INTEGRAL INEQUALITIES SIMILAR TO GRONWALL INEQUALITY

Transcription

1 Electronic Journl of Differentil Equtions, Vol. 2007(2007, No. 176, pp ISSN: URL: or ftp ejde.mth.txstte.edu (login: ftp INTEGRAL INEQUALITIES SIMILAR TO GRONWALL INEQUALITY MOHAMED DENCHE, HASSANE KHELLAF Abstrct. In the present pper, we estblish some nonliner integrl inequlities for functions of one vrible, with further generliztion functions with n independent vribles. We pply our results to system of nonliner differentil equtions for functions of one vrible nd to the nonliner hyperbolic prtil integrodifferentil eqution in n-independent vribles. These results extend the Gronwll type inequlities obtined by Pchptte 6 nd Oguntuse Introduction Integrl inequlities ply big role in the study of differentil integrl eqution nd prtil differentil equtions. They were introduced for by Gronwll in , who gve their pplictions in the study of some problems concerning ordinry differentil eqution. One of the most useful inequlities with one vrible of Gronwll type is stted s follows. Lemm 1.1. Let u, Ψ nd g be rel continuous functions defined in, b, g(t 0 for t, b. Suppose tht on, b we hve the inequlity Then u(t Ψ(t u(t Ψ(t g(su(sds. (1.1 g(sψ(s exp g(σdσ ds. (1.2 Since tht time, the theory of these inequlities knew fst growth nd gret number of monogrphs were devoted to this subject 1, 3, 4, 7. The pplictions of the integrl inequlities were developed in remrkble wy in the study of the existence, the uniqueness, the comprison, the stbility nd continuous dependence of the solution in respect to dt. In the lst few yers, series of generliztions of those inequlities ppered. Among these generliztion, we cn quote Pchptte s work Mthemtics Subject Clssifiction. 26D10, 26D20, 26D15. Key words nd phrses. Integrl inequlity; subdditive nd submultiplictive function; nonliner prtil differentil eqution. c 2007 Texs Stte University - Sn Mrcos. Submitted November 15, Published December 12,

2 2 M. DENCHE, H. KHELLAF EJDE-2007/176 In the present pper we estblish some new nonliner integrl inequlities for functions of one vrible, with further generliztion of these inequlities to function with n independent vribles. These results extend the Gronwll type inequlities obtined by Pchptte 6 nd Oguntuse Mins Results Our min results re given in the following theorems: Theorem 2.1. Let u(t, f(t be nonnegtive continuous functions in rel intervl I =, b. Suppose tht k(t, s nd its prtil derivtives k t (t, s exist nd re nonnegtive continuous functions for lmost every t, s I. Let Φ(u(t be rel-vlued, positive, continuous, strictly non-decresing, subdditive, nd submultiplictive function for u(t 0 nd let W (u(t be rel-vlued, positive, continuous, nd non-decresing function defined for t I. Assume tht (t is positive continuous function nd nondecresing for t I. If u(t (t f(su(sds f(sw ( for τ s t b, then for t t 1, { u(t p(t (t f(sψ 1( Ψ(ζ p(t = 1 ζ = Ψ(x = k(s, τφ(p(τφ( b f(s exp( τ Here Ψ 1 is the inverse of Ψ nd t 1 is chosen so tht Ψ(ζ Proof. Define function z(t by z(t = (t then (2.6 cn be restted s k(s, τφ(p(τφ( k(s, τφ(u(τdτds, (2.1 } f(σdσdτ ds, (2.2 f(σdσds, (2.3 k(b, sφ(p(s(sds, (2.4 ds Φ(W (s, x > 0. (2.5 τ ( f(sw u(t z(t f(σdσdτ Dom(Ψ 1. k(s, τφ(u(τdτ ds, (2.6 f(su(sds. (2.7 Clerly z(t is nonnegtive nd continuous in t I, using lemm 1.1 to (2.7, we get ( u(t z(t f(sz(s exp f(σdσ ds; (2.8

