Scholarly article on topic 'Some new fractional difference inequalities of Gronwall-Bellman type'

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Academic research paper on topic "Some new fractional difference inequalities of Gronwall-Bellman type"

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ORIGINAL RESEARCH Open Access

Some new fractional difference inequalities of Gronwall-Bellman type

Gunturu Venkata Sita Rama Deekshitulu* and Jonnalagadda Jagan Mohan

Abstract

The purpose of the present paper is to establish some important fractional difference inequalities of Gronwall-Bellman type that have a wide range of applications in the study of fractional difference equations.

Keywords: Difference equations, Fractional order, Inequalities, Initial value problem MSC: 39A10,39A99

Introduction

Fractional calculus has gained importance during the past three decades due to its applicability in diverse fields of science and engineering [1]. The notions of fractional calculus may be traced back to the works of Euler, but the idea of fractional difference is very recent.

Diaz and Osler [2] defined the fractional difference by the rather natural approach of allowing the index of differencing, in the standard expression for the nth difference, to be any real or complex number. Later, Hirota [3] defined the fractional order difference operator Va, where a is any real number, using Taylor's series. Nagai [4] adopted another definition for fractional difference by modifying Hirota's [3] definition. Recently, Deekshitulu and Jagan Mohan [5] modified the definition of Nagai [4] and discussed some basic inequalities, comparison theorems, and qualitative properties of the solutions of fractional difference equations [5-10].

Discrete inequalities involving sequences of real numbers, which may be considered as discrete analogues of the Gronwall-Bellman inequality, have been extensively used in the analysis of finite difference equations. In the year 1973, Pachpatte [11] established the following remarkable inequality:

Theorem 1. Let u(n), b(n), and c(n) be nonnegative real valued functions defined on N+ and c > 0 is a constant. If, for n e N+,

Correspondence: dixitgvsr@hotmail.com

Fluid Dynamics Division, School of Advanced Sciences, VIT University, Vellore, Tamil Nadu, 632014, India

u(n) < c + ^ bj j=0

u(j) + J2 c(k)u(k) k=0

u(n) < c

n—1 j—1

1 + E b(»n [1 + b(j) + c(j)] j=0 k=0

Throughout the present paper, we use the following notations [12]: N is the set of natural numbers including zero and Z is the set of integers. N+ = {a, a + 1, a + 2,...} for a e Z. Let u(n) be a real valued function defined on N+.Then,forallm,n2 e N+ ,andm > u(j) = 0

and ]"[;•=m u(j) = 1, i.e., empty sums and products are taken to be 0 and 1, respectively. If n and n + 1are in N+, the backward difference operator V is defined as Vu(n) = u(n) — u(n — 1).

Now, we introduce some basic definitions and results concerning nabla discrete fractional calculus. The extended binomial coefficient Q, (a e R, n e Z), is defined by [4]:

T(a +1)

T(a — n + 1)F(n + 1) 1 0

n = 0 n < 0.

In 2003, Nagai [4] gave the following definition for the fractional order difference operator:

ringer

© 2012 Deekshitulu and Mohan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Definition 1. Let a e R and m be an integer such that m — 1 < a < m. The difference operator V of order a, with step length e, is defined as

Va u(n) =

Va-m[ Vmu(n)] = sm-aJ2

e-a Wa\ (-1)ju(n — j) j=0 \j

a — m

(-l)jVmu(n — j) a > 0

a < 0.

The above definition of Va u(n) given by Nagai [4] contains the V operator and the term (—iy inside the summation index, and hence, it becomes difficult to study the properties of the solution. Deekshitulu and Jagan Mohan [5] modified the above definition for a e (0,1) as follows:

Definition 2. The fractional sum operator of order a (a e R) is defined as

V-a u(n) = J2 j=0

j + a — l

u(n — j)

n — j + a — l n — j

and the fractional difference operator of order a (a e R and 0 < a < 1) is defined as

Definition 3. For any complex numbers a and /,

(a)ß =

F(a+ß) r(a)

undefined

when a and a + /3 are neither zero

nor negative integers when a = // = 0 when a = 0 and / is neither zero

nor a negative integer otherwise.

Remark 2. For any complex numbers a and /, when a, /, and a + / are neither zero nor negative integers,

(a + ß)n = J2 TJ (a)n—k(ß)k k=0

for any positive integer n.

j — a

Vau(n) = )Vu(n — j)

j=0 \ j

n — j — a — l\ in — a — l\

n — j )H n — l )u(0).

