# Nabla discrete fractional Grüss type inequality

N/A
N/A
Protected

Share "Nabla discrete fractional Grüss type inequality"

Copied!
9
0
0

Full text

(1)

## inequality

1

2*

3

### and Kenan Ta¸s

2

*Correspondence:

billur@cankaya.edu.tr

2Department of Mathematics and

Computer Science, Çankaya University, Ankara, 06810, Turkey Full list of author information is available at the end of the article

Abstract

Properties of the discrete fractional calculus in the sense of a backward diﬀerence are introduced and developed. Here, we prove a more general version of the Grüss type inequality for the nabla fractional case. An example of our main result is given.

MSC: Primary 39A12; 34A25; 26A33; secondary 26D15; 26D20

Keywords: nabla discrete fractional calculus; nabla discrete Grüss inequality

1 Introduction

The Grüss inequality is of great interest in diﬀerential and diﬀerence equations as well as many other areas of mathematics [–]. The classical inequality was proved by Grüss in  []: iff andgare continuous functions on [a,b] satisfyingϕf(t)≤andγ

g(t)≤for allt∈[a,b], then

ba b

a

f(t)g(t)dt–  (ba)

b

a

f(t)dt

b

a

g(t)dt≤ 

(ϕ)(γ). ()

The literature on Grüss type inequalities is extensive, and many extensions of the clas-sical inequality () have been intensively studied by many authors in the st century [–]. Here we are interested in Grüss type inequalities in the nabla fractional calculus case.

We begin with basic deﬁnitions and notation from the nabla calculus that are used in this paper. The delta calculus analog has been studied in detail, and a general overview is given in []. The domains used in this paper are denoted byNawherea∈R. This is a discrete time scale with a graininess of , so it is deﬁned as

Na:={a,a+ ,a+ , . . .}.

ForNa, we use the terminology that a function’s domain, in the case studied here, is based ata. We use theρ-function, or backward jump function, from the time scale asρ:Na

Na, given byρ(t) :=max{a,t– }. We deﬁne the backward diﬀerence operator, or the nabla operator (∇), for a functionf :Na→Rby

(∇f)(t) :=f(t) –(t)=f(t) –f(t– ).

(2)

In this paper, we use the convention that

f(t) = (∇f)(t).

We then deﬁne higher order diﬀerences recursively by

nf(t) :=n–f(t)

fort∈Na+nwheren∈N. We take as convention that∇is the identity operator.

Based on these preliminary deﬁnitions, we sayFis an anti-nabla diﬀerence off onNaif and only if∇F(t) =f(t) fort∈Na+. We then deﬁne the deﬁnite nabla integral off :Na→R by

d

c

f(t)∇t:=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d

t=c+f(t), ifc<d,

, ifc=d,

ct=d+f(t), ifd<c,

wherec,d∈Na. We now give the fundamental theorem of nabla calculus.

Theorem .(Fundamental theorem of nabla calculus) Let f :Na→Rand let F be an

anti-nabla diﬀerence of f onNa,then for any c,d∈Nawe have

d

c

f(t)∇t=F(d) –F(c).

The nabla product rule for two functionsu,ν:Na→Randt∈Na+is given by

u(t)ν(t)=u(t)∇v(t) +(t)∇u(t).

This immediately leads to the summation by parts formula for the nabla calculus:

c

s=b+

u(t)∇v(t) =u(t)v(t)|cbc

s=b+

(t)∇u(t).

Discrete fractional initial value problems have been intensively studied by many authors. Some of the very recent results are in [–].

Now that we have established the basic deﬁnitions of the nabla calculus, we will move on to extending these deﬁnitions to the fractional case and establish deﬁnitions for the fractional sum and fractional diﬀerence which are analogues to the continuous fractional derivative.

In order to do this, we remind the reader of the rising factorial function. Forn,t∈N, the rising factorial function is deﬁned by

tn:=t(t+ )· · ·(t+n– ) = (t+n– )! (t– )! .

(3)

Deﬁnition (Rising function) Fort,α∈R, the rising function is deﬁned by

:=(t+α) (t) .

To motivate the deﬁnition of a fractional sum, we look at the deﬁnition of the integral sum derived from the repeated summation rule.

