Some New Generalized Nonlinear Integral Inequalities for Functions of Two Independent Variables

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1 Int. Journal of Math. Analysis, Vol. 7, 213, no. 4, HIKARI Ltd, Some New Generalized Nonlinear Integral Inequalities for Functions of Two Independent Variables H. Khellaf, M. Smakdji and M. Denche Laboratory of Differential Equations University of Constantine 1, Constantine 25, Algeria Copyright c 213 H. Khellaf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to establish some nonlinear Gronwal- Bellman integral inequalities in the case of functions of two independent variables which can be used as handy tools in the theory of partial differential and integral equations. The results extend and improve the earlier publication 8], an application is given to illustrate the efficiency of the obtained results. Mathematics Subject Classification: 26D1, 26D15, 35A5 Keywords: Integral inequality, two independent variables, submultiplicative function, nondecreasing function, partial differential equations 1 INTRODUCTION The Gronwall Bellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, continuation, boundedness and stability and other qualitative properties of the solutions of differential and integral equations. The literature on such inequalities and their applications is vast; see 1, 3, 5, 6, 7, 9, 1] and the references given therein. Many authors have established Gronwall Bellman type integral inequalities in two independent variables; see for example 2, 3, 4]. Recently, A. Khan in 8] obtained a useful upper bound on the following inequality:

2 1962 H. Khellaf, M. Smakdji and M. Denche φ(x, y c A(s, yφ(s, yds H(s, tφ p (x, ydtds, B(x, tφ p (x, tdt (1 and its variants, where 1 >p> and c> are constants and φ(x, y C(R 2, R. However, sometimes we need to study this type of inequality with a function c(x, y in place of the constant term c. Motivated mainly by the work of 8], we discuss, in this paper a more general form of integral Gronwall-Bellman type inequality for the functions with two variables, as the following : u q (x, y c(x, y a i (s, yu p (s, yds d k (x, y, s, tw(u(s, tdsdt, (2 and its variants, where c(x, y is a function and q p> are constants for all (x, y Δ. Furthermore, we show that the results in 8] can be reduced from our inequality (2 in some special cases. We also apply our result to a boundary value problem of a partial differential equation for estimation of its solution. In this paper, we consider the explicit bounds on some general versions of (1 which the constant c> on the right side of (1 is replaced by the function c(x, y and contain some power nonlinear terms with respect to the unknown function u(x; y on the both side of (1. 2 MAIN RESULTS In what follows, R denotes the set of real numbers, R =,, I 1 =,X and I 2 =,Y are the given subsets of R, and Δ = I 1 I 2, E = {(x, y, s, t Δ 2 : s x X; t y Y }. The first-order partial derivatives of a function u(x, y for x, y R with respect to x and y are denoted as usual by D 1 u(x, y and D 2 u(x, y, respectively. The following lemma is useful in our main results.

3 Some new generalized nonlinear integral inequalities 1963 Lemma 1 (Inequality in one variable Let c(x,u(x and a i (x C(I 1, R be nonnegative continuous functions for any x I 1 and i =1, 2,..., n and assume that c(x is nondecreasing for x I 1. Suppose that q p > are constants. If u(x satisfies the inequality u q (x c(x n a i (su p (sds, (3 for s x, then the following conclusions are true (i If p = q, then u(x c(x 1/p 1 n ] x exp a i (sds, (4 p for x I 1. (ii If p<q,then u(x c(x 1/q 1 q p q n c (p q/q (sa i (sds q p, (5 Now, let us list our main results as the follows : Theorem 2 Let c(x, y,u(x, y,a i (x, y and b j (x, y C(Δ, R be nondecreasing functions in each variables, for any i =1, 2,..., n 1 and j =1, 2,..., n 2. Suppose that 1 >p> is a constant. (A1 If u(x, y satisfies u(x, y c(x, y for all (x, y Δ, then for all (x, y Δ. Where a i (s, yu(s, yds b j (x, tu p (x, tdt, (6 u(x, y c(x, ye 1 (x, yq 1 (x, y, (7 E 1 (x, y = exp ( n1 a i (s, yds, (8 Q 1 (x, y = 1(1 p b j (x, tc p 1 (x, te p 1 (x, tdt 1 p. (9