3 EJDE-2007/176 INTEGRAL INEQUALITIES 3 moreover if z(t is nondecresing in t I, we obtin p(t is defined by 2.3. From (2.6, we hve z(t (t v(t = From (2.9 we observe tht ( v(t k(t, sφ p(s (s b ζ k(t, sφ(p(s(sds k(b, sφ(p(s(sds ( k(t, sφ p(s u(t z(tp(t, (2.9 f(sw (v(sds, (2.10 k(t, sφ(u(sds. (2.11 f(τdτ f(τw (v(τdτ ds ( k(t, sφ p(s ( k(t, sφ p(s Φ(W (v(sds. f(τw (v(τdτ ds f(τdτ Φ(W (v(sds (2.12 Where ζ is defined by (2.4. Since Φ is subdditive nd submultiplictive function, W nd v(t re nondecresing. Define r(t s the right side of (2.12, then r( = ζ nd v(t r(t, r(t is positive nondecresing in t I nd ( r (t = k(t, tφ p(t f(τdτ Φ(W (v(t k t (t, sφ(p(s ( Φ(W (r(t k(t, tφ p(t ( k t (t, sφ p(s f(τdτφ(w (v(sds, f(τdτ dividing both sides of (2.13 by Φ(W (r(t we obtin r (t Φ(W (r(t k(t, sφ(p(s f(τdτ ds, (2.13 f(τdτds. (2.14 Note tht for ds Ψ(x = Φ(W (s, x > 0, it follows tht Ψ(r(t r (t = Φ(W (r(t. (2.15 From (2.15 nd (2.14, we hve, Ψ(r(t k(t, sφ(p(s f(τdτds (2.16

4 4 M. DENCHE, H. KHELLAF EJDE-2007/176 integrte (2.16 from to t, leds to then Ψ(r(t Ψ(ζ r(t Ψ 1( Ψ(ζ k(t, sφ(p(s k(t, sφ(p(sφ( f(τdτds, By (2.17, (2.12, (2.10 nd (2.9 we hve the desired result f(τdτds. (2.17 The preceding theorem is generliztion of the result obtined by Pchptte in 6, Theorem 2.1. Theorem 2.2. Let u(t, f(t, b(t, h(t be nonnegtive continuous functions in rel intervl I =, b. Suppose tht h(t C 1 (I, R is nondecresing. Let Φ(u(t, W (u(t nd (t be s defined in Theorem 2.1. If ( u(t (t f(su(sds f(sh(sw b(τφ(u(τdτ ds, for τ s t b, then for t t 2, { u(t p(t (t f(sh(sψ 1( Ψ(ϑ ( b(τφ p(τ τ } f(σh(σdσ dτ ds. Where p(t is defined by (2.3, Ψ is defined by (2.5 nd ϑ = b b(sφ(p(s(sds, the t 2 is chosen so tht Ψ(ϑ b(τφ(p(τ τ f(σh(σdσdτ is in Dom(Ψ 1. The proof of the bove theorem follows similr rguments s the proof of Theorem 2.1; So we omit it. The preceding theorem is generliztion of the result obtined by Oguntuse in 5, Theorem 2.3, 2.9. In this section we use the following clss of function. A function g : R R is sid to belong to the clss S if it stisfies the following conditions, (1 g(u is positive, nondecresing nd continuous for u 0 nd (2 (1/vg(u g(u/v, u > 0, v 1. Theorem 2.3. Let u(t, f(t, (t, k(t, s, Φ nd W be s defined in Theorem 2.1, let g S. If ( u(t (t f(sg(u(sds f(sw k(s, τφ(u(τdτ ds, (2.18 for τ s t b, then for t t 3, { u(t p(t (t f(sψ 1( Ψ(ζ k(s, τφ(p(τφ( τ } f(σdσdτ ds, (2.19