Deﬁnition  For any given positive real numberα, the (nabla) left fractional sum of order α>  is deﬁned by

∇–α

a f(t) =  (α)

t

s=a+

tρ(s)α–f(s),

wheret∈ {a+ ,a+ , . . .}, and

∇–α

a f(t) =  (α)

t

s=a

tρ(s)α–f(s),

wheret∈ {a,a+ ,a+ , . . .}.

Theorem .([]) Let f be a real-valued function deﬁned onNaandα,β> .Then

∇–α

aaβf(t) =∇a–(α+β)f(t) =∇aβaαf(t).

Deﬁnition (Fractional Caputo like nabla diﬀerence) Forμ> ,m–  <μ<m,m=μ (where·is the ceiling function),m∈N,v=mμ, we have the following:

μ

a∗f(t) =∇av

mf(t), tNa.

We also will use the following discrete Taylor formula.

Theorem .([]) Let f :Z→Rbe a function and let a∈Z.Then,for all t∈Zwith t∈Na+mwe have the representation

f(t) = m–

k=

(ta)k

k! ∇

kf(a) +  (m– )!

t

τ=a+

(tτ+ )m–∇mf(τ). ()

The following discrete backward fractional Taylor formula will be useful.

Theorem .([]) Let f:Z→Rbe a function and let a∈Z.Here m–  <μ<m,m=μ, μ> .Then for all t∈Na+mwe have the representation

f(t) = m–

k=

(ta)k

k! ∇

kf(a) +(μ)

t

τ=a+

(4)

Corollary .(To Theorem .) Additionally assume thatkf(a) = ,for k= , , , . . . ,

m– .Then

f(t) =  (μ)

t

τ=a+

(tτ+ )μ–∇(μa+)f(τ), ∀ta+m. ()

Also, we have the following.

Theorem .([]) Let p∈N,ν>p,a∈N.Then

pν

a f(t)

=∇a–(νp)f(t) ()

for t∈Na.

Now, we give the following discrete backward fractional extended Taylor formula.

Theorem .([]) Let f :Z→Rbe a function and let a∈Z+.Here m–  <μ<m,m=

μ,μ> .Consider p∈N:μ>p.Then for all ta+m,t∈N,we have the representation

pf(t) = m–

k=p

(ta)kp (kp)! ∇

kf(a) +(μp)

t

τ=a+

(tτ+ )μp–∇(μa+)f(τ). ()

Corollary .(To Theorem .) Additionally assume thatkf(a) = ,for k=p, . . . ,m– .

Then

pf(t) =(μp)

t

τ=a+

(tτ+ )μp–μ

(a+)∗f(τ), ∀ta+m,t∈N. ()

Remark . Letfbe deﬁned on{am+ ,am+ , . . . ,b}, wherebais an integer. Then () and () are valid only fort∈ {a+m, . . . ,b}. Here we must assume thata+m<b.

2 Main results

We present the following discrete nabla Grüss type inequality.

Theorem . Let m–  <μ<m,m=μnon-integer,μ> ;p,a∈Z+withμ>p.Consider

b∈Nsuch that a+m<b.Let f and g two real-valued functions deﬁned on{am+ ,a

m+ , . . . ,b}.Here j∈ {a+m, . . . ,b}.Assume thatkf(a) =kg(a) = ,for k=p+ , . . . ,m– 

and

m≤ ∇

μ

(a+)∗f(s)≤M, m≤ ∇

μ

(a+)∗g(s)≤M

for s∈ {a+ , . . . ,b},then

bam

b

j=a+m+

pf(j)pg(j)  (bam)

b

j=a+m+

pf(j)

b

j=a+m+

pg(j)

(5)

where m,m,M,and Mare positive constants,and

C:=

Q(bam)(bam+ ) + (bam)μp, C:=

(ba+ )μp+– (m)μp+

μp+  – (bam)(μp+ )

,

Q:=

ba+

s=m+

(s)μp

 

.

Proof By Theorem ., we have

pf(j) =pf(a) +(μp)

j

τ=a+

(jτ+ )μp–μ

(a+)∗f(τ)

,

pg(j) =pg(a) +(μp)

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

forj∈ {a+m+ ,a+m+ , . . . ,b}. We get

pf(j)pg(j)=  [(μp)]

j

τ=a+

(jτ+ )μp–μ

(a+)∗f(τ)

·

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ ∇ pf(a)

(μp)

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ ∇ pg(a)

(μp)

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

+∇pf(a)pg(a).