Remark 1. If we take a = 1 in (2), using the definition of the generalized binomial coefficient, we have

n—l (' — l

J2 j — ) Vu(n — j) =

A Vu(n)+J j l )Vu(n — j)

-1 Vu(n)+ 1 j - 1 Vu(n - j)

= V u(n).

Gray and Zhang [l3] gave the following definition:

Theorem 2. Let u(n) and v(n) be real valued functions defined on N+ and a, / e R such that 0 < a, /, a + / < 1 and c, d are constants. Then,

1. Va [ cu(n) + dv(n)] = cVau(n) + dVav(n).

2. V—aVau(n) = u(n) — u(0).

3. Vau(0) = 0 and Vau(1) = u(1) — u(0) = Vu(1).

Proof. 2 Consider

V—aVau(n) = V—a [Vau(n)\

= J /« — j + a - ^ V» u(j) n —j

n — j + a — l n — j

n — j + a — l n — j

E ^ a )Vu(j — k)

j — i — a

E^ j — r )Vu(i)

j=1 i=1

n — j + a — 1\ 0 — i — a

n — j

j — i

Recently, the authors have established the following fractional discrete Gronwall-Bellman-type inequality [10]:

yy F(n —j + a)__^ — i — a + 1) Vu(i)

= T(n — j + 1)T(a) r(j — i + 1)F(—a + 1) v ;

Theorem 3. Let u(n), a(n), and b(n) be nonnegative real valued functions defined on N+. If, for n e N+,

T(n — j — i + a) T(j — a + 1)

i=1 j=0

T(n — j — i + 1)T(a) F(j + 1)r(—a + 1)

Vau(n + 1) < a(n)u(n) + b(n),

V u(i)

T(n — i + 1)

= r(n — i + 1) j=0 r(n — i — j + 1)r(j + 1) Y(n — i — j + a) T(j — a + 1)

r(—a + 1)

^(n — i + 1) jgl jl)(a)n—i—j(—a + 1)j

T(n — i + 1)

A T(n — i + 1)

y----—Vu(i) = u(n) — u(0).

^ Y(n — i + 1)r(1)

The proofs of 1 and 3 are clear from Definition 2.

u(n) < u(0) n [1 + B(n — 1, a; j)a(j)] j=0

n— 1 n—1

B(n — 1, a; j)b(j)Y\ [1+B(n— 1, a; k)a(k)]. j=0 k=j+1

Main Results

In this section, we shall establish some new fractional order difference inequalities of Gronwall-Bellman type analogous to the inequality (Theorem 1) given by Pach-patte [11]. Let u(n), b(n), c(n),p(n), and q(n) be nonnegative real valued functions defined on N+ and u(0) > 0be a constant.

Definition 4. Let f (n, r) : N+ x R ^ R. Then, a nonlinear difference equation of order a,0 < a < 1, together with an initial condition is of the form

Theorem 4. Ifa(n) is a positive, monotonic, and nonde-creasingreal valued function defined on N+ and

Vau(n + 1) = f (n, u(n)), u(0) = u0. (3) n—1

u(n) < a(n) + EB(n — 1, a; j)b(j) Now, we consider (1) and replace u(n) by Vau(n), and j=0

we have

n — j + a — 1

V—a [ Va u(n)] = E(" V" Va u(j)]

n — j

u(j) + E B(j — 1, a; k)c(k)u(k) k=0

or u(n) — u(0) = E jn+—a. ^ [ Vau(j)]

for n e N+, then

n—1 , • a 2\

u(n) = u0 ^E \ J-i )[ V a u(j + 1)] u(n) < a(n) i—n V j /

j=0 n—1

or u(n) = u0 + EB(n — 1, a; j)f (j, u(j)), (4) j=0

where B(n, a; j) = (n —j+—'j—1) for 0 < j < n. The above recurrence relation shows the existence of the solution of (3).

1 + E B(n — 1, a; j)b(j) j=0 j—1

x n [! + B(j— 1 a; k) [ b(k) + c(k)] ]

for n e N+.