5 EJDE-2007/176 INTEGRAL INEQUALITIES 5 p(t = Ω 1( Ω(1 ζ = b Ω(δ = δ f(sds, (2.20 k(b, sφ(p(s(sds, (2.21 ε ds, δ ε > 0. (2.22 g(s Here Ω 1 is the inverse function of Ω, nd Ψ, Ψ 1 re defined in theorem 2.1, t 3 is chosen so tht Ω(1 f(sds is in the domin of Ω 1, nd Ψ(ζ is in the domin of Ψ 1. Proof. Define the function z(t = (t Then (2.18 cn be restted s k(s, τφ(p(τφ ( τ f(σdσ dτ, u(t z(t ( f(sw k(s, τφ(u(τdτ ds. (2.23 f(sg(u(sds. (2.24 When z(x is positive, continuous, nondecresing in x I nd g S, then it cn be restted s u(t t z(t 1 f(sg( u(s ds. (2.25 z(s The inequlity (2.25 my be treted s one-dimensionl Bihri-L Slle inequlity (see 1, which implies u(t p(tz(t, (2.26 p(t is defined by (2.20. By (2.23 nd (2.26 we get u(t p(t (t v(s = f(sw (v(sds, k(s, τφ(u(τdτ. Now, by following the rgument s in the proof of Theorem 2.1, we obtin the desired inequlity in (2.19. Theorem 2.4. Let u(t, f(t, b(t, h(t, Φ(u(t, W (u(t nd (t be s defined in Theorem 2.2, let g S. If u(t (t f(sg(u(sds f(sh(sw ( b(τφ(u(τdτds,

6 6 M. DENCHE, H. KHELLAF EJDE-2007/176 for τ s t b, then for t t 4, { u(t p(t (t f(sh(sψ 1( Ψ(ϑ b(τφ(p(τ τ } f(σh(σdσdτ ds. Here p(t is defined by (2.20, Ψ is defined by (2.5 nd ϑ = b b(sφ(p(s(sds, the vlue t 4 is chosen so tht Ψ(ϑ b(τφ(p(τ τ f(σh(σdσdτ Dom(Ψ 1. The proof of the bove theorem follows similr rguments s in the proof of Theorem 2.3, we omit it. 3. Integrl Inequlities in severl vribles In wht follows we denote by R the set of rel numbers, nd R = 0,. All the functions which pper in the inequlities re ssumed to be rel vlued of n vribles which re nonnegtive nd continuous. All integrls re ssumed to exist on their domins of definitions. Throughout this pper, we ssume tht I = ; b in ny bounded open set in the dimensionl Eucliden spce R n nd tht our integrls re on R n (n 1, = ( 1, 2,..., n, b = (b 1, b 2,..., b n R n. For x = (x 1, x 2,... x n, t = (t 1, t 2,... t n I, we shll denote = n n... dt n... dt 1. Furthermore, for x, t R n, we shll write t x whenever t i x i, i = 1, 2,..., n nd 0 x b, for x I, nd D = D 1 D 2... D n, D i = x i for i = 1, 2,..., n. Let C(I, R denote the clss of continuous functions from I to R. The following theorem dels with n-independent vribles versions of the inequlities estblished in Pchptte 6, Theorem 2.3. Theorem 3.1. Let u(x, f(x, (x be in C(I, R nd let K(x, t, D i k(x, t be in C(I I, R for ll i = 1, 2,..., n, nd let c be nonnegtive constnt. (1 If u(x c f(s u(s k(s, τu(τdτ ds, (3.1 for x I nd τ s b, then ( u(x c 1 f(t exp (f(s k(b, sds dt (2 If u(x (x for x I nd τ s b, then u(x (x e(x 1 f(s u(s (3.2 k(s, τu(τdτ ds, (3.3 ( f(t exp (f(s k(b, sds dt, (3.4

