fractional difference inequalities of volterra type

International Journal of Pure and Applied Mathematics————————————————————————–Volume 70 No. 2 2011, 137-149

FRACTIONAL DIFFERENCE INEQUALITIES OF

VOLTERRA TYPE

G.V.S.R. Deekshitulu1 §, J. Jagan Mohan2, P.V.S. Anand3

1Department of MathematicsKakinada Institute of Engineering and Technology - II

Korangi, Kakinada, 533 461, INDIA2Department of Mathematics

Vignan’s Institute of Information TechnologyDuvvada, Visakhapatnam, 530 049, INDIA3Center for Mathematical Sciences - DST

C.R. Rao Advanced Institute of Mathematics,Statistics and Computer Science (AIMSCS), UOH Campus

Professor C.R. Rao Road, Gachibowli Hyderabad, 500 046 A.P., INDIA

Abstract: In this work, fractional difference equation of Volterra type is con-sidered and some fundamental inequalities and comparison results are obtained.

AMS Subject Classification: 39A10, 39A99Key Words: fractional difference equations, inequalities, Volterra equations

1. Introduction

Fractional calculus has gained importance during the past three decades due toits applicability in diverse fields of science and engineering and lot of literatureis available on the applications of fractional calculus in modeling mechanicaland electrical properties of real materials. But the study of Theory of fractionaldifferential equations was initiated and existence and uniqueness of solutions fordifferent types of fractional differential equations have been established recently[8]. Much of literature is not available on fractional integro differential equationsalso, though theory of integro differential equations [9] has been almost all

Received: November 27, 2010 c© 2011 Academic Publications, Ltd.§Correspondence author

138 G.V.S.R. Deekshitulu, J.J. Mohan, P.V.S. Anand

developed parallel to theory of differential equations. Very little progress hasbeen made to develop the theory of analogous fractional difference equations,particularly of Volterra type.

In this direction, Diaz and Osler [5] defined the fractional difference bythe rather natural approach of allowing the index of differencing, in the stan-dard expression for the nth difference, to be any real or complex number.Later, R.Hirota [12], defined the difference operator ∇α using Taylor’s series.G.V.S.R.Deekshitulu and J.Jagan Mohan [3] modified the definition of AtsushiNagai [1] for 0 < α ≤ 1 in such a way that the expression for ∇α does notinvolve any difference operator and derived some basic inequalities and com-parison theorems.

The inequalities and comparison principles which provide explicit bounds onunknown functions play a very important role in the theory of finite differenceequations. In the present paper, the authors considered an initial value problemof volterra type of fractional order and some basic difference inequalities andcomparison results are obtained.

2. Preliminaries

Let un be any function defined for n ∈ N+0 where N

+a = {a, a + 1, a + 2, ...} for

a ∈ Z. Hirota [12] took the first n terms of Taylor series of ∆α−n = ε−α(1−B)α

and gave the following definition.

Definition 1. Let α ∈ R. Then difference operator of order α is definedby

∆α−nun =

ε−α

n−1∑

j=0

(

α

j

)

(−1)jun−j, α 6= 1,2,...,

ε−m

m∑

j=0

(

m

j

)

(−1)jun−j , α = m ∈ Z>0.

(1)

Here(

an

)

, (a ∈ R, n ∈ Z) stands for a binomial coefficient defined by

(

a

n

)

=

Γ(a + 1)

Γ(a − n + 1)Γ(n + 1), n > 0,

1, n = 0,0, n < 0.

(2)

In 2002, Atsushi Nagai [1] introduced another definition of fractional differencewhich is a slight modification of Hirota’s fractional difference operator.

FRACTIONAL DIFFERENCE INEQUALITIES OF... 139

Definition 2. Let α ∈ R and m be an integer such that m − 1 < α ≤ m.The difference operator ∆∗,−n of order α is defined as

∆α∗,−nun = ∆α−m

−n ∆m−nun = εm−α

n−1∑

j=0

(

α − m

j

)

(−1)j∆m−(n−j)un−j. (3)

G.V.S.R.Deekshitulu and J.Jagan Mohan [3] gave a more convenient formof the above definition by taking m = 1 and reorganizing the terms as follows.