4 1964 H. Khellaf, M. Smakdji and M. Denche (A2 If u(x, y satisfies u(x, y c(x, y for all (x, y I, then for all (x, y Δ. Where a i (s, yu p (s, yds b j (x, tu(x, tdt, (1 u(x, y c(x, ye 2 (x, yq 2 (x, y, (11 E 2 = exp Q 2 (x, y = 1(1 p for all (x, y Δ. ( n2 b j (x, tdt, (12 a i (s, yc p 1 (s, ye p 2(s, yds The proof of the theorem will be given in the next section. 1 p, (13 Theorem 3 Let c(x, y,u(x, y,a i (x, y and b j (x, y C(Δ, R be defined as in Theorem 2. Suppose that q p> are constants. If u(x, y satisfies the inequality : u q (x, y c(x, y a i (s, yu p (s, yds b j (x, tu p (x, tdt, (14 for all (x, y Δ, then the following conclusions are true (B1 If p = q, then for all (x, y Δ, where u(x, y c 1 p (x, ye 1 p 1 (x, yq 1 p 3 (x, y, (15 Q 3 (x, y = exp and E 1 (x, y is defined in (8. (B2 If p<q,then ( n2 b j (x, te 1 (x, tdt, (16 u(x, y c 1/q (x, ye 4 (x, yq 4 (x, y, (17

5 Some new generalized nonlinear integral inequalities 1965 for all (x, y Δ, where Q 4 (x, y = 1 q p q z (p q/q (x, tb j (x, tdt q p, (18 and E 4 (x, y = 1 q p q c (p q/q (s, ya i (s, yq p 4 (s, yds q p. (19 Where z(x, y c(x, ye q 4 (x, y for all (x, y Δ. Remark 4 If we take b j (x, y =for any j =1, 2,..., n 2 and keeping y fixed, then Theorem 3 reduce exactly to Lemma 1. Using Theorems 2 and 3, we can get some more generalized results as follow: Theorem 5 Let p, c(x, y,u(x, y,a i (x, y and b j (x, y be defined as in Theorem 2, let d k (x, y, s, t C(E,R be nondecreasing function in x and y for each s I 1 and t I 2 for all k =1, 2,..., n 3. Let w C(R, R be nondecreasing and submultiplicative function with w(u > for u >. (Ã1 If u(x, y satisfies u(x, y c(x, y for all (x, y Δ, then a i (s, yu(s, yds b j (x, tu p (x, tdt d k (x, y, s, tw(u(s, tdsdt, (2 u(x, y M 1 (x, ye 1 (x, y Q 1 (x, y, (21 for all x x 1, y y 1.Where ] y M 1 (x, y G G(c(x, 1 y d k (x, y, s, tw (E 1 (s, t w( Q 1 (s, tdsdt, (22 for all x x 1, y y 1, and G(u = u u dt w(t,u u >. (23

6 1966 H. Khellaf, M. Smakdji and M. Denche Where E 1 (x, y is defined in (8 and Q 1 (x, y = 1(1 p b j (x, tm p 1 1 (x, te p 1(x, tdt 1 p. (24 Here, where G 1 is the inverse function of G and the real numbers x 1,y 1 R are chosen so that G(c(x, y n 3 d k (x, y, s, tw (E 1 (s, t w( Q 1 (s, tdsdt Dom(G 1. (Ã2 If u(x, y satisfies u(x, y c(x, y n 3 for all (x, y Δ, then for all x x 2, a i (s, yu p (s, yds b j (x, tu(x, tdt d k (x, y, s, tw(u(s, tdsdt, (25 u(x, y M 2 (x, ye 2 (x, y Q 2 (x, y, (26 y y 2.Where M 2 (x, y G G(c(x, 1 y d k (x, y, s, tw ] ( E 2 (s, t Q 2 (s, t dsdt for all x x 2, y y 2,G and E 2 are defined in (23 and (12, with Q 2 (x, y = 1(1 p a i (s, ym p 1 2 (s, ye p 2(s, yds 1 p (27. (28 Here, where G 1 is the inverse function of G and the real numbers x 2,y 2 R are chosen so that G(c(x, y n 3 d k (x, y, s, tw (E 2 (s, t w( Q 2 (s, tdsdt Dom(G 1, Remark 6 If we take d k (x, y, s, t =for any k =1, 2,..., n 3, then Theorem 5 reduce to Theorem 2.