7 EJDE-2007/176 INTEGRAL INEQUALITIES 7 e(x = f(s (s k(s, τ(τdτ ds. (3.5 Proof. (1 The inequlity (3.1 implies the estimte u(x c f(s u(s k(b, τu(τdτ ds. We define the function z(x = c f(s u(s Then z( 1, x 2,..., x n = c, u(x z(x nd Dz(x = f(x u(x Define the function f(x z(x v(x = z(x k(b, τu(τdτ ds. k(b, su(sds, k(b, sz(sds. k(b, sz(sds, then z( 1, x 2,..., x n = v( 1, x 2,..., x n = c, Dz(x f(xv(x nd z(x v(x, nd we hve Dv(x = Dz(x k(b, xz(x (f(x k(b, xv(x. (3.6 Clerly v(x is positive for ll x I, hence the inequlity (3.6 implies the estimte tht is v(xdv(x v 2 (x v(xdv(x v 2 (x f(x k(b, x; f(x k(b, x (D nv(x(d 1 D 2... D n 1 v(x v 2 ; (x hence ( D1 D 2... D n 1 v(x D n f(x k(b, x. v(x Integrting with respect to x n from n to x n, we hve thus D 1 D 2... D n 1 v(x v(x v(xd 1 D 2... D n 1 v(x v 2 (x Tht is, n n f(x 1,..., x n 1, t n k(b, x 1,..., x n 1, t n dt n ; n n f(x 1,..., x n 1, t n k(b, x 1,..., x n 1, t n dt n (D n 1v(x(D 1 D 2... D n 2 v(x v 2. (x D n 1 (D 1 D 2... D n 2 v(x v(x n n f(x 1,..., x n 1, t n k(b, x 1,..., x n 1, t n dt n,

8 8 M. DENCHE, H. KHELLAF EJDE-2007/176 integrting with respect to x n 1 from n 1 to x n 1, we hve D 1 D 2... D n 2 v(x v(x Continuing this process, we obtin D 1 v(x v(x n n 1 n n 1 n f(x1,..., x n 2, t n 1, t n k(b, x 1,..., x n 2, t n 1, t n dt n dt n 1. n f(x 1, t 2, t 3,..., t n k(b, x 1, t 2, t 3,..., t n dt n... dt 2. Integrting with respect to x 1 from 1 to x 1, we hve tht is, log v(x v( 1, x 2,..., x n f(t k(b, tdt; ( v(x c exp f(t k(b, tdt. (3.7 Substituting (3.7 into Dz(x f(xv(x, we hve ( Dz(x cf(x exp f(t k(b, tdt, (3.8 integrting (3.8 with respect to the x n component from n to x n, then with respect to the n 1 to x n 1, nd continuing until finlly 1 to x 1, nd noting tht z( 1, x 2,..., x n = c, we hve ( z(x c 1 f(t exp f(s k(b, sds dt. This completes the proof of the first prt. (2 Define function z(x by z(x = f(s u(s k(s, τu(τdτ ds. (3.9 Then from (3.3, u(x (x z(x nd using this in (3.9, we get z(x f(s (s z(s k(s, τ(τ z(τdτ ds, e(x f(s z(s k(s, τz(τdτ ds, (3.10 e(x is defined by (3.5. Clerly e(x is positive, continuous n nondecresing for ll x I. From (3.10 it is esy to observe tht z(x x z(s s e(x 1 f(s e(s k(s, τ z(τ e(τ dτ ds. Now, by pplying the inequlity in prt 1, we hve ( z(x e(x 1 f(t exp (f(s k(b, sds dt. (3.11 The desired inequlity in (3.4 follows from (3.11 nd the fct tht u(x (x z(x.

