 (x)=

_a m

m

2_b(2)_a mx

bamx x 2₊ 1

mamma(1) + 2a (1)₍₁₎

a(1)

b(1)_a mx

bamx x+

1 m2

a(1)_{(1) +a}(2)₍₁₎ a(1) ,

Lme3 (x)=

_a m

m

3_b(3)_amx bamx x

3₊ 1 m

_a m

m

2

3a(1) + 3a(1)₍₁₎ a(1)

b(2)_amx bamx x

2

+_m1₂a_mma(1) + 6a

(1)_{(1) + 3a}(2)₍₁₎

a(1)

b(1)_a mx

bamx x+

1 m3

a(1)_{+ 3a}(2)_{(1) +a}(3)₍₁₎ a(1) ,

Lme4 (x)=

_a m

m

4_b(4)_amx bamx x

4₊ 1 m

_a m

m

3

6a(1) + 4a(1)₍₁₎ a(1)

b(3)_amx bamx x

3

+_m1₂

_a m

m

2

7a(1) + 18a(1)_{(1) + 6a}(2)₍₁₎ a(1)

b(2)_a mx

bamx x 2

+ 1

m3 am

m

a(1) + 14a(1)_{(1) + 18a}(2)_{(1) + 4a}(3)₍₁₎ a(1)

b(1)_amx bamx x

+_m1₄a

(1)_{(1) + 7a}(2)_{(1) + 6a}(3)_{(1) +a}(4)₍₁₎

a(1)

(3.14)

for anyx∈Jandm∈N.

Proof. The relation (3.13) results from (3.8). The proof of relations (3.14) follows imme-diately by differentiating (3.6) with respect tou, and after that take 1 foruandamxfor

Lemma 3.7. Forx∈Jandm∈N, the following hold

T0Lm (x)=1,

T1Lm (x)= −m

1−a_mmb(1)

amx

bamx

x+a (1)₍₁₎

a(1) ,

(3.15)

_T

2Lm (x)= −m2

1− _a

m

m

2_b(2)_a mx

bamx

x2

+m2

1−a_mmb(1)

amx

bamx

2x2−_m1 a(1) + 2_a a(2)(1)

(1) x

+mx+a

(1)_{(1) +a}(2)₍₁₎

a(1) ,

(3.16)

_T

4Lm (x)= −m4

1− _a

m

m

4_b(4)_a mx

bamx

x4

+m4

1− _a

m

m

3_b(3)_amx bamx

4x4₋ 1 m

6a(1) + 4a(1)₍₁₎ a(1) x

3

+m4

1− _a

m

m

2_b(2)_amx bamx

−6x4+ 4_m1 3a(1) + 3a (1)₍₁₎

a(1) x 3

−_m1₂7a(1) + 18a_a(1)(1) + 6a(2)(1)

(1) x

2

+m4

1−a_mmb(1)

amx

bamx

4x4₋₆1 m

a(1) + 2a(1)₍₁₎ a(1) x

3

+ 4_m1₂a(1) + 6a

(1)_{+ 3a}(2)₍₁₎

a(1) x 2

−_m1₃a(1) + 14a(1)+ 18_a a(2)(1) + 4a(3)(1)

(1) x

+ 3m2x2+a(1) + 10a

(1)_{(1) + 6a}(2)₍₁₎

a(1) mx

+a

(1)_{(1) + 7a}(2)_{(1) + 6a}(3)_{(1) +a}(4)₍₁₎

a(1) .

(3.17)

Proof. The proof follows immediately from (3.12) andLemma 3.6.

Theorem 3.8. Let f :R0→Rbe a function, f ∈E(w). Ifx∈R0, f is continuous inx,α2

andm(0)∈Nexist such that

1≤α2<2 (3.18)

andm2−α2|₁−₍a_m/m₎i₍b(i)₍a_mx₎/b₍a_mx₎₎|_{is bounded for any}m∈N_,m≥m_{(0), where}

i∈ {1, 2}, then

lim

m→∞

Assume that f is continuous onR0 and a compact intervalK⊂R0exists, such that there

existm(0)∈Nandli(K) so that for anym∈N,m≥m(0), andx∈K, one has

m2−α2₁− _a

m

m

i_b(i)_a mx

bamx

≤li(K), (3.20)

wherei∈ {1, 2}.

Then, the convergence given in (3.19) is uniform inKand

_{Lmf (x)−f(x)≤M(K)ωf;} 1

√

m2−α2

(3.21)

for anyx∈Kand anym≥m(0), whereM(K) is a constant depending onK.

Proof. Because m2−α2|₁−₍a_m/m₎i₍b(i)₍a_mx₎/b₍a_mx₎₎| _{is bounded for any}m∈N_,m≥

m(0), it results that (T2Lm)(x)/mα2_{is bounded for any}m∈N_,m≥m_{(0). Taking relation}

(3.16) into account, we apply now theCorollary 2.9. The proof is similar on a compact

intervalK.

Theorem 3.9. Let f :R0→Rbe a function, f ∈E(w). Ifx∈R0, f is a two times diff

eren-tiable function inxwith f(2)_{continuous inx,α}

2,α4andm(2)∈Nexist such that

1≤α2<2, (3.22)

2≤α4< α2+ 2, (3.23)

m4−α4|₁−₍a_m/m₎i₍b(i)₍a_mx₎/b₍a_mx₎₎| _{is bounded for any} m∈N_,m≥m_{(2), where}i∈

{1, 2, 3, 4}, then

lim

m→∞m

2−α2L

mf (x)−f(x)−_m1T1Lm (x)f(1)(x)− 1 2m2

T2Lm (x)f(2)(x)

=0.

