International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 1 (2015), 79-95
http://www.etamaths.com

ANALYSIS OF NONLINEAR FRACTIONAL NABLA

DIFFERENCE EQUATIONS

JAGAN MOHAN JONNALAGADDA

Abstract. In this paper, we establish sufficient conditions on global existence

and uniqueness of solutions of nonlinear fractional nabla difference systems and
investigate the dependence of solutions on initial conditions and parameters.

1. Introduction

Discrete fractional calculus deals with sums and differences of arbitrary order-
s. Looking into the literature of fractional difference calculus, two approaches are
found: one using the ∆ - point of view (called the fractional delta difference ap-
proach) and another using the ∇ - perspective (called the nabla fractional difference
approach). The theory for fractional nabla difference calculus was initiated by Gray
and Zhang [18], Atici and Eloe [9] and Anastassiou [17], where basic approaches,
definitions and properties of fractional sums and differences were reported. Recent-
ly, a series of papers continuing research on fractional nabla difference equations
has appeared [10, 11, 12, 14, 16, 19, 20, 21, 22, 23, 26]. But a very little progress
has been made to develop fractional nabla difference systems [11, 24].

In the following example, we illustrate the advantage of fractional order nabla
difference system over integer order nabla difference system.

Example 1. Consider the following two systems.

∇0u(t) = βtβ−1, 0 < β < 1, t ∈ N1,(1.1)

∇α0∗u(t) = βt
β−1, 0 < α < β < 1, t ∈ N1,(1.2)

where ∇α0∗ is the Caputo type fractional nabla difference operator.

The solution of (1.1) is given by

(1.3) u(t) = u(0) + tβ, t ∈ N0,

Clearly (1.3) tends to ∞ as t →∞ for 0 < β < 1 and thus it is unstable. But the
solution of (1.2) is given by

(1.4) u(t) = u(0) +
Γ(1 −β)

Γ(1 −β + α)
tα−β, t ∈ N0.

2010 Mathematics Subject Classification. 39A10, 39A99.
Key words and phrases. fractional order; nabla difference; fixed point; global existence; u-

niqueness; stability.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

79



80 JONNALAGADDA

Clearly (1.4) tends to 0 as t →∞ for 0 < α < β < 1 and therefore it is stable, which
implies that the fractional order system may have additional attractive feature over
the integer order system.

On the other hand, several authors [13, 15, 31, 32] used fixed point theorems to
discuss existence, uniqueness and stability properties of fractional differential sys-
tems. Motivated by this fact, in this paper, we initiate the study on global existence
and uniqueness of solutions of nonlinear fractional nabla difference systems.

The present paper is organized as follows: Section 2 contains preliminaries on
nabla discrete fractional calculus and functional analysis. We consider a system
of nonlinear fractional nabla difference equations and obtain sufficient conditions
on global existence and uniqueness of solutions and the dependence of solutions on
initial conditions and parameters in sections 3 and 4 respectively.

2. Preliminaries

We shall use the following notations, definitions and known results of discrete
fractional calculus [8, 9, 24, 29] throughout this article. For any a, b ∈ R, Na =
{a,a + 1,a + 2, ...........}, Na,b = {a,a + 1,a + 2, ...........,b} where a < b.

Definition 2.1. For any α, t ∈ R, the α rising function is defined by

tα =
Γ(t + α)

Γ(t)
, t ∈ R\{......,−2,−1, 0}, 0α = 0.

We observe the following properties of rising factorial function.

Lemma 2.1. Assume the following factorial functions are well defined.

(1) tα(t + α)β = tα+β.
(2) If t ≤ r then tα ≤ rα.
(3) If α < t ≤ r then r−α ≤ t−α.

Definition 2.2. Let u : Na → R, α ∈ R+ and choose N ∈ N1 such that N − 1 <
α < N.

(1) (Nabla Difference) The first order backward difference or nabla difference
of u is defined by

∇u(t) = u(t) −u(t− 1), t ∈ Na+1,

and the Nth - order nabla difference of u is defined recursively by

∇Nu(t) = ∇(∇N−1u(t)), t ∈ Na+N.

In addition, we take ∇0 as the identity operator.

(2) (Fractional Nabla Sum) The αth - order fractional nabla sum of u is given
by

(2.1) ∇−αa u(t) =
1

Γ(α)

t∑
s=a+1

(t−ρ(s))α−1u(s), t ∈ Na

where ρ(s) = s− 1. Also, we define the trivial sum by ∇−0a u(t) = u(t) for
t ∈ Na.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 81

(3) (R - L Nabla Fractional Difference) The αth - order Riemann - Liouville
type nabla fractional difference of u is given by

(2.2) ∇αau(t) = ∇
N
[
∇−(N−α)a u(t)

]
, t ∈ Na+N.