Definition 3. Let α ∈ R such that 0 < α ≤ 1. The difference operator ∇of order α is defined as

∇αun =

n−1∑

j=0

(

j − α

j

)

∇un−j. (4)

Remark 1. For any α ∈ R (0 < α ≤ 1),

∇−αun =

n−1∑

j=0

(

j + α

j

)

∇un−j. (5)

Further ∇αun and ∇−αun can be expressed in the terms of the argumentsof un as for any α ∈ R (0 < α ≤ 1),

∇αun = un −

(

n − 1 − α

n − 1

)

u0 − α

n−1∑

j=1

1

(j − α)

(

j − α

j

)

un−j (6)

and

∇−αun = un −

(

n − 1 + α

n − 1

)

u0 + α

n−1∑

j=1

1

(j + α)

(

j + α

j

)

un−j (7)

i.e.

∇−αun =

n∑

j=1

(

n − j + α − 1

n − j

)

uj −

(

n − 1 + α

n − 1

)

u0. (8)

Remark 2. The difference operator ∇ of order α satisfies the followingproperties.

i For any real numbers α and β, ∇α∇βun = ∇α+βun.


ii For any constant ’c’, ∇α[cun +vn] = c∇αun+∇αvn where vn be any functiondefined for n ∈ N

+0 .

iii For α ∈ R, ∇α(unvn) =∑n−1

m=0

(

αm

)

[∇α−mun−m][∇mvn].

iv ∇αu0 = 0 and ∇αu1 = u1 − u0 = ∇u1.

v ∇α∇−αun = ∇−α∇αun = un − u0.

vi ∇α∇−α(un − u0) = ∇−α∇α(un − u0) = un − u0.

Definition 4. Let f(n, r, s) be a function defined for n ∈ N+0 , 0 ≤ r < ∞,

0 ≤ s < ∞ and g(n,m, r) be a function defined for n, m ∈ N+0 , m ≤ n,

0 ≤ r < ∞. Let vn be a function defined for n ∈ N+0 . Then a nonlinear

fractional difference equation of Volterra type is of the form

∇αvn+1 = f(n, vn,

n−1∑

m=0

g(n,m, vm)), v(0) = v0. (9)

3. Main Results

In this section we deal with a nonlinear fractional difference equation of Volterratype of the form (9) where f(n, r, s) is a nonnegative and nondecreasing functionwith respect to r and s, 0 ≤ r, s < ∞ for any fixed n ∈ N

+0 and g(n,m, r) is a

nonnegative and nondecreasing function with respect to r, 0 ≤ r < ∞ for anyfixed n, m ∈ N

+0 .

Theorem 5. Let vn and wn be any two nonnegative functions defined forn ∈ N

+0 . Suppose that for n ∈ N

+0 and 0 < α ≤ 1, the inequalities

∇αvn+1 ≤ f(n, vn,

n−1∑

m=0

g(n,m, vm)), (10)

∇αwn+1 ≥ f(n,wn,

n−1∑

m=0

g(n,m,wm)) (11)


hold. Then v0 ≤ w0 implies

vn ≤ wn (12)

for all n ∈ N+0 .

Proof. Suppose that (12) is not true. Then, because of v0 ≤ w0, there existsa k ∈ N

+0 such that vm ≤ wm for m ≤ k and vk+1 > wk+1. From the monotone

properties of f and g, for m ≤ k,

vm ≤ wm ⇒ g(k,m, vm) ≤ g(k,m,wm)

⇒

k−1∑

m=0

g(k,m, vm) ≤

k−1∑

m=0

g(k,m,wm) (since g is nonnegative)

⇒ f(k, vk,

k−1∑

m=0

g(k,m, vm)) ≤ f(k,wk,

k−1∑

m=0

g(k,m,wm)).

