Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 2, Article 15, 21 ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES B.G. PACHPATTE DEPARTMENT OF MATHEMATICS, MARATHWADA UNIVERSITY AURANGABAD 431 4 (MAHARASHTRA INDIA. bgpachpatte@hotmail.com Received 8 November, 2; accepted 2 Januar, 21 Communicated b D. Bainov ABSTRACT. The aim of the present paper is to establish some new integral inequalities in two independent variables and their discrete analogues which provide eplicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theor of certain partial differential and finite difference equations. Ke words and phrases: Integral inequalities, discrete analogues, two independent variables, partial differential and difference equations, nondecreasing, nonincreasing, terminal value, non-self-adjoint hperbolic partial differential equation. 2 Mathematics Subject Classification. 26D1, 26D15. 1. INTRODUCTION The integral and finite difference inequalities involving functions of one and more than one independent variables which provide eplicit bounds on unknown functions pla a fundamental role in the development of the theor of differential and finite difference equations. During the past few ears, man such new inequalities have been discovered, which are motivated b certain applications (see 1] 1]. In the qualitative analsis of some classes of partial differential and finite difference equations, the bounds provided b the earlier inequalities are inadequate and it is necessar to seek some new inequalities in order to achieve a diversit of desired goals. Our main objective here is to establish some useful integral inequalities involving functions of two independent variables and their discrete analogues which can be used as read and powerful tools in the analsis of certain classes of partial differential and finite difference equations. 2. STATEMENT OF RESULTS In what follows, R denotes the set of real numbers and R + =,, N = {, 1, 2,... } are the given subsets of R. The first order partial derivatives of a function z (, defined for, R with respect to and are denoted b z (, and z (, respectivel. We ISSN (electronic: 1443-5756 c 21 Victoria Universit. All rights reserved. 46-
2 B.G. PACHPATTE use the usual conventions that empt sums and products are taken to be and 1 respectivel. Throughout the paper, all the functions which appear in the inequalities are assumed to be realvalued and all the integrals, sums and products involved eist on the respective domains of their definitions. We need the inequalities in the following lemma, which are the slight variants of the inequalities given in 5, pp. 12, 28]. Lemma 2.1. Let u (t, a (t, b (t be nonnegative and continuous functions defined for t R +. (α 1 Assume that a (t is nondecreasing for t R +. If u (t a (t + t b (s u (s ds, for t R +, then ( t u (t a (t ep b (s ds, for t R +. (α 2 Assume that a (t is nonincreasing for t R +. If u (t a (t + t b (s u (s ds, for t R +, then ( u (t a (t ep b (s ds, t for t R +. The proofs of the inequalities in (α 1, (α 2 can be completed as in 5, pp. 12, 28, 325-326] (see also 4]. Here we omit the details. Our main results on integral inequalities are established in the following theorems. Theorem 2.2. Let u (,, a (,, b (,, c (, be nonnegative continuous functions defined for, R +. (a 1 If (2.1 u (, a (, + b (, for, R +, then ( (2.2 u (, a (, + b (, e (, ep for, R +, where (2.3 e (, = for, R +. (a 2 If (2.4 u (, a (, + b (, c (s, t u (s, t dtds, c (s, t a (s, t dtds, for, R +, then ( (2.5 u (, a (, + b (, ē (, ep c (s, t b (s, t dtds, c (s, t u (s, t dtds, c (s, t b (s, t dtds, J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 