For α = 0, we set ∇0au(t) = u(t), t ∈ Na.

(4) (Caputo Fractional Nabla Difference) The αth - order Caputo type frac-
tional nabla difference of u is given by

(2.3) ∇αa∗u(t) = ∇
−(N−α)
a

[
∇Nu(t)

]
, t ∈ Na+N.

For α = 0, we set ∇0a∗u(t) = u(t), t ∈ Na.

Theorem 2.2. (Power Rule) Let α > 0 and µ > −1. Then,
(1) ∇−αa (t−a)µ =

Γ(µ+1)
Γ(µ+α+1)

(t−a)µ+α, t ∈ Na.

(2) ∇αa (t−a)µ =
Γ(µ+1)

Γ(µ−α+1) (t−a)
µ−α, t ∈ Na+N .

Let f : Na × R → R, u : Na → R and 0 < α < 1. Consider a nonautonomous
fractional nabla difference equation of Riemann - Liouville type together with an
initial condition of the form

∇αa−1u(t) = f(t,u(t)), t ∈ Na+1,(2.4)

∇−(1−α)a−1 u(t)
∣∣∣
t=a

= u(a) = u0.(2.5)

Then, from [30], u is a solution of the initial value problem (2.4) - (2.5) if and only
if it has the following representation

(2.6) u(t) =
(t−a + 1)α−1

Γ(α)
u0 +

1

Γ(α)

t∑
s=a+1

(t−ρ(s))α−1f(s,u(s)), t ∈ Na.

If we consider a nonautonomous fractional nabla difference equation of Caputo type
together with an initial condition of the form

∇αa∗u(t) = f(t,u(t)), t ∈ Na+1,(2.7)
u(a) = u0.(2.8)

Then, u is a solution of the initial value problem (2.7) - (2.8) if and only if it has
the following representation

(2.9) u(t) = u0 +
1

Γ(α)

t∑
s=a+1

(t−ρ(s))α−1f(s,u(s)), t ∈ Na.

Now we present some important definitions and theorems of functional analysis
[3, 7] which will be useful in establishing main results.

Definition 2.3. Rn is the space of all ordered n-tuples of real numbers. Clearly,
Rn is a Banach space with respect to the supremum norm. A closed ball with radius
r centered at the origin of Rn is defined by

B∞0 (r) = {u = (u1,u2, ...,un) ∈ R
n : ‖u‖∞ ≤ r}.



82 JONNALAGADDA

Definition 2.4. l∞ = l∞(R) is the space of all real sequences defined on the set
of positive integers where any individual sequence is bounded with respect to the
usual supremum norm. Clearly l∞ is a Banach space under the supremum norm.
A closed ball with radius r centered on the null sequence of l∞ is defined by

B∞0 (r) = {u = {u(t)}
∞
t=0 ∈ l

∞ : ‖u‖∞ ≤ r}.

Definition 2.5. A subset S of l∞ is uniformly Cauchy (or equi - Cauchy), if for
every � > 0, there exists k ∈ N1 such that |u(t1) −u(t2)| < � whenever t1, t2 ∈ Nk+1,
for any u = {u(t)}∞t=0 in S.

Theorem 2.3. (Discrete Arzela - Ascoli’s Theorem) A bounded uniformly Cauchy
subset S of l∞ is relatively compact.

Theorem 2.4. (Krasnoselskii’s Fixed Point Theorem) Let S be a nonempty, closed,
convex and bounded subset of a Banach space X, and let A : X → X and B : S → X
be two operators such that

(1) A is a contraction with constant L < 1,
(2) B is continuous, BS resides in a compact subset of X,
(3) [x = Ax + By, y ∈ S] =⇒ x ∈ S.

Then the operator equation Ax + Bx = x has a solution in S.

Theorem 2.5. (Generalized Banach Fixed Point Theorem) Let S be a nonempty,
closed subset of a Banach space (X,‖.‖), and let a γn ≥ 0 for every n ∈ N0 and such∑∞
n=0 γn converges. Moreover, let the mapping T : S → S satisfy the inequality

‖Tnu−Tnv‖≤ γn‖u−v‖
for every n ∈ N1 and any u,v ∈ S. Then, T has a uniquely defined fixed point u∗.
Furthermore, for any u0 ∈ S, the sequence (Tnu0)∞n=1 converges to this fixed point
u∗.

