A note on some discrete inequalities *.

Abstract

In this note we establish some new iterated discrete inequalities which can be used as tools in the study of certain classes of sum-difference equations.

Keywords and Phrases: Discrete inequalities, Iterated sums, Sum-difference equations, Explicit bounds, Empty sums and product, Estimate on the solution, Uniqueness of solutions.

1. Introduction

The discrete inequalities which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of difference equations and numerical analysis. It is hardly imagine these theories without such inequalities. In 1979 Rab [4] (see also [5]) has given the explicit upper bound on the following useful iterated integral inequality 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

for t [member of] [[alpha], [beta]] under some suitable conditions on the functions involved in (1.1). It is natural to expect that the discrete generalizations of the above inequality would be equally important in certain new applications. Motivated and inspired by the above inequality (see also [1,2,3]), in the present note we establish some new discrete generalizations of the above inequality which can be used as tools in the study of certain classes of sum-difference equations. Some applications are also given to illustrate the usefulness of one of our results.

2. Statement of Results

In what follows R denotes the set of real numbers. Let [R.sub.+] = [0, [infinity]) and [N.sub.0] = {0, 1, 2, ...} be the given subsets of R. For i = 1, ..., n, let [J.sub.i] = {([t.sub.1], ..., [t.sub.i]) : ([t.sub.1], ..., [t.sub.i]) [member of] [N.sup.i.sub.0]}. For any functions w (t) : [N.sub.0] [right arrow] [R.sub.+], [k.sub.i] ([t.sub.1], ..., [t.sub.i]) : [J.sub.i] [right arrow] [R.sub.+], first we give the following notations used to simplify the details of presentation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We define [DELTA]w (t) = w (t + 1) - w(t) and use the usual conventions that the empty sums and products are taken to be 0 and 1 respectively.

Our main results are established in the following theorems.

Theorem 1. Let u(t) and [k.sub.i] ([t.sub.1], ... [t.sub.i]) for i = 1, ..., n be real-valued nonnegative functions defined on [N.sub.0] and [J.sub.i] respectively.

([a.sub.1]) Let c [greater than or equal to] 0 be a constant. If

U(t) [less than or equal to] c + [n.summation over (i=1)][B.sub.1][u](t), (2.1)

for t [member of] [N.sub.0], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

for t [member of] [N.sub.0].

([a.sub.2]) Let a(t) be a real-valued, nonnegative and nondecreasing function on [N.sub.0]. If

u(t) [less than or equal to] a(t) + [n.summation over (i=1)][B.sub.i][u](t), (2.3)

for t [member of] [N.sub.0], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

for t [member of] [N.sub.0].

Theorem 2. Let u(t), [k.sub.i] ([t.sub.1], ... [t.sub.i]), i = 1, ..., n be as in Theorem 1. ([b.sub.1]) Let [??](t) be a real-valued nonnegative function defined on [N.sub.0] and nondecreasing. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

for t [member of] [N.sub.0], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

for t [member of] [N.sub.0].

186 B. G. Pachpatte

([b.sub.2]) Let a(t), b(t) be real-valued nonnegative functions defined on [N.sub.0]. If

u(t) [less than or equal to] a(t) + b(t) [n.summation over (i=1)][B.sub.i][u](t), (2.7)

for t [member of] [N.sub.0], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

for t [member of] [N.sub.0].

3. Proofs of Theorems 1 and 2

([a.sub.1]) Define a function z(t) by the right hand side of (2.1), i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then z(0) = c, u(t) [less than or equal to] z(t) , z(t) is nondecreasing for t [member of] [N.sub.0] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e.

[DELTA]z(t) [less than or equal to] G [1] (t)z(t). (3.1)

Now a suitable application of Theorem 1.2.1 given in [2] to (3.1) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

Using (3.2) in u(t) [less than or equal to] z(t) we get the desired inequality in (2.2).

([a.sub.2]) The proof can be completed by closely looking at the proof of Theorem 1.2.4 given in [2,p.14] and by making use of the inequality established in ([a.sub.1]).

([b.sub.1]) From the hypotheses on [??](t) we observe that [DELTA][??](t) [greater than or equal to] 0 for t [member of] [N.sub.0] . Define a function z(t) by the right hand side of (2.5). Then z(0) = [??](0), u(t) [less than or equal to] z(t), z(t) is nondecreasing for t [member of] [N.sub.0] and as in the proof of part ([a.sub.1]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3)

Now a suitable application of Theorem 1.2.1 given in [2] to (3.3) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

Using (3.4) in u(t) [less than or equal to] z(t) we get the required inequality in (2.6).