3 for, R +, where (2.6 ē (, = for, R +. c (s, t a (s, t dtds, Theorem 2.3. Let u (,, a (,, b (,, c (, be nonnegative continuous functions defined for, R +. (b 1 Assume that a (, is nondecreasing in R +. If (2.7 u (, a (, + for, R +, then (2.8 u (, p (, (2.9 (2.1 b (s, u (s, ds + ( a (, + A (, ep for, R +, where ( p (, = ep A (, = b (s, ds, c (s, t p (s, t a (s, t dtds, for, R +. (b 2 Assume that a (, is nonincreasing in R +. If (2.11 u (, a (, + b (s, u (s, ds + for, R +, then ( (2.12 u (, p (, a (, + Ā (, ep (2.13 (2.14 for, R +, where ( p (, = ep for, R +. Ā (, = b (s, ds, c (s, t p (s, t a (s, t dtds, c (s, t u (s, t dtds, ] c (s, t p (s, t dtds, c (s, t u (s, t dtds, ] c (s, t p (s, t dtds, Theorem 2.4. Let u (,, a (,, b (, be nonnegative continuous functions defined for, R + and F : R 3 + R + be a continuous function which satisfies the condition F (,, u F (,, v K (,, v (u v, for u v, where K (,, v is a nonnegative continuous function defined for,, v R +. (c 1 Assume that a (, is nondecreasing in R +. If (2.15 u (, a (, + b (s, u (s, ds + F (s, t, u (s, t dtds, J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
4 B.G. PACHPATTE for, R +, then (2.16 u (, p (, a (, + B (, ( ep for, R +, where (2.17 B (, = ] K (s, t, p (s, t a (s, t p (s, t dtds, F (s, t, p (s, t a (s, t dtds, for, R + and p (, is defined b (2.9. (c 2 Assume that a (, is nonincreasing in R +. If (2.18 u (, a (s, + b (s, u (s, ds + for, R +, then (2.19 u (, p (, a (, + B (, ( ep for, R +, where (2.2 B (, = F (s, t, u (s, t dtds, ] K (s, t, p (s, t a (s, t p (s, t dtds, F (s, t, p (s, t a (s, t dtds, for, R + and p (, is defined b (2.13. We require the following discrete version of Lemma 2.1 to establish the discrete analogues of Theorems 2.3 and 2.4. Lemma 2.5. Let u (n, a (n, b (n be nonnegative functions defined for n N. (β 1 Assume that a (n is nondecreasing for n N. If for n N, then n 1 u (n a (n + b (s u (s, s= n 1 u (n a (n b (s], for n N. (β 2 Assume that a (n is nonincreasing for n N. If u (n a (n + b (s u (s, for n N, then for n N. u (n a (n s= s=n+1 s=n+1 b (s], J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 5 The proof of (β 1 can be completed b following the proof of Theorem 1 o in 6, p. 256] and closel looking at the proof of Theorem 4.2.2 in 5, p. 326]. For the proof of (β 2, see 1] and also 4]. The discrete analogues of Theorems 2.2 2.4 are given in the following theorems. Theorem 2.6. Let u (m, n, a (m, n, b (m, n, c (m, n be nonnegative functions defined for m, n N. (p 1 If (2.21 u (m, n a (m, n + b (m, n for m, n N, then (2.22 u (m, n a (m, n + b (m, n f (m, n for m, n N, where (2.23 f (m, n = for m, n N. (p 2 If s= (2.24 u (m, n a (m, n + b (m, n for m, n N, then (2.25 u (m, n a (m, n + b (m, n f (m, n for m, n N, where (2.26 f (m, n = for m, n N. s= s= c (s, t u (s, t, c (s, t a (s, t, s=m+1 s=m+1 s=m+1 c (s, t u (s, t, c (s, t a (s, t, c (s, t b (s, t ] c (s, t b (s, t Theorem 2.7. Let u (m, n, a (m, n, b (m, n, c (m, n be nonnegative functions defined for m, n N. (q 1 Assume that a (m, n is nondecreasing in m N. If (2.27 u (m, n a (m, n + for m, n N, then (2.28 u (m, n q (m, n s= b (s, n u (s, n + a (m, n + G (m, n s= s=m+1, ] c (s, t u (s, t, c (s, t q (s, t ]],, J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