Theorem 2.6. (Schauder Fixed Point Theorem) Let S be a nonempty, closed and
convex subset of a Banach space X. Let T : S → S be a continuous mapping such
that TS is a relatively compact subset of X. Then T has at least one fixed point in
S. That is, there exists an x ∈ S such that Tx = x.

Definition 2.6. Let X be a Banach space with respect to a norm ‖.‖. Define the
set

S = S(X) = {u : u = {u(t)}∞t=0, u(t) ∈ X}.
Then, S is a linear space of sequences of elements of X under obvious definition of
addition and scalar multiplication. Now we employ the notation

u = {u(t)}∞t=0, ‖u‖∞ = sup
t∈N0
|u(t)|,

and define the set

S∞(X) = {u : u ∈ S(X) with ‖u‖∞ < ∞}.
Clearly S∞(X) is a Banach space consisting of elements of S(X), with respect to
the supremum norm.

Definition 2.7. From Definitions 2.4 and 2.6, we observe that l∞ = l∞(R) =
S∞(R). Now we choose X = Rn in Definition 2.6 to define

l∞ = l∞(Rn) = S∞(Rn) = {u : u = {u(t)}∞t=0, u(t) ∈ R
n with ‖u‖∞ < ∞}.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 83

Thus, l∞ denotes the Banach space comprising sequences of vectors with respect
to the supremum norm ‖.‖∞ defined by

‖u‖∞ = sup
t∈N0
‖u(t)‖.

A closed ball with radius r centered on the null sequence in l∞ is defined by

B∞0 (r) = {u = {u(t)}
∞
t=0 ∈ l

∞ : ‖u‖∞ ≤ r}.

3. Existence & Uniqueness

In this section we prove existence and uniqueness theorems pertaining to the ini-
tial value problems associated with a system of fractional nabla difference equations
of the form

(3.1) ∇α−1u(t) = f (t, u(t)), ∇
−(1−α)
−1 u(t)

∣∣∣
t=0

= u(0) = c, 0 < α < 1, t ∈ N1

and

(3.2) ∇α0∗u(t) = f (t, u(t)), u(0) = c, 0 < α < 1, t ∈ N1,
where ∇α−1 and ∇α0∗ are the Riemann - Liouville and Caputo type fractional differ-
ence operators, u(t) is an n-vector whose components are functions of the variable t,
c is a constant n-vector and f (t, u(t)) is an n-vector whose components are functions
of the variable t and the n-vector u(t).

Let u : N0 → l∞ and f : N0 × l∞ → l∞. Analogous to (2.6), u = {u(t)}∞t=0 ∈ l∞
is any solution of the initial value problem (3.1) if and only if

(3.3) u(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0.

Analogous to (2.9), u = {u(t)}∞t=0 ∈ l∞ is any solution of the initial value problem
(3.2) if and only if

(3.4) u(t) = c +
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0.

Define the operators

Tu(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0,(3.5)

T ′u(t) = c +
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0,(3.6)

Au(t) =
(t + 1)α−1

Γ(α)
c, t ∈ N0,(3.7)

Bu(t) =
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0.(3.8)

It is evident from (3.3) - (3.6) that u is a fixed point of T if and only if u is a
solution of (3.1) and u is a fixed point of T ′ if and only if u is a solution of (3.2).

First we use Krasnoselskii’s fixed point theorem (Theorem 2.4) to establish global
existence of solutions of (3.1). Clearly A is a contraction mapping with constant 0,
implies condition (1) of Theorem 2.4 holds.



84 JONNALAGADDA

Theorem 3.1. (Global Existence) If f is continuous with respect to the second
variable and there exist constants β1 ∈ [α, 1) and L1 ≥ 0 such that

(3.9) ‖f (t, u(t))‖≤ L1t−β1, t ∈ N1,

then the nonautonomous initial value problem (3.1) has at least one bounded solu-
tion in l∞.

Proof. To prove condition (2) of Theorem 2.4, we define a set

S1 = {u : ‖u(t)‖≤‖c‖ + L1Γ(1 −β1), t ∈ N1}.

Clearly S1 is a nonempty, closed, bounded and convex subset of l
∞. First, we show

that B maps S1 into S1. Using Lemma 2.1, Theorem 2.2 and (3.9), we have

‖Bu(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1 ‖f (s, u(s))‖

≤
L1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−β1

= L1∇−α0 t
−β1

=
L1Γ(1 −β1)

Γ(1 −β1 + α)
t−(β1−α)

≤
L1Γ(1 −β1)

Γ(1 −β1 + α)
(1)−(β1−α)

= L1Γ(1 −β1)
≤ ‖c‖ + L1Γ(1 −β1), t ∈ N1,

implies BS1 ⊂ S1. Next, we show that B is continuous on S1. Let � > 0 be given.
Then there exists m ∈ N1 such that, for t ∈ Nm+1,

L1Γ(1 −β1)
Γ(1 −β1 + α)

t−(β1−α) <
�

2
.