Thus

∇αvk+1 ≤ ∇αwk+1. (13)

Now consider

∇αwk+1 = wk+1 −

(

k − α

k

)

w0 − α

k∑

j=1

1

(j − α)

(

j − α

j

)

wk+1−j

< vk+1 −

(

k − α

k

)

v0 − α

k∑

j=1

1

(j − α)

(

j − α

j

)

vk+1−j

= ∇αvk+1

which is a contradiction to (13). Hence the proof.

Corollary 6. Let f(n,r) be a nonnegative function defined for n ∈ N+0 ,

0 ≤ r < ∞ and g(n,m, r) be as above. Let vn and wn be any two nonnegativefunctions defined for n ∈ N

+0 . Suppose that for n ∈ N

+0 and 0 < α ≤ 1, the

inequalities

∇αvn+1 ≤ f(n, vn) +

n−1∑

m=0

g(n,m, vm), (14)

∇αwn+1 ≥ f(n,wn) +n−1∑

m=0

g(n,m,wm) (15)


hold. If for any n ∈ N+0 and 0 < α ≤ 1,

f(n, vn) − f(n,wn) ≤ −α(vn − wn) (16)

then v0 ≤ w0 implies

vn ≤ wn (17)

for all n ∈ N+0 .

Proof. Suppose that (17) is not true. Then because of v0 ≤ w0 there existsa k ∈ N

+0 such that vm ≤ wm for m ≤ k and

vk+1 > wk+1. (18)

From the monotone property of g, for m ≤ k,

vm ≤ wm ⇒ g(k,m, vm) ≤ g(k,m,wm)

⇒

k−1∑

m=0

g(k,m, vm) ≤

k−1∑

m=0

g(k,m,wm).

Now using Remark (1) and (16),

vk+1 ≤(

k − α

k

)

v0 + α

k∑

j=1

1

(j − α)

(

j − α

j

)

vk+1−j + f(k, vk) +

k−1∑

m=0

g(k,m, vm)

=

(

k − α

k

)

v0 + α

k∑

j=2

1

(j − α)

(

j − α

j

)

vk+1−j + αvk + f(k, vk) +

k−1∑

m=0

g(k,m, vm)

≤

(

k − α

k

)

w0 + α

k∑

j=2

1

(j − α)

(

j − α

j

)

wk+1−j + αwk + f(k,wk)

+k−1∑

m=0

g(k,m,wm)

=

(

k − α

k

)

w0 + α

k∑

j=1

1

(j − α)

(

j − α

j

)

wk+1−j + f(k,wk) +k−1∑

m=0

g(k,m,wm)

=wk+1,

which is a contradiction to (18). Hence the proof.


Theorem 7. Let wn be solution of the difference equation

∇αwn+1 = f(n,wn,

n−1∑

m=0

g(n,m,wm)), w(0) = w0 (19)

for all n, m ∈ N+0 and 0 < α ≤ 1. Suppose that the inequality

∇αvn+1 ≤ f(n, vn,

n−1∑

m=0

g(n,m, vm)) (20)

satisfied for all n ∈ N+0 and 0 < α ≤ 1, where vn is a nonnegative function

defined for n ∈ N+0 such that v0 ≤ w0. Then

vn ≤ wn (21)

for all n ∈ N+0 .

Proof. Consider (19) and (20). Applying Theorem 5, since v0 ≤ w0 weobtain vn ≤ wn.