7 Some new generalized nonlinear integral inequalities 1967 Remark 7 If we take a i (s, y =b j (x, t =for i =1,.., n 1,,.., n 2, some interesting new Gronwall-Bellman type inequalities of two variables can be obtained in the form: u(x, y c(x, y n 3 d k (x, y, s, tw(u(s, tdsdt. By choosing suitable functions for w, some interesting new Gronwall-Bellman type inequalities of two variables can be obtained from Theorem 5. For example if we take w(s =s r, the following interesting inequalities are easily obtained. Corollary 8 Let p, c(x, y,u(x, y,a 1 (x, y, b 1 (x, y and d 1 (x, y, s, t be defined as in Theorem 5. Suppose <r<1 is a constant. (C1 If u(x, y satisfies u(x, y c(x, y for (x, y Δ, then for all (x, y Δ, where m 1 (x, y for (x, y Δ, and a 1 (s, yu(s, yds b 1 (x, tu p (x, tdt d 1 (x, y, s, tu r (s, tdsdt, (29 u(x, y m 1 (x, ye 1 (x, y q 1 (x, y, c 1 r (x, y(1 r q 1 (x, y = 1(1 p y ( x e 1 (x, y = exp a 1 (s, yds (C2 If u(x, y satisfies d 1 (x, y, s, te r 1 (s, t qr 1 b 1 (x, tm p 1. 1 (x, te p 1(x, tdt 1 r (s, tdsdt, 1 p. then u(x, y c(x, y a i (s, yu p (s, yds b(x, tu(x, tdt d 1 (x, y, s, tu r (s, tdsdt, (3 u(x, y m 2 (x, ye 2 (x, y q 2 (x, y,

8 1968 H. Khellaf, M. Smakdji and M. Denche for all (x, y Δ, where m 2 (x, y c 1 r (x, y(1 r d 1 (x, y, s, te r 2 (s, t qr 2 1 r (s, tdsdt, for (x, y Δ, and q 2 (x, y = 1(1 p a 1 (s, ym p 1 2 (s, ye p 1 p 2(s, yds, x ( y e 2 (x, y = exp b 1 (x, tdt. Remark 9 (i If r =, by using Theorem 2, an estimation of (29 and (3 can be easily obtained. (ii If r =1, an estimation of (29 and (3 can be obtained in the following Corollary 11. Remark 1 (i Corollary 8 (C1 reduces to Theorem 2.3 of the main results in 8], when c(x, y =c (constant, d 1 (x, y, s, t =d(s, t, = =and r = p. (ii Under some suitable conditions, Corollary 8 (C2 is also a generalization of the main result in 8, Theorem 2.4] Corollary 11 Let p, c(x, y,u(x, y,a 1 (x, y, b 1 (x, y and d 1 (x, y, s, t be defined as in Theorem 5. ( C1 If u(x, y satisfies u(x, y c(x, y a 1 (s, yu(s, yds b 1 (x, tu p (x, tdt y x d 1 (x, y, s, tu(s, tdsdt, for (x, y Δ, then u(x, y m 1 (x, yẽ 1 (x, y q 1 (x, y, for all (x, y Δ.Where ( m 1 (x, y c(x, y exp d 1 (x, y, s, tẽ 1 (s, t q 1 (s, tdsdt, for (x, y Δ, and q 1 (x, y = 1(1 p b 1 (x, t m p 1 1 (x, tẽ p 1 p 1(x, tdt. y ( x ẽ 1 (x, y = exp a 1 (s, yds.