6 B.G. PACHPATTE (2.29 (2.3 for m, n N, where q (m, n = G (m, n = s= b (s, n], s= c (s, t q (s, t a (s, t, for m, n N. (q 2 Assume that a (m, n is nonincreasing in m N. If (2.31 u (m, n a (m, n + b (s, n u (s, n + for m, n N, then (2.32 u (m, n q (m, n (2.33 (2.34 for m, n N, where for m, n N. q (m, n = Ḡ (m, n = s=m+1 a (m, n + Ḡ (m, n s=m+1 s=m+1 b (s, n], s=m+1 s=m+1 c (s, t q (s, t a (s, t, c (s, t u (s, t, c (s, t q (s, t Theorem 2.8. Let u (m, n, a (m, n, b (m, n be nonnegative functions defined for m, n N and L : N 2 R + R + be a function which satisfies the condition L (m, n, u L (m, n, v M (m, n, v (u v, for u v, where M (m, n, v is a nonnegative function for m, n N, v R +. (r 1 Assume that a (m, n is nondecreasing in m N. If (2.35 u (m, n a (m, n + s= b (s, n u (s, n + for m, n N, then (2.36 u (m, n q (m, n a (m, n + H (m, n for m, n N, where (2.37 H (m, n = s= s= for m, n N and q (m, n is defined b (2.29. s= L (s, t, q (s, t a (s, t, L (s, t, u (s, t, ]] M (s, t, q (s, t a (s, t q (s, t, ]], J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 7 (q 2 Assume that a (m, n is nonincreasing in m N. If (2.38 u (m, n a (m, n + for m, n N, then s=m+1 b (s, n u (s, n + (2.39 u (m, n q (m, n a (m, n + H (m, n for m, n N, where (2.4 H (m, n = s=m+1 s=m+1 for m, n N and q (m, n is defined b (2.33. s=m+1 L (s, t, u (s, t, M (s, t, q (s, t a (s, t q (s, t L (s, t, q (s, t a (s, t, ]], 3. PROOFS OF THEOREMS 2.2 2.4 Since the proofs resemble one another, we give the details for (a 1, (b 1 and (c 1 ; the proofs of (a 2, (b 2 and (c 2 can be completed b following the proofs of the above mentioned results with suitable changes. (a 1 Define a function z (, b (3.1 z (, = Then (2.1 can be restated as c (s, t u (s, t dtds. (3.2 u (, a (, + b (, z (,. From (3.1 and (3.2 we have (3.3 z (, = e (, + c (s, t a (s, t + b (s, t z (s, t] dtds c (s, t b (s, t z (s, t dtds, where e (, is defined b (2.3. Clearl, e (, is nonnegative, continuous, nondecreasing in and nonincreasing in for, R +. First we assume that e (, > for, R +. From (3.3 it is eas to observe that (3.4 z (, e (, c (s, t b (s, t z (s, t e (s, t dtds. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
8 B.G. PACHPATTE Define a function v (, b the right hand side of (3.4, then v (, = v (, = 1, z(, e(, v (,, v (, is nonincreasing in, R + and (3.5 v (, = v (, c (, t b (, t z (, t e (, t dt c (, t b (, t v (, t dt c (, t b (, t dt. Treating, R + fied in (3.5, dividing both sides of (3.5 b v (,, setting = s and integrating the resulting inequalit from to, R + we get ( (3.6 v (, ep c (s, t b (s, t dtds. Using (3.6 in z(, v (,, we have e(, ( (3.7 z (, e (, ep c (s, t b (s, t dtds. The desired inequalit (2.2 follows from (3.2 and (3.7. If e (, is nonnegative, we carr out the above procedure with e (, +ε instead of e (,, where ε > is an arbitrar small constant, and then subsequentl pass to the limit as ε to obtain (2.2. (b 1 Define a function z (, b (3.8 z (, = Then (2.7 can be restated as (3.9 u (, z (, + c (s, t u (s, t dtds. b (s, u (s, ds. Clearl z (, is a nonnegative, continuous and nondecreasing function in, R +. Treating, R + fied in (3.9 and using part (α 1 of Lemma 2.1 to (3.9, we get (3.1 u (, z (, p (,, where p (, is defined b (2.9. From (3.1 and (3.8 we have (3.11 u (, p (, a (, + v (, ], where (3.12 v (, = From (3.11 and (3.12 we get v (, = A (, + c (s, t u (s, t dtds. c (s, t p (s, t a (s, t + v (s, t] dtds c (s, t p (s, t v (s, t dtds, J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 9 where A (, is defined b (2.1. Clearl, A (, is nonnegative, continuous, nondecreasing in, R + and nonincreasing in, R +. Now, b following the proof of (a 1, we obtain ( (3.13 v (, A (, ep c (s, t p (s, t dtds. Using (3.13 in (3.11 we get the required inequalit in (2.8. (c 1 Define a function z (, b (3.14 z (, = a (, + Then (2.15 can be restated as (3.15 u (, z (, + F (s, t, u (s, t dtds. b (s, u (s, ds. Clearl, z (, is a nonnegative, continuous and nondecreasing function in, R +. Treating, R + fied in (3.15 and using part (α 1 of Lemma 2.1 to (3.15, we obtain (3.16 u (, z (, p (,, where p (, is defined b (2.9. From (3.16 and (3.15 we have (3.17 u (, p (, a (, + v (, ], where (3.18 v (, = F (s, t, u (s, t dtds. From (3.18, (3.17 and the hpotheses on F it follows that (3.19 v (, F (s, t, p (s, t (a (s, t + v (s, t F (s, t, p (s, t a (s, t + F (s, t, p (s, t a (s, t]dtds B (, + K (s, t, p (s, t a (s, t p (s, t v (s, t dtds. Clearl, B (, is nonnegative, continuous and nondecreasing in and nonincreasing in for, R +. B following the proof of (a 1, we get ( (3.2 v (, B (, ep K (s, t, p (s, t a (s, t p (s, t dtds. The required inequalit (2.16 follows from (3.17 and (3.2. 4. PROOFS OF THEOREMS 2.6 2.8 We give the proofs of (p 1, (q 1, (r 1 onl; the proofs of (p 2, (q 2, (r 2 can be completed b following the proofs of the above mentioned inequalities. (p 1 Define a function z (m, n b (4.1 z (m, n = Then (2.21 can be stated as s= c (s, t u (s, t. (4.2 u (m, n a (m, n + b (m, n z (m, n. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
1 B.G. PACHPATTE From (4.1 and (4.2 we have (4.3 z (m, n s= = f (m, n + c (s, t a (s, t + b (s, t z (s, t] s= c (s, t b (s, t z (s, t, where f (m, n is defined b (2.23. Clearl, f (m, n is nonnegative, nondecreasing in m and nonincreasing in n for m, n N. First, we assume that f (m, n > for m, n N. From (4.3 we observe that Define a function v (m, n b z (m, n f (m, n (4.4 v (m, n = then z(m,n f(m,n v (m, n and s= s= c (s, t b (s, t c (s, t b (s, t z (s, t f (s, t. z (s, t f (s, t, (4.5 v (m + 1, n v (m, n] v (m + 1, n + 1 v (m, n + 1] z (m, n + 1 = c (m, n + 1 b (m, n + 1 f (m, n + 1 c (m, n + 1 b (m, n + 1 v (m, n + 1. From (4.5 and using the facts that v (m, n >, v (m, n + 1 v (m, n for m, n N, we observe that (4.6 v (m + 1, n v (m, n] v (m, n v (m + 1, n + 1 v (m, n + 1] v (m, n + 1 c (m, n + 1 b (m, n + 1. Keeping m fied in (4.6, set n = t and sum over t = n, n + 1,..., r 1 (r n + 1 is arbitrar in N to obtain (4.7 v (m + 1, n v (m, n] v (m, n v (m + 1, r v (m, r] v (m, r r c (m, t b (m, t, Noting that lim r v (m, r = lim r v (m + 1, r = 1 and b letting r in (4.7 we get i.e., v (m + 1, n v (m, n] v (m, n (4.8 v (m + 1, n c (m, t b (m, t c (m, t b (m, t, ] v (m, n. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 11 Now, b keeping n fied in (4.8 and setting m = s and substituting s =, 1, 2,..., m 1 successivel and using the fact that v (, n = 1, we get ] (4.9 v (m, n c (m, t b (m, t. Using (4.9 in z(m,n f(m,n s= v (m, n we have (4.1 z (m, n f (m, n s= c (s, t b (s, t The required inequalit in (2.22 follows from (4.2 and (4.1. If f (m, n is nonnegative, then we carr out the above procedure with f (m, n + ε instead of f (m, n, where ε > is an arbitrar small constant, and subsequentl pass to the limit as ε to obtain (2.22. (q 1 Define a function z (m, n b (4.11 z (m, n = a (m, n + then (2.27 can be restated as (4.12 u (m, n z (m, n + s= s= c (s, t u (s, t, b (s, n u (s, n. Clearl, z (m, n is nonnegative and nondecreasing in m, m N. Treating n, n N fied in (4.12 and using part (β 1 of Lemma 2.5 to (4.12, we obtain (4.13 u (m, n z (m, n q (m, n, where q (m, n is defined b (2.29. From (4.13 and (4.11 we have (4.14 u (m, n q (m, n a (m, n + v (m, n], where (4.15 v (m, n = From (4.14 and (4.15, it is eas to see that v (m, n G (m, n + s= s= c (s, t u (s, t. ] c (s, t q (s, t v (s, t, where G (m, n is as defined b (2.3. The rest of the proof of (2.28 can be completed b following the proof of (p 1 given above, and we omit the further details. (r 1 The proof follows b closel looking at the proofs of (p 1, (q 1 and (c 1 given above. Here we leave the details to the reader.. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
