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R E S E A R C H

Open Access

Nabla discrete fractional Grüss type

inequality

A Feza Güvenilir

1

, Billur Kaymakçalan

2*

, Allan C Peterson

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– ).

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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– )! .

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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+

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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)

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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

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=  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

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ba

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af(x)g(x)dx

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References

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