9 EJDE-2007/176 INTEGRAL INEQUALITIES 9 The following theorem dels with n-independent vribles versions of the inequlities estblished in Theorem 2.3. We need the inequlities in the following lemm (see 4. Lemm 3.2. Let u(x nd b(x be nonnegtive continuous functions, defined for x I, nd let g S. Assume tht (x is positive, continuous function, nondecresing in ech of the vribles x I. Suppose tht u(x c holds for ll x I with x, then u(x G 1 G(c b(tg(u(tdt, (3.12 b(tdt, (3.13 for ll x I such tht G(c b(tdt Dom(G 1, G(u = u u 0 dz/g(z, u > 0(u 0 > 0. Theorem 3.3. Let u(x, f(x, (x nd k(x, t be s defined in Theorem 3.1. Let Φ(u(x be rel-vlued, positive, continuous, strictly non-decresing, subdditive nd submultiplictive function for u(x 0 nd let W (u(x be rel-vlued, positive, continuous nd non-decresing function defined for x I. Assume tht (x is positive continuous function nd nondecresing for x I. If ( u(x (x f(tg(u(tdt f(tw k(t, sφ(u(sds dt, (3.14 for s t x b, then for x x, { u(x β(x (x f(tw Ψ 1( Ψ(η k(b, sφβ(s β(x = G 1 (G(1 η = b u } f(τdτds dt, (3.15 f(sds, (3.16 k(b, sφ(β(s(sds, (3.17 G(u = 1/g(z dz, u > 0(u 0 > 0, (3.18 u 0 x ds Ψ(x = Φ(W (s, x > 0. (3.19 Here G 1 is the inverse function of G, nd Ψ is the inverse function of Ψ 1, x is chosen so tht G(1 f(sds is in the domin of G 1, nd Ψ(η is in the domin of Ψ 1. k(b, sφ β(s f(τdτ ds,

10 10 M. DENCHE, H. KHELLAF EJDE-2007/176 Proof. Define the function z(x = (x Then 3.14 cn be restted s u(x z(x ( f(tw k(t, sφ(u(sds dt. (3.20 f(tg(u(tdt. We hve z(x is positive, continuous, nondecresing in x I nd g S. Then the bove inequlity cn be restted s By Lemm 3.2 we hve u(x z(x 1 β(x is defined by (3.16. By (3.20, we hve z(x = (x v(x = f(tg ( u(t dt. (3.21 z(t u(x z(xβ(x, (3.22 By (3.24 nd (3.22, we observe tht ( v(x k(b, tφ β(t (t η k(b, sφ(β(s(sds k(b, sφ(β(s f(tw (v(tdt, (3.23 k(x, tφ(u(tdt. (3.24 k(b, sφβ(s f(sw (v(sds dt f(τw (v(τdτds, f(τdτφ(w (v(sds. (3.25 Where η is defined by (3.17. Since Φ is subdditive nd submultiplictive function, W nd v(x re nondecresing for ll x I. Define r(x s the right side of (3.25, then r( 1, x 2,..., x n = η nd v(x r(x, r(x is positive nd nondecresing in ech of the vribles x 1, x 2, x 3,... x n. Hence Since D n (D 1... D n 1 r(x Φ(W (r(x Dr(x k(b, xφβ(x Φ(W (r(x f(sds. Dr(x = Φ(W (r(x D nφ(w (r(xd 1... D n 1 r(x Φ 2 (W (r(x the bove inequlity implies (D 1... D n 1 r(x Dr(x D n Φ(W (r(x Φ(W (r(x, nd (D 1... D n 1 r(x D n k(b, xφθ(x, Φ(W (r(x,