Let {uk}, (k = 1, 2, .....) be a sequence in S1 such that uk → u in S1. Then, we
have ‖uk − u‖∞ → 0 as k → ∞. Since f is continuous with respect to the second
variable, we get ‖f (t, uk) − f (t, u)‖∞ → 0 as k →∞. For t ≤ m,

‖Buk(t) −Bu(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, uk(s)) − f (s, u(s))‖

≤
[ 1

Γ(α)

t∑
s=1

(t−ρ(s))α−1
][

sups∈{1,2,......,m}‖f (s, uk(s)) − f (s, u(s))‖
]

=
tα

Γ(α + 1)
‖f (s, uk) − f (s, u)‖∞

→ 0 as k →∞.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 85

For t ∈ Nm+1,

‖Buk(t) −Bu(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1[‖f (s, uk(s))‖ + ‖f (s, u(s))‖]

≤
2L1Γ(1 −β1)
Γ(1 −β1 + α)

t−(β1−α) < �.

Thus we have, ‖Buk −Bu‖∞ → 0 as k → ∞, implies B is continuous. Now, we
show that BS1 is relatively compact. Let t1, t2 ∈ Nm+1 such that t2 > t1. Then,
we have

‖Bu(t1) −Bu(t2)‖ ≤
1

Γ(α)

t1∑
s=1

(t1 −ρ(s))α−1‖f (s, u(s))‖ +
1

Γ(α)

t2∑
s=1

(t2 −ρ(s))α−1‖f (s, u(s))‖

≤
L1Γ(1 −β1)

Γ(1 −β1 + α)
t
−(β1−α)
1 +

L1Γ(1 −β1)
Γ(1 −β1 + α)

t
−(β1−α)
2 < �.

Thus {Bu : u ∈ S1} is a bounded and uniformly Cauchy subset of l∞. Hence, by
Theorem 2.3, BS1 is relatively compact.

Now we prove condition (3) of Theorem 2.4. Let us suppose, for a fixed v ∈ S1,
u = Au + Bv. Using Lemma 2.1, Theorem 2.2 and (3.9), we have

‖u(t)‖ ≤ ‖Au(t)‖ + ‖Bv(t)‖

≤
(t + 1)α−1

Γ(α)
‖c‖ +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, v(s))‖

≤
(1)α−1

Γ(α)
‖c‖ +

L1Γ(1 −β1)
Γ(1 −β1 + α)

t−(β1−α)

≤ ‖c‖ +
L1Γ(1 −β1)

Γ(1 −β1 + α)
(1)−(β1−α)

= ‖c‖ + L1Γ(1 −β1), t ∈ N1.

Thus u ∈ S1. According to Theorem 2.4, T has a fixed point in S1 which is a
solution of (3.1). Hence the proof. �

Theorem 3.2. (Global Existence) If f is continuous with respect to the second
variable and there exist constants β2 ∈ [α, 1) and L2 ≥ 0 such that

(3.10) ‖f (t, u(t))‖≤ L2t−β2 ‖u(t)‖ , t ∈ N1,

then the nonautonomous initial value problem (3.1) has at least one bounded solu-
tion in l∞ provided that

(3.11) L2Γ(1 −β2) < 1.

Proof. Define

S2 =
{

u : ‖u(t)‖≤
‖c‖

[1 −L2Γ(1 −β2)]
, t ∈ N1

}
.



86 JONNALAGADDA

Clearly S2 is a nonempty, closed, bounded and convex subset of l
∞. First, we show

that B maps S2 into S2. Using Lemma 2.1, Theorem 2.2 and (3.10), we have

‖Bu(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1 ‖f (s, u(s))‖

≤
L2

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−β2 ‖u(s)‖

≤
L2‖c‖

[1 −L2Γ(1 −β2)]
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−β2

=
L2‖c‖

[1 −L2Γ(1 −β2)]
∇−α0 t

−β2

=
L2‖c‖

[1 −L2Γ(1 −β2)]
Γ(1 −β2)

Γ(1 −β2 + α)
t−(β2−α)

≤
L2‖c‖

[1 −L2Γ(1 −β2)]
Γ(1 −β2)

Γ(1 −β2 + α)
(1)−(β2−α)

=
L2‖c‖Γ(1 −β2)

[1 −L2Γ(1 −β2)]

=
‖c‖

[1 −L2Γ(1 −β2)]
−‖c‖

≤
‖c‖

[1 −L2Γ(1 −β2)]
, t ∈ N1,

implies BS2 ⊂ S2. The remaining proof of condition (2) is similar to that of
Theorem 3.1 and we omit it.