Theorem 8. Let f1(n, r, s) and f2(n, r, s) be two nonnegative and nonde-creasing function with respect to r and s, 0 ≤ r, s < ∞ for any fixed n ∈ N

+0

and g1(n,m, r) and g2(n,m, r) be a nonnegative and nondecreasing functionwith respect to r, 0 ≤ r < ∞ for any fixed n, m ∈ N

+0 . Let yn be a function

defined for n ∈ N+0 and that

f1(n, xn,

n−1∑

m=0

g1(n,m, xm)) ≤ ∇αyn+1 ≤ f2(n, yn,

n−1∑

m=0

g2(n,m, ym)) (22)

for all n ∈ N+0 and 0 < α ≤ 1. Let vn and wn be the solutions of the difference

equations

∇αvn+1 = f1(n, xn,

n−1∑

m=0

g1(n,m, xm)), v(0) = v0, (23)

∇αwn+1 = f2(n, yn,

n−1∑

m=0

g2(n,m, ym)), w(0) = w0. (24)

and suppose that v0 ≤ y0 ≤ w0. Then

vn ≤ yn ≤ wn, n ∈ N+0 . (25)


Proof. Consider the second part of (22) and (24). i.e.

∇αyn+1 ≤ f2(n, yn,

n−1∑

m=0

g2(n,m, ym)),

∇αwn+1 = f2(n, yn,

n−1∑

m=0

g2(n,m, ym)).

Applying Theorem 7, since y0 ≤ w0 we obtain the right half of the inequality in(25) i.e yn ≤ wn. A similar argument yields the left half of the inequality (25).

Theorem 9. Let xn and yn be solutions of the difference equations

∇αxn+1 = f1(n, xn,

n−1∑

m=0

g1(n,m, xm)), x(0) = x0, (26)

∇αyn+1 = f2(n, yn,

n−1∑

m=0

g2(n,m, ym)), y(0) = y0 (27)

where the functions xn, yn, g1(n,m, r), g2(n,m, r), f1(n, r, s) and f2(n, r, s) aredefined for n, m ∈ N

+0 , m ≤ n, 0 ≤ r < ∞, 0 ≤ s < ∞ and satisfy the

conditions

|g1(n,m, xm) − g2(n,m, ym)| ≤ g(n,m, |xm − ym|), (28)

|f1(n, xm, um) − f2(n, ym, vm)| ≤ f(n, |xm − ym|, |um − vm|) (29)

for all n, m ∈ N+0 and m ≤ n. Here un and vn are any nonnegative functions

defined for n ∈ N+0 . Let wn be solution of the difference equation

∇αwn+1 = f(n,wm,

n−1∑

m=0

g(n,m,wm), w(0) = w0 (30)

for all n, m ∈ N+0 and 0 < α ≤ 1. If

|x0 − y0| ≤ w0 (31)

then

|xn − yn| ≤ wn (32)

for all n ∈ N+0 .


Proof. Define a function zn by zn = |xn − yn|. Then z0 = |x0 − y0| ≤ w0.On account of the monotonicity of f(n, r, s), we obtain, using Remark 2(v),

z1 = |x1 − y1| = |x0 + f1(0, x0, 0) − y0 − f2(0, y0, 0)|

≤ |x0 − y0| + |f1(0, x0, 0) − f2(0, y0, 0)|

≤ |x0 − y0| + f(0, |x0 − y0|, 0)

≤ w0 + f(0, w0, 0)

= w0 + ∇αw1 = w1.

If the inequality zn ≤ wn is fulfilled for n = 1, 2, ...k, it follows by the mono-tonicity of f(n, r, s) that

zk+1 = |

(

k − α

k

)

x0 + α

k∑

j=1

1

(j − α)

(

j − α

j

)

xk+1−j

+f1(k, xk,

k−1∑

m=0

g1(k,m, xm))

−

(

k − α

k

)

y0 − α

k∑

j=1

1

(j − α)

(

j − α

j

)

yk+1−j

−f2(k, yk,

k−1∑

m=0

g2(k,m, ym))|

= |

(

k − α

k

)

(x0 − y0) + α

k∑

j=1

1

(j − α)

(

j − α

j

)

(xk+1−j − yk+1−j)

+ f1(k, xk,

k−1∑

m=0

g1(k,m, xm)) − f2(k, yk,

k−1∑

m=0

g2(k,m, ym))|

≤ |

(

k − α

k

)

(x0 − y0)| + |α

k∑

j=1

1

(j − α)