([b.sub.4]) Define a function z(t) by

z(t) = [n.summation over (i=1)][B.sub.i][u](t).

Then z(0) = 0, z(t) is nondecreasing for t [member of] [N.sub.0] , (2.7) can be restated as

u(t) [less than or equal to] a(t) + b(t) z(t) (3.5)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

Now an application of Theorem 1.2.1 given in [2] with z(0) = 0 to (3.6) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7)

Using (3.7) in (3.5) we get the required inequality in (2.8).

4. Applications

In this section, we present applications of the inequality established in Theorem 1, part ([a.sub.1]) to study certain properties of solutions of the nonlinear sum-difference equation of the form

x(t) = f(t) + [n.summation over (i=1)][A.sub.i][x](t), (4.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for t [member of] [N.sub.0], [F.sub.i] : [J.sub.i] x R [right arrow] R for i = 1, ...,n and f : [N.sub.0] [right arrow] R.

The following theorem deals with the estimate on the solution of equation (4.1).

Theorem 3. Suppose that the functions f, [F.sub.i] in equation (4.1) satisfy the conditions

[absolute value of(t)] [less than or equal to] c, (4.2)

[absolute value of [F.sub.i]([t.sub.1], ...,[t.sub.i], x([t.sub.i]))] [less than or equal to] [k.sub.i] ([t.sub.1], ..., [t.sub.i]) [absolute value of x([t.sub.i])], (4.3)

for i = 1, ..., n , where c , [k.sub.i] are as in Theorem 1, part ([a.sub.1]). If x(t) is any solution of equation (4.1) on [N.sub.0], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

for t [member of] [N.sub.0].

Proof. Let x(t) be a solution of equation (4.1) on [N.sub.0]. Using (4.2), (4.3) in (4.1) we have

[absolute value of x(t)] [less than or equal to] c + [n.summation over (i=1)][B.sub.i][[absolute value of x]](t). (4.5)

Now a suitable application of the inequality given in Theorem 1, part ([a.sub.1]) to (4.5) yields the required estimate in (4.4).

The next result deals with the uniqueness of solutions of equation (4.1).

Theorem 4. Suppose that the functions [F.sub.i], i = 1, ..., n in equation (4.1) satisfy the conditions

[absolute value of [F.sub.i]([t.sub.1], ...,[t.sub.i], x([t.sub.i])) - [F.sub.i] ([t.sub.1], ..., y([t.sub.i]))] [less than or equal to] [k.sub.i]([t.sub.1], ..., [t.sub.i]) [absolute value of x([t.sub.i]) - y([t.sub.i])], (4.6)

where [k.sub.i] are as in Theorem 1. Then the equation (4.1) has at most one solution on [N.sub.0].

Proof. Let y(t) and z(t) be two solutions of equation (4.1) on [N.sub.0]. Using the facts that y(t) and z(t) are the solutions of equation (4.1) and the conditions (4.6) we have

[absolute value of y(t) - z(t)] [less than or equal to] [n.summation over (i=1)] [B.sub.i][[absolute value of y - z]](t). (4.7)

Now a suitable application of the inequality given in Theorem 1, part ([a.sub.1]) to (4.7) (when c = 0) yields y(t) = z(t), i.e. there is at most one solution of equation (4.1) on [N.sub.0].

In concluding, we note that, one can very easily obtain the explicit bound on the following inequality

u(t) [less than or equal to] c + [n.summation over (i=1)][B.sub.i][g(u)](t),

and its further generalizations, under some suitable conditions on the functions involved therein (see also [2]). We leave the formulation of such results to the reader to fill in where needed. Various applications of the other inequalities established here will be given elsewhere.

Received March 30, 2004, Accepted December 31, 2004.

References

[1] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.

[2] B. G. Pachpatte, Inequalities for Finite Difference Equations, Marcel Dekker, Inc., New York, 2002.

[3] B. G. Pachpatte, Some new finite difference inequalities, Computer Math. Appl. 28 (1994), 227-241.

[4] M. Rab, Linear integral inequalities, Arch. Math. 1. Scripta Fac. Sci. Nat. Ujep Brunensis 15 (1979), 37-46.

[5] E. C. Young, On integral inequalities of Gronwall-Bellman type, Proc. Amer. Math. Soc. 94 (1985), 636-640.

* 1991 Mathematics SubjectClassifications. 26D10, 26D15.

([dagger]) E-mail:[e-mail]

B. G. Pachpatte ([dagger])

Shri Niketan Colony, Near Abhinay Talkies,

Aurangabad 431 001 (Maharashtra) India