Proof. Since a(n) is a positive, monotonic, and nonde-creasing real valued function, from (6), we observe that

u(n) a(n)

< 1 + B(n — 1, a; j)b(j)

u()) , 1 /\ n\u(k)

+ B() — 1, a; k)c(k)

< 1 +J2 B(n — 1, a; ))b()) )=0

u()) , ST^nr 1 M n\U(k)

"TT + B() — 1, a;k) c(k) —¡-a(j) a(k)

Define a function

z(n) = 1 + J2 B(n — 1, a; j)b(j) )=0

u()) , 1 / \ n\u(k)

-FK + l^ B() — 1 a; k)c(k)-r-a()) a(k)

Then, z(0) = 1, Un < z(n), and using (4), we get

Va z(n + 1) = b(n) < b(n)

u(n) , 1 /\ n\u(k)

—- + > B(n — 1,a;k)c(k)—— a(n) k=0 a(k)

z(n) + J2 B(n — 1, a; k)c(k)z(k) k=0

Now, again by application of Theorem 3, we get

z(n) < z(0) + B(n — 1, a;)) )=0 )—1

b(J)Y\ [1 + B() — 1, a; k) [ b(k) + c(k)] ]

Using (13) in Un < z(n), we get the required inequality (7). □

Theorem 5. Ifa(n) is a nonnegativefunction defined on N+ and for n e N+,

u(n) < u(0) + B(n — 1, a; f)a(j) )=0

u()) + b()) +J2 B() — 1, a; k)c(k)u(k)

u(n) < u(0) + J2B(n — 1, a;))a())[ b()) + A())], )=0

v(n) = z(n) +J2 B(n — 1, a; k)c(k)z(k). (10)

Then, v(0) = z(0) = 1, z(n) < v(n), and Vav(n + 1) = Vaz(n + 1) + c(n)z(n) < [ b(n) + c(n)] v(n). Now, an application of Theorem 3 yields

v(n) < [] [1 + B(n — 1,a;))[b()) + c(f)]]. (11) )=0

Then, from (9) and (11), we have

Va z(n+1) < b(n)H [1 + B(n — 1, a;))[ b()) + c())]]. )=0

A(n) = u(0)H [1 + B(n — 1, a;))[ a()) + c())] ] )=0 n—1

x ^ B(n — 1, a; ))a())b()) )=0 n—1

x Y[ [1 + B(n — 1,a;k)[a(k) + c(k)]]. k=)+1

Proof. Define a function

z(n) = u(0) + J2 B(n — 1, a; ))a()) )=0

U()) + b()) +J2 B) — 1, a; k)c(k)u(k)

Then, z(0) = u(0), u(n) < z(n), and

If [1 - B(n - 1, a; j)a(j)] > 0 and [1 + B(n - 1, a; j) [ a(j) - b(j)] ] > 0for all 0 < j < (n -1), then,forn e N+,

Vaz(n + 1) < a(n)b(n) + a(n)

z(n) +J2 B(n - 1, a; k)c(k)z(k)

u(n) < u(0)

v(n) = z(n) + J2B(n - 1, a; k)c(k)z(k).

n [1 - B(n - 1, a; j)a(j)] j=0

x ^B(n - 1, a; j)a(j)C(j) j=0 n—1

x EI [1 - B(n - 1, a; k)a(k)] k=j+1

Then, v(0) = z(0), z(n) < v(n), and Vav(n + 1) = Vaz(n+1)+c(n)z(n) < a(n)b(n)+[a(n)+c(n)] v(n).Now, an application of Theorem 3 yields

v(n) < v(0^ [1 + B(n - 1,a;j)[a(j) + c(j)] ] j=0 n—1

+ J2B(n - 1, a; j)a(j)b(j) j=0 n—1

x Yl [1 + B(n - 1,a;k)[a(k) + c(k)]] = A(n) k=j+1

Then, from (17) and (19), we have

Vaz(n + 1) < a(n)[b(n) + A(n)]. (20) Now, again by application of Theorem 3, we get

z(n) < z(0) + J2B(n - 1, a; j)a(j)[ b(j) + A(j)]. (21) j=0

Using (21) in u(n) < z(n), we get the required inequality (15). □

Theorem 6. Leta(n) be a nonnegative real valued function defined on N+. Suppose the following inequality holds for all n e N+:

B(n) ^ [1 + B(n - 1,a;j)[a(j) + b(j) + c(j)] ] j=0

C(n) ^ f] [1 + B(n - 1,a;j)[a(j) - b(j)] ] j=0 n—1

x J2B(n - 1, a; j)b(j)B(j) j=0 n—1

x H [1 + B(n - 1,a;k)[a(k) - b(k)]]. k=j+1

Proof. Define a function

z(n) = u(0) + J2B(n - 1, a; j)a(j) j=0 -k-1

J^B(k - 1, a; l)c(l)u(l)

J^B(j - 1, a; k)b(k) k=0

Then, z(0) = u(0), u(n) < z(n), and

u(n) < u(0) + B(n - 1, a; j)a(j) j=0 rk-1

J^B(k - 1, a; l)c(l)u(l)

J^B(j - 1, a; k)b(k) k=0

Vaz(n + 1) < a(n)