(

j − α

j

)

(xk+1−j − yk+1−j)|

+ |f1(k, xk,

k−1∑

m=0

g1(k,m, xm)) − f2(k, yk,

k−1∑

m=0

g2(k,m, ym))|

≤

(

k − α

k

)

|x0 − y0| + α

k∑

j=1

1

(j − α)

(

j − α

j

)

|xk+1−j − yk+1−j|

+ f(k, |xk − yk|, |k−1∑

m=0

g1(k,m, xm) −k−1∑

m=0

g2(k,m, ym)|)


is satisfied for all n ∈ N+0 and 0 < α ≤ 1. Then v0 ≤ w0 implies

vn ≤ wn (34)

where wn is solution of the difference equation

∇αwn+1 = pn +n−1∑

m=0

h(n,m)f(m,wm,

m−1∑

j=0

g(m, j,wj)), w0 = p0 (35)

for all n ∈ N+0 and 0 < α ≤ 1.

Theorem 11. Let f be a nonnegative and nondecreasing function withrespect to its arguments. Let wn be solution of the difference equation

∇αwn+1 = f(wn,

n−1∑

j=0

wj,

n−2∑

j=0

wj) (36)

for all n ∈ N+0 and 0 < α ≤ 1. Suppose that the inequality

∇αvn+1 ≤ f(vn,

n−1∑

j=0

vj,

n−2∑

j=0

vj) (37)

is satisfied for all n ∈ N+0 and 0 < α ≤ 1, where vj (j = 0, 1, 2...) is a positive

sequence of functions defined for n ∈ N+0 such that v0 ≤ w0. Then

vn ≤ wn (38)

for all n ∈ N+0 .

Proof. The proof of the present theorem is by induction on n. For n = 0the claim is true. Suppose the theorem is true for n = k. Then vj ≤ wj forj ≤ k. If possible, suppose vk+1 > wk+1. By the monotone property of f , wehave

vk ≤ wk,

k−1∑

j=0

vj ≤k−1∑

j=0

wj ,

k−2∑

j=0

vj ≤k−2∑

j=0

wj

⇒ f(vk,

k−1∑

j=0

vj ,

k−2∑

j=0

vj) ≤ f(wk,

k−1∑

j=0

wj ,

k−2∑

j=0

wj).

Now

∇αvk+1 ≤ f(vk,

k−1∑

j=0

vj ,

k−2∑

j=0

vj) ≤ f(wk,

k−1∑

j=0

wj ,

k−2∑

j=0

wj) = ∇αwk+1. (39)


Consider

∇αwk+1 = wk+1 −

(

k − α

k

)

w0 − α

k∑

j=1

1

(j − α)

(

j − α

j

)

wk+1−j

< vk+1 −

(

k − α

k

)

v0 − α

k∑

j=1

1

(j − α)

(

j − α

j

)

vk+1−j

= ∇αvk+1

which is a contradiction to (38). Hence the theorem is true for n = k+1. Thusby the property of mathematical induction the theorem is true for all n ∈ N

+0 .

Corollary 12. Let f be a nonnegative and nondecreasing function withrespect to its arguments. Let wn be solution of the difference equation

∇αwn+1 = f(wn, wn−1, ..., wn−k) (40)

for all n ∈ N+0 and 0 < α ≤ 1. Suppose that the inequality

∇αvn+1 ≤ f(vn, vn−1, ..., vn−k) (41)

is satisfied for all n ∈ N+0 and 0 < α ≤ 1, where vj (j = 0, 1, 2, ...) is a positive

sequence of functions defined for n ∈ N+0 such that vj ≤ wj , j = 0, 1, ..., k.

Then

vn ≤ wn (42)

for all n ∈ N+0 .

Proof. Using the monotone property of f the proof is clear.

Acknowledgments

This work is supported by DST-CMS project Lr.No.SR/S4/MS:516/07, Dt.21-04-2008.

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fractional difference inequalities of volterra type

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