Now we prove condition (3) of Theorem 2.4. Let us suppose, for a fixed v ∈ S2,
u = Au + Bv. Using Lemma 2.1, Theorem 2.2 and (3.10), we have

‖u(t)‖ ≤ ‖Au(t)‖ + ‖Bv(t)‖

≤
(t + 1)α−1

Γ(α)
‖c‖ +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, v(s))‖

≤
(1)α−1

Γ(α)
‖c‖ +

L2‖c‖Γ(1 −β2)
[1 −L2Γ(1 −β2)]

≤ ‖c‖ +
L2‖c‖Γ(1 −β2)

[1 −L2Γ(1 −β2)]

=
‖c‖

[1 −L2Γ(1 −β2)]
, t ∈ N1.

Thus u ∈ S2. According to Theorem 2.4, T has a fixed point in S2 which is a
solution of (3.1) - (3.2). Hence the proof. �

Now we apply Schauder fixed point theorem (Theorem 2.6) to establish global
existence of solutions of (3.2).

Theorem 3.3. (Global Existence) If f satisfies the hypothesis of Theorem 3.1, then
the nonautonomous initial value problem (3.2) has at least one bounded solution in
l∞.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 87

Proof. Define a set

S3 = {u : u(0) = c, ‖u(t) − c‖≤ L1Γ(1 −β1), t ∈ N1}.

Clearly S3 is a nonempty, closed, bounded and convex subset of l
∞. First, we show

that T ′ maps S3 into S3. Using Lemma 2.1, Theorem 2.2 and (3.9), we have

‖T ′u(t) − c‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1 ‖f (s, u(s))‖

≤
L1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−β1

= L1∇−α0 t
−β1

=
L1Γ(1 −β1)

Γ(1 −β1 + α)
t−(β1−α)

≤
L1Γ(1 −β1)

Γ(1 −β1 + α)
(1)−(β1−α)

= L1Γ(1 −β1), t ∈ N1,

and T ′u(0) = c, implies T ′S3 ⊂ S3. Next, we show that T ′ is continuous on S3.
Let � > 0 be given. Then there exists m ∈ N1 such that, for t ∈ Nm+1,

(3.12)
L1Γ(1 −β1)

Γ(1 −β1 + α)
t−(β1−α) <

�

2
.

Let {uk}, (k = 1, 2, .....) be a sequence in S3 such that uk → u in S3. Then, we
have ‖uk − u‖∞ → 0 as k → ∞. Since f is continuous with respect to the second
variable, we get ‖f (t, uk) − f (t, u)‖∞ → 0 as k →∞. For t ≤ m,

‖T ′uk(t) −T ′u(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, uk(s)) − f (s, u(s))‖

≤
[ 1

Γ(α)

t∑
s=1

(t−ρ(s))α−1
][

sups∈{1,2,......,m}‖f (s, uk(s)) − f (s, u(s))‖
]

=
tα

Γ(α + 1)
‖f (s, uk) − f (s, u)‖∞

→ 0 as k →∞.

For t ∈ Nm+1,

‖T ′uk(t) −T ′u(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1[‖f (s, uk(s))‖ + ‖f (s, u(s))‖]

≤
2L1Γ(1 −β1)
Γ(1 −β1 + α)

t−(β1−α) < �.

Thus we have, ‖T ′uk −T ′u‖∞ → 0 as k → ∞, implies T
′ is continuous. Now, we

show that T ′S3 is relatively compact. Let t1, t2 ∈ Nm+1 such that t2 > t1. Then,



88 JONNALAGADDA

we have

‖T ′u(t1) −T ′u(t2)‖ ≤
1

Γ(α)

t1∑
s=1

(t1 −ρ(s))α−1‖f (s, u(s))‖ +
1

Γ(α)

t2∑
s=1

(t2 −ρ(s))α−1‖f (s, u(s))‖

≤
L1Γ(1 −β1)

Γ(1 −β1 + α)
t
−(β1−α)
1 +

L1Γ(1 −β1)
Γ(1 −β1 + α)

t
−(β1−α)
2 < �.