(3.24)

In addition, if the limit lim_m

→∞((T2Lm)(x)/m

α2_{) exists and}

lim

m→∞

T2Lm (x)

mα2 =B2(x)∈R, (3.25)

then

lim

m→∞m

2−α2L_mf ₍x₎−f₍x₎− 1

m

T1Lm (x)f(1)(x)

=1

2B2(x)f

(2)_{(x). (3.26)}

Assume that f is a two-times differentiable function onR0with f(2)continuous onR0and

a compact intervalK⊂R0exists, such that there existm(2)∈Nandli(K) so that for any m≥m(2) andx∈K, one has

m4−α4₁− _a

m

m

i_b(i)_amx bamx

≤li(K), (3.27)

Proof. From (3.23), it results that 4−α4>2−α2, and then we have that m2−α2|1− (am/m)i(b(i)(amx)/b(amx))|,i∈{1, 2}are bounded for anym≥m(2). So (T2Lm)(x)/mα2

is bounded for anym≥m(2). Using the same idea from the proof ofTheorem 3.8, we have that (T2Lm)(x)/mα2and (T4Lm)(x)/mα4are bounded for anym∈N,m≥m(2), and

then we applyCorollary 2.10.

Now, we give some applications wheream=mfor anym∈N. In the following, by

par-ticularization and applying Theorems3.8and3.9, we can obtain approximation theorems and Voronovskaja-type theorems for some known operators. Because every application is a simple substitute in the theorems of this section, we will not replace anything.

Application 3.10. Ifa(x)=1 andb(x)=ex_,x∈R

0, we obtain the Mirakjan-Favard-Sz´asz operators (see [3–5]).

Application 3.11. Ifa(x)=g(x)=∞n=0anxnandb(x)=ex,x∈R0, we obtain the opera-tors considered by Jakimovski and Leviatan in the paper [1].

Application 3.12. Ifa(x)=g(x)=1 andb(x)=coshx=∞k=0(1/(2k)!)x2k,x∈R0, then we get the operators considered by Le´sniewicz and Rempulska in the paper [6].

Application 3.13. Ifa(x)=g(x)=1 andb(x)=sinhx=∞k=0(1/(2k+ 1)!)x2k+1,x∈R0, we get the operators

Amf (x)=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

1 sinhmx

∞

k=0

(mx)2k+1 (2k+ 1)!f

2k+ 1 m

ifx >0,

f(0) ifx=0,

(3.28)

wherem∈Nandx∈R0. The operators of this type are introduced and studied by Rem-pulska and Skorupka in the paper [7].

Application 3.14. Ifa(x)=b(x)=g(x)=coshx,x∈R0, we obtain the operators studied by Ciupa in [8].

Application 3.15. Ifa(x)=g(x)=∞n=0anxnandb(x)=coshx,x∈R0, we get the opera-tors constructed by Ciupa in the paper [9], and studied in [9,10].

Application 3.16. Ifa(x)=1 andb(x)=bm((1/m)x),x∈R0 andm∈N, we obtain the operators studied in the paper [11].

References

[1] A. Jakimovski and D. Leviatan, “Generalized Sz´asz operators for the approximation in the infi-nite interval,” Mathematica, vol. 11 (34), pp. 97–103, 1969.

[2] O. T. Pop, “About some linear and positive operators defined by infinite sum,” Demonstratio

Mathematica, vol. 39, no. 2, pp. 377–388, 2006.

[3] J. Favard, “Sur les multiplicateurs d’interpolation,” Journal de Math´ematiques Pures et Appliqu´ees, vol. 23, pp. 219–247, 1944.

[4] G. M. Mirakjan, “Approximation of continuous functions with the aid of polynomials,” Doklady

Akademii Nauk SSSR, vol. 31, pp. 201–205, 1941 (Russian).

[5] O. Sz´asz, “Generalization of S. Bernstein’s polynomials to the infinite interval,” Journal of

[6] M. Le´sniewicz and L. Rempulska, “Approximation by some operators of the Sz´asz-Mirakjan type in exponential weight spaces,” Glasnik Matematiˇcki, vol. 32(52), no. 1, pp. 57–69, 1997. [7] L. Rempulska and M. Skorupka, “Approximation theorems for some operators of the

Sz´asz-Mirakjan type in exponential weight spaces,” Publikacije Elektrotehniˇckog Fakulteta. Univerzitet

u Beogradu, vol. 7, pp. 36–44, 1996.

[8] A. Ciupa, “Approximation by a generalized Sz´asz type operator,” Journal of Computational

Anal-ysis and Applications, vol. 5, no. 4, pp. 413–424, 2003.

[9] A. Ciupa, “A positive linear operator for approximation in exponential weight spaces,” in

Math-ematical Analysis and Approximation Theory. Proceedings of the 5th Romanian-German seminar on approximation theory and Its applications, pp. 85–96, Burg, Sibiu, Romania, 2002.

[10] A. Ciupa, “A Voronovskaya-type theorem for a positive linear operator,” International Journal of

Mathematics and Mathematical Sciences, vol. 2006, Article ID 42368, 7 pages, 2006.

[11] O. T. Pop, “About some linear operators defined by infinite sum,” Analele S¸tiintifice ale

Universi-tatii Ovidius Constant¸a, vol. 15, no. 2, 2007.

Ovidiu T. Pop: Department of Mathematics, National College “Mihai Eminescu”, 5 Mihai Eminescu Street, 440014 Satu Mare, Romania; Department of Mathematics and Informatics, Vest University “Vasile Goldis¸” of Arad, Branch of Satu Mare, 26 Mihai Viteazul Street, 440030 Satu Mare, Romania