J^B(n - 1, a; k)b(k) k-1

J^B(k - 1, a; l)c(l)z(l)

Adding a(n)z(n) to both sides of the above inequality, we have

Va z(n + 1) + a(n)z(n) < a(n)

z(n) + J2 B(n — 1, a; k)b(k) k=0 k—1

Y^B(k — 1, a; l)c(l)z(l) Ll=0

v(n) = z(n)+Y^ B(n—1, a; k)b(k)

EB(k — 1, a; l)c(l)z(l) ,l=0 (28)

Then, v(0) = z(0), z(n) < v(n), and

Vav(n+1) = Vaz(n+1)+b(n)J2B(n—1, a; l)c(l)z(l).

Using the facts that Vaz(n + 1) < a(n)v(n) and z(n) < v(n), we get

Vav(n + 1) < a(n)v(n) + b(n)J2B(n — 1,a;l)c(l)z(l).

Adding b(n)v(n) to both sides of the above inequality, we have

Vav(n + 1) + b(n)v(n) < a(n)v(n)

+ b(n)

v(n) + J2 B(n — 1, a; l)c(l)z(l)

w(n) = v(n) +J2 B(n — 1, a; l)c(l)z(l). l=0

Then, w(0) = v(0), v(n) < w(n), and

Vaw(n + 1) <Vav(n + 1) + c(n)w(n). (33)

Now, from (30) and (31), we have

Vav(n+1) < Vav(n+1)+b(n)v(n) < [ a(n)+b(n)] w(n).

Using (34) in (33), we get Vaw(n + 1) < [ a(n) + b(n) + c(n)] w(n). (35)

Now, an application of Theorem 3 yields

w(n) < w(0) n [1 + B(n — 1,a;j)[a(j) + b(j) + c(j)] ] j=0

= w(0)B(n). (36)

Then, from (31) and (35), we have

Vav(n + 1) < [ a(n) — b(n)] v(n) + w(0)b(n)B(n). Now, again by application of Theorem 3, we get

v(n) < v(0) n [1 + B(n — 1,a;j)[a(j) — b(j)] ] j=0 n—1

+w(0)J2 B(n — 1, a; j)b(j)B(j) j=0

x [1 + B(n — 1,a;k)[a(k) — b(k)]] k=j+1

= v(0)C(n). (37)

Then, from (27) and (37), we get

Vaz(n + 1) < —a(n)z(n) + v(0)a(n)C(n). Now, again by application of Theorem 3, we get

z(n) < z(0) H [1 — B(n — 1, a; j)a(j)] j=0 n—1

+ v(0)E B(n — 1, a; j)a(j)C(j) j=0

x n [1 — B(n — 1, a; k)a(k)]. (38)

Using (38) in u(n) < z(n), we get the required inequality (23). □

Theorem 7. Let a(n) be a nonnegative real valuedfunc-tion defined on N+. Suppose the following inequality holds for all n e N+:

J^B(j — 1, a; k)b(k) k=0

u(n) < u(0) +EB(n — 1, a; j)a(j) j=0 rk—1

J^B(k — 1, a; l)[ c(l)u(l) + p(l)] .l=0

If [1 - B(n - 1, a; j)a(j)] > 0 and [1 + B(n - 1, a; j) [ a(j) - b(j)] ] > 0 for all 0 < j < (n -1), then, for n e N+

u(n) < u(0) ^ [1 - B(n - 1, a; j)a(j)] j=0 n-1

+ J2B(n - 1, a; j)a(j)E(j) j=0 n-1

x EI [1 - B(n - 1, a; k)a(k)], (40)

D(n) = u(0) [] [1 + B(n - 1, a; j)[ a(j) + b(j) + c(j)] ] j=0 n—1 n—1

+ Ep(j) n [1+B(n-1, a;k)[ a(k)+b(k)+c(k)] ]

j=0 k=j+1

E(n) = u(0) ^ [1 + B(n - 1,a;j)[a(j) - b(j)] ] j=0 n-1

+ J2B(n - 1, a; j)b(j)D(j)

j=0 n-1

x EI [1 + B(n - 1, a; k)[ a(k) - b(k)]]. k=j+1

Theorem 8. Leta(n) be a nonnegative real valued function defined on N+. Suppose the following inequality holds for all n e N+:

u(n) < p(n) + q(n)^2, B(n - 1, a; j)a(j)

J^B(j - 1, a; k)b(k)

J^B(k - 1, a; l)c(l)u(l)