If we take the sum froma+m+  tobwe get

b

j=a+m+

pf(j)pg(j)=pf(a)pg(a)

+ ∇ pf(a)

(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ ∇ pg(a)

(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–μ

(a+)∗f(τ)

+ 

[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

·

j

τ=a+

(jτ+ )μp–μ

(a+)∗g(τ)

(6)

Then

bam

b

j=a+m+

pf(j)pg(j)

= 

bam

pf(a)pg(a)

+ 

(bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

·

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ ∇

pf(a)

(bam)(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ ∇

pg(a)

(bam)(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

. ()

On the other hand,

bam

b

j=a+m+

pf(j)

=∇pf(a)

+ 

(bam)(μp) b

j=a+m+

j

τ=a+

(jτ+ )μp–μ

(a+)∗f(τ)

and

bam

b

j=a+m+

pg(j)

=∇pg(a)

+ 

(bam)(μp) b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

.

Multiplying the above two terms, we get

 (bam)

b

j=a+m+

pf(j)

b

j=a+m+

pg(j)

=∇pf(a)∇pg(a)

+ 

(bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

(7)

·

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ (∇

pf(a))

(bam)(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)g(τ)

+ (∇

pg(a))

(bam)(μp)

b

j=a+m+

j

τ=a+

(jτ+ )μp–∇(μa+)f(τ)

. ()

Using () and (), we obtain

 (bam)

b

j=a+m+

pf(j)pg(j)

– 

(bam)

b

j=a+m+

pf(j)

b

j=a+m+

pg(j)

MM

(bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–

j

τ=a+

(jτ+ )μp–

mm

(bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–

·

b

j=a+m+

j

τ=a+

(jτ+ )μp–

= MM (bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–

mm

(bam)[(μp)]

b

j=a+m+

j

τ=a+

(jτ+ )μp–

.

Next, in order to calculate the last two sums, we ﬁrst observe that

j

τ=a+

(jτ+ )μp–=

j

a

(jτ+ )μp–∇τ

=  μp

(ja+  +μp)

(ja+ ) –(μp+ )

.

Therefore,

b

j=a+m+

j

τ=a+

(jτ+ )μp–

=  μp

b

j=a+m+

(ja+  +μp)

(ja+ ) –(μp+ )

=  μp

(m+  +μp) (m+ ) +· · ·+

(ba+  +μp) (ba+ )

bam

(8)

=  μp

(m+ )μp+ (m+ )μp+· · ·+ (ba+ )μp

bam

μp (μp+ )

=  μp

ba+

s=m+

(s)μpbam

μp (μp+ )

=  μp

ba+

m+

τμpτbam

μp (μp+ )

=  μp

(ba+ )μp+– (m+ )μp+

μp+ 

bam

μp (μp+ ).

Similarly, we get

b

j=a+m+

j

τ=a+

(jτ+ )μp–

≤ 

(μp)

b

j=a+m+

(jτ+ )μp

+ 

(μp)

μp(bam),

which by use of Hölder’s inequality transforms into

b

j=a+m+

j

τ=a+

(jτ+ )μp–

≤(bam+ )/

(μp)

ba+

s=m+

(s)μp

/

+bam (μp)

μp

.

By hypothesis,

b

j=a+m+

j

τ=a+

(jτ+ )μp–

Q(bam+ )/

(μp) +

(bam)[μp]

(μp) .

Consequently, we get

bam

b

j=a+m+

pf(j)pg(j)

– 

(bam)

b

j=a+m+

pf(j)

b

j=a+m+

pg(j)

MMC–mmC

(bam)[(μp+ )].

Example Letμ= .,m= ,p= ,b= ,a=  in Theorem .. Deﬁne

(9)

wheret∈ {–, –, , , . . . , }. Herej∈ {, , }. Using Theorem ., we obtain

 

t=

∇(t– )(t– )

t=

∇(t– )

t=

∇(t– )

MMC–mmC

[(.)] ,

whereM= ,M= ,m= ,m=  andC= ,,C= ,, andQ= ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AFG conceived the study, and participated in its design and coordination. BK carried out the mathematical studies and participated in the sequence alignment and drafted the manuscript. ACP participated in the design of the study. KT conceived the study, and participated in its design and coordination. All authors read and approved the ﬁnal manuscript.