Thus {T ′u : u ∈ S3} is a bounded and uniformly Cauchy subset of l∞. Hence, by
Theorem 2.3, T ′S3 is relatively compact. According to Theorem 2.6, T

′ has a fixed
point in S3 which is a solution of (3.2). Hence the proof. �

We use generalized Banach fixed point theorem (Theorem 2.5) to prove the
uniqueness of solutions of (3.1) and (3.2).

Theorem 3.4. (Global Uniqueness) If f is continuous with respect to the second
variable and there exist constants γ ∈ [α, 1) and M ≥ 0 such that

(3.13) ‖f (t, u) − f (t, v)‖∞ ≤ Mt
−γ‖u − v‖∞, t ∈ N1,

for any pair of elements u and v in l∞. Then the initial value problems (3.1) and
(3.2) have unique bounded solution in l∞ provided that

(3.14) c = MΓ(1 −γ) < 1.

Proof. Let us define the iterates of operator T as follows:

T 1 = T, Tn = ToTn−1, n ∈ N1.

It is sufficient to prove that Tn is a contraction operator for sufficiently large n.
Actually, we have

(3.15) ‖Tnu −Tnv‖∞ ≤ cn‖u − v‖∞

where the constant c depends only on M and γ. In fact, using Lemma 2.1, Theorem
2.2 and (3.13), we get

‖Tu(t) −Tv(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖

≤
M

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−γ‖u − v‖∞

= M∇−α0 t
−γ‖u − v‖∞

=
MΓ(1 −γ)

Γ(1 −γ + α)
t−(γ−α)‖u − v‖∞

≤
MΓ(1 −γ)

Γ(1 −γ + α)
(1)−(γ−α)‖u − v‖∞

= c‖u − v‖∞,

implies

(3.16) ‖Tu −Tv‖∞ ≤ c‖u − v‖∞.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 89

Therefore (3.15) is true for n = 1. Assuming (3.15) is valid for n, we obtain similarly

‖Tn+1u(t) −Tn+1v(t)‖ = ‖(ToTn)u(t) − (ToTn)v(t)‖

≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s,Tnu(s)) − f (s,Tnv(s))‖

≤
M

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−γ‖Tnu −Tnv‖∞

≤ Mcn∇−α0 t
−γ‖u − v‖∞

=
McnΓ(1 −γ)
Γ(1 −γ + α)

t−(γ−α)‖u − v‖∞

≤
McnΓ(1 −γ)
Γ(1 −γ + α)

(1)−(γ−α)‖u − v‖∞

= cn+1‖u − v‖∞.

Thus, by the principle of mathematical induction on n, the statement (3.15) is true
for each n ∈ N1. Since c < 1, the geometric series

∑∞
n=0 c

n converges. Hence T has
a uniquely defined point u∗ in S1 (or S2). This completes the proof. Similarly we
can prove that T ′ has a uniquely defined point u∗ in S3. �

4. Dependence of Solutions on Initial Conditions and Parameters

The initial value problems (3.1) and (3.2) describes a model of a physical problem
in which often some parameters such as lengths, masses, temperature, etc. are
involved. The values of these parameters can be measured only up to a certain
degree of accuracy. Thus, in (3.1) and (3.2), the initial value c, the order of the
difference operator α and the function f , may be subject to some errors either
by necessity or for convenience. Hence, it is important to know how the solution
changes when these parameters are slightly altered. We shall discuss this question
quantitatively in the following theorems.

Theorem 4.1. Assume that f is continuous and satisfies (3.13) with respect to the
second variable. Suppose u and v are the solutions of the initial value problems

∇α+�−1 u(t) = f (t, u(t)), ∇
−(1−α−�)
−1 u(t)

∣∣∣
t=0

= u(0) = c, t ∈ N1,(4.1)

∇α−1v(t) = f (t, v(t)), ∇
−(1−α)
−1 v(t)

∣∣∣
t=0

= v(0) = c, t ∈ N1,(4.2)

respectively, where � > 0 and 0 < α < α + � < 1. Then

(4.3) ‖u − v‖∞ = O(�)

provided that (3.14) holds.

Proof. We have

u(t) =
(t + 1)α+�−1

Γ(α + �)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α+�−1f (s, u(s)), t ∈ N0,

v(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, v(s)), t ∈ N0.