12 B.G. PACHPATTE 5. SOME APPLICATIONS In this section we present some immediate applications of part (a 2 of Theorem 2.2 to stud certain properties of solutions of the following terminal value problem for the hperbolic partial differential equation (5.1 (5.2 u (, = h (,, u (, + r (,, u (, = σ (, u (, = τ (, u (, = d, where h : R 2 + R R, r : R 2 + R, σ, τ : R + R are continuous functions and d is a real constant. The following theorem deals with the estimate on the solution of (5.1 (5.2 Theorem 5.1. Suppose that (5.3 (5.4 h (,, u c (, u, σ ( + τ ( d + r (s, t dtds a (,, where a (,, c (, are as defined in part (a 2 of Theorem 2.2. Let u (, be a solution of (5.1 (5.2 for, R +, then ( (5.5 u (, a (, + ē (, ep c (s, t dtds, for, R +, where ē (, is defined b (2.6. Proof. If u (, is a solution of (5.1 (5.2, then it can be written as (see 1, p. 8] (5.6 u (, = σ ( + τ ( d + for, R +. From (5.6, (5.3, (5.4 we get (5.7 u (, a (, + h (s, t, u (s, t + r (s, t] dtds, c (s, t u (s, t dtds. Now, a suitable application of part (a 2 of Theorem 2.2 to (5.7 ields the required estimate in (5.5. Our net result deals with the uniqueness of the solutions of (5.1 (5.2. Theorem 5.2. Suppose that the function h in (5.1 satisfies the condition (5.8 h (,, u h (,, v c (, u v, where c (, is as defined in Theorem 2.2. Then the problem (5.1 (5.2 has at most one solution on R 2 +. Proof. The problem (5.1 (5.2 is equivalent to the integral equation (5.6. Let u (,, v (, be two solutions of (5.1 (5.2. From (5.6, (5.8 we have (5.9 u (, v (, c (s, t u (s, t v (s, t dtds. Now a suitable application of part (a 2 of Theorem 2.2 ields u (, = v (,, i.e., there is at most one solution to the problem (5.1 (5.2. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/
ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES 13 We note that the inequalit given in part (b 2 of Theorem 2.3 can be used to obtain the bound and uniqueness of the solutions of the following non-self-adjoint hperbolic partial differential equation (5.1 u (, = (r (, u (, + h (,, u (,, with the given terminal value conditions in (5.2, under some suitable conditions on the functions involved in (5.1 (5.2. We also note that the inequalities given here have man applications, however, various applications of other inequalities is left for another time. REFERENCES 1] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992. 2] S.S. DRAGOMIR, On Gronwall tpe lemmas and applications, Monografii Matematice Univ. Timişoara, No. 29, 1987. 3] S.S. DRAGOMIR AND N.M. IONESCU, On nonlinear integral inequalities in two independent variables, Studia Univ. Babeş-Bolai, Math., 34 (1989, 11 17. 4] A. MATE AND P. NEVAI, Sublinear perturbations of the differential equation (n = and of the analogous difference equation, J. Differential Equations, 52 (1984, 234 257. 5] B.G. PACHPATTE, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998. 6] B.G. PACHPATTE, On some new discrete inequalities and their applications, Proc. Nat. Acad. Sci., India, 46(A (1976, 255 262. 7] B.G. PACHPATTE, On certain new finite difference inequalities, Indian J. Pure Appl. Math., 24 (1993, 373 384. 8] B.G. PACHPATTE, Some new finite difference inequalities, Computer Math. Appl., 28 (1994, 227 241. 9] B.G. PACHPATTE, On some new discrete inequalities useful in the theor of partial finite difference equations, Ann. Differential Equations, 12 (1996, 1 12. 1] B.G. PACHPATTE, On some fundamental finite difference inequalities, Tamkang J. Math., (to appear. J. Inequal. Pure and Appl. Math., 2(2 Art. 15, 21 http://jipam.vu.edu.au/