If [1 - B(n - 1, a; j)a(j)] > 0 and [1 + B(n - 1, a; j) [ a(j) - b(j)]] > 0 for all 0 < j < (n -1), then, for n e N+,

u(n) < p(n) + q(n)

J^B(n - 1,a;j)a(j)G(j) j=0

x EI [1 - B(n - 1, a; k)a(k)] k=j+1

F (n) =J2 B(n — 1, a;))[ c())p())] )=0 n—1

x EI [1+B(n— 1,a;k)[a(k) +b(k)+ c(k)q(k)]] k=)+1

G(n) =J2 B(n — 1, a; ))b())F ()) )=0 n—1

x EI [1 + B(n — 1,a;k)[a(k) — b(k)] ]. k=)+1

Proof. Define a function z(n) by

z(n) = ^^ B(n — 1, a; j)a(j)

Y2,B(j - 1,a;k)b(k)

^2,B(k - 1, a; l)c(l)u(l) ,l=0

Then, z(0) = 0, u(n) < p(n) + q(n)z(n), and using the same argument as in the proof of Theorem 6, we obtain

"n—1

J2B(n — 1, a; k)b(k) .k=0

'k—1

J2,B(k — 1, a; l)[ c(l)p(l) +c(l)q(l)z(l)]

Vaz(n+1) < a(n)

Adding a(n)z(n) to both sides of the above inequality, we have

Va z(n + 1) + a(n)z(n)

< a(n)

z(n) + J2B(n - 1, a; k)b(k)

J^B(k-1, a; l) [ c(l)p(l)+c(l)q(l)z(l)]

v(n) = z(n) + J2B(n - 1, a; k)b(k)

J^B(k - 1, a; l)[ c(l)p(l) + c(l)q(l)z(l)]

Then, v(0) = z(0), z(n) < v(n), and Va v(n + 1) =Va z(n + 1)

n— 1

+ b(n)J2 B(n — 1, a; l)[ c(l)p(l) + c(l)q(l)z(l)].

Using the facts that Vaz(n + 1) < a(n)v(n) and z(n) < v(n), we get

Va v(n + 1) < a(n)v(n)

n— 1

+ b(n)J2 B(n — 1, a; l) [ c(l)p(l) + c(l)q(l)z(l)].

Adding b(n)v(n) to both sides of the above inequality, we have

Va v(n + 1) + b(n)v(n) < a(n)v(n)

n—1 -

+b(n) v(n) +EB(n—1, a; l)[c(l)p(l)+c(l)q(l)z(l)]

Now, again by application of Theorem 3, we get

v(n) < J2B(n — 1, a;j)b(j)F(j)

w(n) = v(n) + J2B(n — 1, a; l)[ c(l)p(l) + c(l)q(l)z(l)]. l=0

Then, w(0) = v(0), v(n) < w(n), and

Vaw(n +1) < Vav(n +1)+[ c(n)p(n) + c(n)q(n)w(n)].

Now, from (53) and (54), we have

Vav(n+1) < Vav(n+1)+b(n)v(n) <[a(n)+b(n)] w(n).

Using (56) in (55), we get

Vaw(n +1) <[a(n) + b(n) + c(n)q(n)] w(n) + c(n)p(n).

Now, an application of Theorem 3 yields

w(n) < £B(n — 1, a;j)[ c(j)p(j)]

x Y\ [1 + B(n — 1, a; k)[ a(k) + b(k) + c(k)q(k)]]

= F (n). (58)

Then, from (53) and (58), we have Vav(n + 1) < [ a(n) — b(n)] v(n) + b(n)F(n).

j=0 n—1

x EI C1 + B(n — 1, a; k) [ a(k) — b(k)] ] = G(n). k=j+1

Then, from (49) and (59), we get

Vaz(n + 1) < —a(n)z(n) + a(n)G(n). Now, again by application of Theorem 3, we get

z(n) <J2B(n — 1,a;j)a(j)G(j) j=0

x EI [! — B(n — 1, a; k)a(k)]. (60)

Using (60) in u(n) < p(n)+q(n)z(n), we get the required inequality (44). □

Conclusions

In this paper, some new Gronwall-Bellman-type fractional difference inequalities are established which provide explicit bounds for the solutions of fractional difference equations.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Both authors gave excellent contributions to the final manuscript. Both authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the referees for their suggestions and comments which considerably helped improve the content of this paper.

Received: 28 August 2012 Accepted: 11 November 2012 Published: 10 December 2012

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doi:10.1186/2251-7456-6-69

Cite this article as: Deekshitulu and Mohan: Some newfraetional difference inequalities of Gronwall-Bellman type. Mathematical Sciences 2012 6:69.

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