90 JONNALAGADDA

Consider

‖u(t) − v(t)‖ ≤
∣∣∣(t + 1)α+�−1

Γ(α + �)
−

(t + 1)α−1

Γ(α)

∣∣∣‖c‖
+
∥∥∥ 1

Γ(α + �)

t∑
s=1

(t−ρ(s))α+�−1f (s, u(s)) −
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, v(s))
∥∥∥

≤
∣∣∣ Γ(α)
Γ(α + �)

(t + α)� − 1
∣∣∣(t + 1)α−1

Γ(α)
‖c‖∞

+
∥∥∥ 1

Γ(α + �)

t∑
s=1

(t−ρ(s))α+�−1[f (s, u(s)) − f (s, v(s))]
∥∥∥

+
∥∥∥ 1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, v(s))
[
1 −

Γ(α)

Γ(α + �)
(t−s + α)�

]∥∥∥

≤
∣∣∣ Γ(α)
Γ(t + α)

Γ(� + t + α)

Γ(� + α)
− 1
∣∣∣(2)α−1

Γ(α)
‖c‖∞

+
1

Γ(α + �)

t∑
s=1

(t−ρ(s))α+�−1‖f (s, u(s)) − f (s, v(s))‖

+
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, v(s))‖
∣∣∣1 − Γ(α)

Γ(t−s + α)
Γ(� + t−s + α)

Γ(� + α)

∣∣∣, t ∈ N1.
(4.4)

Since

lim
�→0

1

�

[ Γ(α)
Γ(t + α)

Γ(� + t + α)

Γ(� + α)
− 1
]

= C1 (a constant independent of �)

and

lim
�→0

1

�

[
1 −

Γ(α)

Γ(t−s + α)
Γ(� + t−s + α)

Γ(� + α)

]
= C2 (a constant independent of �),

we have

[ Γ(α)
Γ(t + α)

Γ(� + t + α)

Γ(� + α)
− 1
]

= O(�),(4.5) [
1 −

Γ(α)

Γ(t−s + α)
Γ(� + t−s + α)

Γ(� + α)

]
= O(�).(4.6)



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 91

Using (4.5) and (4.6) in (4.4), we get

‖u(t) − v(t)‖ ≤ O(�)α‖c‖∞ + M‖u − v‖∞
1

Γ(α + �)

t∑
s=1

(t−ρ(s))α+�−1s−γ

+O(�)‖f‖∞
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−γ

= O(�)α‖c‖∞ + M‖u − v‖∞∇
−(α+�)
0 t

−γ + O(�)‖f‖∞∇−α0 t
−γ

= O(�)α‖c‖∞ + M‖u − v‖∞
Γ(1 −γ)

Γ(1 + α + �−γ)
tα+�−γ + O(�)‖f‖∞

Γ(1 −γ)
Γ(1 + α−γ)

tα−γ

≤ O(�)α‖c‖∞ + M‖u − v‖∞
Γ(1 −γ)

Γ(1 + α + �−γ)
(1)α+�−γ + O(�)‖f‖∞

Γ(1 −γ)
Γ(1 + α−γ)

(1)α−γ

= O(�)α‖c‖∞ + M‖u − v‖∞Γ(1 −γ) + O(�)‖f‖∞Γ(1 −γ), t ∈ N1.
Then, we have the relation

‖u − v‖∞ ≤
[α‖c‖∞ + ‖f‖∞Γ(1 −γ)]

[1 −MΓ(1 −γ)]
O(�)

implies

‖u − v‖∞ = O(�).
�

Corollary 1. Assume that f is continuous and satisfies (3.13) with respect to the
second variable. Suppose u and v are the solutions of the initial value problems

∇α+�0∗ u(t) = f (t, u(t)), u(0) = c, t ∈ N1,(4.7)
∇α0∗v(t) = f (t, v(t)), v(0) = c, t ∈ N1,(4.8)

respectively, where � > 0 and 0 < α < α + � < 1. Then

(4.9) ‖u − v‖∞ = O(�)
provided that (3.14) holds.

Theorem 4.2. Assume that f is continuous and satisfies (3.13) with respect to the
second variable. Suppose u and v are the solutions of the initial value problems

∇α−1u(t) = f (t, u(t)), ∇
−(1−α)
−1 u(t)

∣∣∣
t=0

= u(0) = c, t ∈ N1,(4.10)

∇α−1v(t) = f (t, v(t)), ∇
−(1−α)
−1 v(t)

∣∣∣
t=0

= v(0) = d, t ∈ N1,(4.11)

respectively, where 0 < α < 1. Then

(4.12) ‖u − v‖∞ = O(‖c − d‖∞)
provided that (3.14) holds.

Proof. We have

u(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0,

v(t) =
(t + 1)α−1

Γ(α)
d +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, v(s)), t ∈ N0.