11 EJDE-2007/176 INTEGRAL INEQUALITIES 11 θ(x = β(x f(sds. Integrting with respect to x n from n to x n, we hve D 1... D n 1 r(x xn k(b, x 1, x 2,..., x n 1, s n Φθ(x 1, x 2,..., x n 1, s n ds n. Φ(W (r(x n Repeting this rgument, we find tht D 1 r(x Φ(W (r(x n 1 n n 1 n k(b, x 1, s 2,..., s n Φθ(x 1, s 2,..., s n ds n ds n 1... ds 2. Integrting both sides of the bove inequlity with respect to x 1 from 1 to x 1, we hve nd From this we obtin Ψ(r(x Ψ(η r(x Ψ 1( Ψ(η v(x r(x Ψ 1( Ψ(η k(b, sφθ(sds, k(b, sφ β(s k(b, sφβ(s f(τdτ ds. f(τdτds. (3.26 By (3.22, (3.23 nd (3.26 we obtin the desired inequlity in ( Some pplictions In this section, our results re pplied to the qulittive nlysis of two ppliction. The first is the system of nonliner differentil equtions for one vrible functions. The second is nonliner hyperbolic prtil integrodifferentil eqution of n-independent vribles. First we consider the system of nonliner differentil equtions du ( dt = F 1 t, u(t, K 1 (t, u(sds, (4.1 for t I =, t R, u C(I, R n, F 1 C(I R n R n, R n nd K 1 C(I R n, R n. In wht follows, we shll ssume tht the Cuchy problem du dt = F 1(t, u(t, K 1 (t, u(sds, x I, u( = u 0 R n, (4.2 hs unique solution, for every I nd u 0 R n. We shll denote this solution by u(.,, u 0. The following theorem dels the estimte on the solution of the nonliner Cuchy problem (4.2. Theorem 4.1. Assume tht the functions F 1 nd K 1 in (4.2 stisfy the conditions K 1 (t, u h(tφ( u, t I, (4.3 F 1 (t, u, v u v, u, v R n, (4.4

12 12 M. DENCHE, H. KHELLAF EJDE-2007/176 h nd Φ re s defined in Theorem 2.2. Then we hve the estimte, for t t 2, u(t,, u 0 e ( u t t0 0 h(se 1 (s, u 0 ds, (4.5 E 1 (t, u 0 = Ψ 1( Ψ(ϑ Φ ( e τ x0 Ψ(t = ϑ = τ h(σdσ dτ, (4.6 ds, t > 0, (4.7 Φ(s u 0 Φ(e s t0 ds, (4.8 nd t 2 is chosen so tht Ψ(ϑ Φ(e τ t0 τ h(σdσdτ is in Dom(Ψ 1 Proof. Let I, u 0 R n nd u(.,, u 0 be the solution of the Cuchy problem (4.2. Then we hve ( u(t,, u 0 = u 0 s, u(s,, u 0, K 1 (s, u(τ,, u 0 dτ ds. (4.9 F 1 Using (4.3 nd (4.4 in (4.9, we hve u(t,, u 0 u 0 f(s u(s,, u 0 ( u 0 f(s u(s,, u 0 h(s K 1 (s, u(τ,, u 0 dτ ds, Φ( u(τ,, u 0 dτ ds. (4.10 Now, suitble ppliction of Theorem 2.2 with (t = u 0, f(t = b(t = 1 nd W (u = u to (4.10 yields (4.5. If, in ddition, we ssume tht the function F 1 stisfies the generl condition F 1 (t, u, v f(t(g( u W ( v, (4.11 f, g nd W re s defined in Theorem 2.4, we obtin n estimte for u(.,, u 0, nd from ny prticulr conditions of (4.11 nd (4.3, we cn get some useful results similr to Theorem 4.1. Secondly, we shll demonstrte the usefulness of the inequlity estblished in Theorem 3.3 by obtining pointwise bounds on the solutions of certin clss of nonliner eqution in n-independent vribles. We consider the nonliner hyperbolic prtil integrodifferentil eqution n u(x x 1 x 2... x n = F ( x, u(x, K(x, s, u(sds G(x, u(x (4.12 for ll x I = ; x R n, x = (x 1, x 2,..., x n, = ( 1, 2,..., n, x = (x 1, x 2,..., x n re in R n nd u C(I, R, F C(I R R, R, K C(I I R, R nd G C(I R, R. With suitble boundry conditions, the solution of (4.12 is of the form ( u(x = l(x F s, u(s, K(s, t, u(tdt ds G(s, u(sds. (4.13