Author details

1Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey.2Department of Mathematics and

Computer Science, Çankaya University, Ankara, 06810, Turkey. 3Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA.

Received: 20 February 2013 Accepted: 24 January 2014 Published:20 Feb 2014

References

1. Boros, G, Moll, V: Irresistible Integrals: Symbols, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004)

2. Cerone, P, Dragomir, SS: A reﬁnement of the Grüss inequality and applications. Tamkang J. Math.38, 37-49 (2007) 3. Grüss, G: Über das Maximum des absoluten Betrages von 1

ba

b

af(x)g(x)dx

1 (ba)2

b af(x)dx

b

ag(x)dx. Math. Z.39,

215-226 (1935)

4. Mercer, AMD: An improvement of the Grüss inequality. JIPAM. J. Inequal. Pure Appl. Math.6(4), Article ID 93 (2005) (electronic)

5. Mitrinovi´c, DS, Pe˘cari´c, JE, Fink, AM: Classical and New Inequalities in Analysis. Kluwer Acadademic, Dordrecht (1993) 6. Anastassiou, GA: Multivariate Fink type identity and multivariate Ostrowski, comparison of means and Grüss type

inequalities. Math. Comput. Model.46, 351-374 (2007)

7. Bohner, M, Matthews, T: The Grüss inequality on time scales. Commun. Math. Anal.3, 1-8 (2007) (electronic) 8. Dragomir, SS: A Grüss type discrete inequality in inner product spaces and applications. J. Math. Anal. Appl.250,

494-511 (2000)

9. Liu, Z: Notes on a Grüss type inequality and its applications. Vietnam J. Math.35, 121-127 (2007) 10. Peri´c, I, Raji´c, R: Grüss inequality for completely bounded maps. Linear Algebra Appl.390, 292-298 (2004)

11. Pe˘cari´c, JE, Tepe˘s, B: On the Grüss type inequalities of Dragomir and Fedotov. J. Inequal. Pure Appl. Math.4(5), Article ID 91 (2003) (electronic)

12. Pe˘cari´c, JE, Tepe˘s, B: A note on Grüss type inequality in terms of-seminorms. Pril. - Maked. Akad. Nauk. Umet., Odd. Mat.-Teh. Nauki23/24, 29-35 (2004)

13. Goodrich, CS: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl.59, 3489-3499 (2010)

14. Goodrich, CS: Existence and uniqueness of solutions to a fractional diﬀerence equation with nonlocal conditions. Comput. Math. Appl.61, 191-202 (2011)

15. Goodrich, CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl.385, 111-124 (2012) 16. Abdeljawad, T, Atici, FM: On the deﬁnitions of nabla fractional operators. Abstr. Appl. Anal.2012, Article ID 406757,

Special Issue (2012)

17. Anderson, DR: Taylor polynomials for nabla dynamic equations on time scales. Panam. Math. J.12, 17-27 (2002) 18. Anastassiou, GA: Nabla discrete fractional calculus and nabla inequalities. Math. Comput. Model.51, 562-571 (2010)

10.1186/1029-242X-2014-86

References

Related documents

In particular we highlight three areas with the intention to improve communication between those researching lipreading; the effects of interchanging between speech read- ing

Figure 2a: Histogram of the results for TF-Ag plasma levels in the reference group... Figure 2b: Histogram of the results for MP-TF activity plasma levels in the reference group.

In 1978 Pollard developed an algorithm using “Monte-Carlo” method to solve ECDLP called Pollard Rho attack which is a random method to compute discrete

A likely reason for the diminished transcription of Wnt target genes in ROSA26::  -catenin mutant embryos is the reduced expression levels of Tcf1 and Lef1.. Since

When the saphenous vein is absent or not suitable for bypass graft- ing, polytetraﬂ uoroethylene or Dacron is a good alterna- tive for femoropopliteal bypass material above the

The average values of ecological factors listed in Table 2, including annual air temperature, relative humidity, annual precipitation, annual average solar

The utilization of two macrophytes ( Lemna and Wolffia ) in their quality as a biofilter in RAS increased dissolved oxygen in the water, significantly decreased the

Abstract: This survey was carried out in the Dormaa Ahenkro District of Brong Ahafo Region in Ghana to assess the postharvest handling practices and the use of different