92 JONNALAGADDA

Consider

‖u(t) − v(t)‖ ≤ ‖c − d‖
(t + 1)α−1

Γ(α)
+

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖

≤ ‖c − d‖∞
(2)α−1

Γ(α)
+ M‖u − v‖∞

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−γ

= α‖c − d‖∞ + M‖u − v‖∞∇−α0 t
−γ

= α‖c − d‖∞ + M‖u − v‖∞
Γ(1 −γ)

Γ(1 + α−γ)
tα−γ

≤ α‖c − d‖∞ + M‖u − v‖∞
Γ(1 −γ)

Γ(1 + α−γ)
(1)α−γ

= α‖c − d‖∞ + M‖u − v‖∞Γ(1 −γ), t ∈ N1.
Then, we have the relation

‖u − v‖∞ ≤
α‖c − d‖∞

[1 −MΓ(1 −γ)]
implies

‖u − v‖∞ = O(‖c − d‖∞).
�

Corollary 2. Assume that f is continuous and satisfies (3.13) with respect to the
second variable. Suppose u and v are the solutions of the initial value problems

∇α0∗u(t) = f (t, u(t)), u(0) = c, t ∈ N1,(4.13)
∇α0∗v(t) = f (t, v(t)), v(0) = d, t ∈ N1,(4.14)

respectively, where 0 < α < 1. Then

(4.15) ‖u − v‖∞ = O(‖c − d‖∞)
provided that (3.14) holds.

Theorem 4.3. Assume that f and g are continuous and satisfies (3.13) with re-
spect to the second variable. Suppose u and v are the solutions of the initial value
problems

∇α−1u(t) = f (t, u(t)), ∇
−(1−α)
−1 u(t)

∣∣∣
t=0

= u(0) = c, t ∈ N1,(4.16)

∇α−1v(t) = g(t, v(t)), ∇
−(1−α)
−1 v(t)

∣∣∣
t=0

= v(0) = c, t ∈ N1,(4.17)

respectively, where 0 < α < 1. Then

(4.18) ‖u − v‖∞ = O(‖f − g‖∞)
provided that (3.14) holds.

Proof. We have

u(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1f (s, u(s)), t ∈ N0,

v(t) =
(t + 1)α−1

Γ(α)
c +

1

Γ(α)

t∑
s=1

(t−ρ(s))α−1g(s, v(s)), t ∈ N0.



NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS 93

Consider

‖u(t) − v(t)‖ ≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, u(s)) − g(s, v(s))‖

=
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s)) + f (s, v(s)) − g(s, v(s))‖

≤
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖

+
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1‖f (s, v(s)) − g(s, v(s))‖

≤ [M‖u − v‖∞ + ‖f − g‖∞]
1

Γ(α)

t∑
s=1

(t−ρ(s))α−1s−γ

= [M‖u − v‖∞ + ‖f − g‖∞]∇−α0 t
−γ

= [M‖u − v‖∞ + ‖f − g‖∞]
Γ(1 −γ)

Γ(1 + α−γ)
tα−γ

≤ [M‖u − v‖∞ + ‖f − g‖∞]
Γ(1 −γ)

Γ(1 + α−γ)
(1)α−γ

= [M‖u − v‖∞ + ‖f − g‖∞]Γ(1 −γ), t ∈ N1.
Then, we have the relation

‖u − v‖∞ ≤
Γ(1 −γ)

[1 −MΓ(1 −γ)]
‖f − g‖∞

implies
‖u − v‖∞ = O(‖f − g‖∞).

�

Corollary 3. Assume that f and g are continuous and satisfies (3.13) with re-
spect to the second variable. Suppose u and v are the solutions of the initial value
problems

∇α0∗u(t) = f (t, u(t)), u(0) = c, t ∈ N1,(4.19)
∇α0∗v(t) = g(t, v(t)), v(0) = c, t ∈ N1,(4.20)

respectively, where 0 < α < 1. Then

(4.21) ‖u − v‖∞ = O(‖f − g‖∞)
provided that (3.14) holds.

Definition 4.1. A solution ũ ∈ l∞ is said to be stable, if given � > 0 and t0 ≥ 0,
there exists δ = δ(�,t0) such that ‖u(t0) − ũ(t0)‖∞ < δ ⇒ ‖u − ũ‖∞ < � for all
t ≥ t0.

Theorem 4.4. Assume that f is continuous and satisfies (3.13) with respect to
the second variable. Then the solutions of (3.1) and (3.2) are stable provided that
(3.14) holds.

Proof. The proof is a direct consequence of Theorem 5.2 and Corollary 2. �



94 JONNALAGADDA

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Department of Mathematics, Birla Institute of Technology and Science Pilani, Hy-
derabad Campus, Hyderabad - 500078, Telangana, India