13 EJDE-2007/176 INTEGRAL INEQUALITIES 13 The following theorem gives the bound of the solution of (4.12. Theorem 4.2. Assume tht the functions l, F, K nd G in (4.12 stisfy the conditions K(s, t, u(t k(s, tφ( u(t, t, s I, u R, (4.14 F (t, u, v 1 u v, u, v R, t I, ( G(s, u 1 u, 2 s I, u R, (4.16 l(x (x, x I, (4.17, f, k nd Φ re s defined in Theorem 2.2, with f(x = b(x e(x for ll x I b, e C(I, R, then we hve the estimte, for x x, ( n ( u(x exp (x i i (x E(tdt. (4.18 Here E(t = Ψ 1( Ψ(η η = i=1 k(x, sφ exp ( n (s i i i=1 k(x, sφ ( (s exp ( Ψ(x = f(τdτ ds, (4.19 n (s i i ds, (4.20 i=1 ds Φ(s, x x0 > 0, (4.21 x is chosen so tht Ψ(η k(x, sφexp( n i=1 (s i i f(τdτds, is in the domin of Ψ 1. Proof. Using the conditions (4.14, (4.17 in (4.13, we hve u(x (x G(s, u(s ds f(s u(s K(s, t, u(t dt ds, x 0 x ( (x u(s k(s, tφ( u(t dt ds. (4.22 Now, suitble ppliction of Theorem 3.3 with f(s = 1, g(u = u nd W (u = u to (4.22 yields (4.18. Remrks. If we ssume tht the functions F nd G stisfy the generl conditions F (t, u, v f(t(g( u W ( v, (4.23 G(t, u f(tg( u, for t I, u R, (4.24 we cn obtin n estimtion of u(x. From the prticulr conditions of (4.14, (4.23 nd (4.24, we cn obtin some results similr to Theorem 3.3. To sve spce, we omit the detils here. Under some suitble conditions, the uniqueness nd continuous dependence of the solutions of (4.1 nd (4.12 cn lso be discussed using our results. Acknowledgments. The uthors re grteful to the nonymous referee for his/her suggestions on the originl mnuscript. The second uthor would like to thnk Prof. Julio G. Dix for his help, to my dvisor Prof. M. Denche for his ssistnce.

14 14 M. DENCHE, H. KHELLAF EJDE-2007/176 References 1 D. Binov nd P. Simeonov; Integrl Inequlities nd Applictions, Kluwer Acdemic Publishers, Dordrecht, ( T. H. Gronwll; Note on the derivtives with respect to prmeter of solutions of system of differentil equtions, Ann. of Mth.,0 (1919 4, pp I. Gyori; A Generliztion of Bellmn s Inequlity for Stieltjes Integrls nd Uniqueness Theorem, Studi Sci. Mth. Hungr, 6 (1971, pp H. Khellf; On integrl inequlities for functions of severl independent vribles, Electron. J. Diff. Eqns., Vol (2003, No.123, pp J. A. Oguntuse; On n inequlity of Gronwll, J. Ineq. Pure nd Appl. Mth., 2(1 (2001, No B. G. Pchptte; Bounds on Certin Integrl Inequlities, J. Ineq. Pure nd Appl. Mth., 3(3 (2002, Article No C. C. Yeh nd M-H. Shim; The Gronwll-Bellmn Inequlity in Severl Vribles, J. Mth. Anl. Appl., 86 (1982, No. 1, pp Mohmed Denche University of Mentouri, Fculty of Science, Deprtment of Mthemtics, Constntine 25000, Algeri E-mil ddress: denech@wissl.dz Hssne Khellf University of Mentouri, Fculty of Science, Deprtment of Mthemtics, Constntine 25000, Algeri E-mil ddress: khellfhssne@yhoo.fr