9 Some new generalized nonlinear integral inequalities 1969 ( C2 If u(x, y satisfies u(x, y c(x, y for (x, y Δ, then for all (x, y Δ, where ( m 2 (x, y c(x, y exp for (x, y Δ, and q2 (x, y = a 1 (s, yu p (s, yds d 1 (x, y, s, tu(s, tdsdt, u(x, y m 2 (x, yẽ 2 (x, y q 2 (x, y, 1(1 p ( ẽ 2 (x, y = exp b 1 (x, tdt b 1 (x, tu(x, tdt d 1 (x, y, s, tẽ 2 (s, t q 2 (s, tdsdt, 1 p. a 1 (s, y m p 1 2 (s, yẽ p 2 (s, yds Using Theorem 3, we can get some more generalized results as follow: Theorem 12 Let c(x, y,u(x, y,a i (x, y, b j (x, y d k (x, y, s, t and w(u be defined as in Theorem 5. Suppose that q p > are constants. If u(x, y satisfies the inequality u q (x, y c(x, y a i (s, yu p (s, yds b j (x, tu p (x, tdt. d k (x, y, s, tw(u(s, tdsdt, (31 for all (x, y Δ, then the following conclusions are true : ( B1 If p = q, then for all x x 3, u(x, y N 1/p 1 (x, ye 1/p 1 (x, yq 1/p 3 (x, y, (32 y y 3. Where N 1 (x, y H H(c(x, 1 y d k (x, y, s, tw ] ( E 1/p 1 (s, tq 1/p 3 (s, t dsdt,

10 197 H. Khellaf, M. Smakdji and M. Denche for all x x 3, y y 3, with u dt H(u = u w(t 1/q, u u >, (33 where E 1 (x, y and Q 3 (x, y are defined in (8 and (16. Here, H 1 is the inverse function of H and the real numbers x 3,y 3 are chosen so that H(c(x, y n 3 ( y d k (x, y, s, tw E 1/p 1 (s, t w(q 1/p 3 (s, tdsdt Dom(H 1. ( B2 If p<q,then u(x, y N 1/q 2 (x, yẽ4(x, y Q 4 (x, y, (34 for all x x 4, y y 4.Where N 2 (x, y H H(c(x, 1 y d k (x, y, s, tw for all x x 4, y y 4 and H is defined in (33, and Q 4 (x, y = 1 q p z (p q/q (x, tb j (x, tdt q and Ẽ 4 (x, y = 1 q p q ] (Ẽ4 (s, t Q 4 (s, t dsdt, q p N (p q/q 2 (s, ya i (s, y Q p 4(s, yds (35, (36 q p. (37 for all x x 4, y y 4, where z(x, y N 2 (x, yẽq 4(x, y. with H 1 is the inverse function of H and the real numbers x 4,y 4 R are chosen so that H(c(x, y n 3 d k (x, y, s, tw(ẽ4(s, tw( Q 4 (s, tdsdt Dom(H 1. Remark 13 If we take p =1, w(s =s, c(x, y =c> (constant, n 1 = n 2 = n 3 =1, = =and d 1 (x, y, s, t =d 1 (s, t, then the inequality established in Theorem 12 ( B2 reduces to the Theorem 2.1 in 8]. Remark 14 Considering q =1,w(s =s p, n 1 = n 2 = n 3 =1, = =, d 1 (x, y, s, t =d 1 (s, t and c(x, y =c (constant in Theorem 12 ( B2, we obtain the Theorem 2.5 in 8]. Remark 15 By choosing suitable functions for w for example w(s =s r with q>r or q = r>(where p = q or q>p, using similar arguments in the proof of Theorem 12, we can obtain many interesting new integral inequalities, but, for space-saving, the details are omitted here.

11 Some new generalized nonlinear integral inequalities PROOF OF THEOREMS Since the proofs resemble each other, we give the details for (A1, (B2 and ( B2 only; the proofs of the remaining inequalities can be completed by following the proofs of the above-mentioned inequalities. Proof. Theorem 2 (A1 We define a function z(x, y by z(x, y =c(x, y by substituting (38 in (6, we get u(x, y z(x, y b j (x, tu p (x, tdt, (38 a i (s, yu(s, yds, (x, y Δ, (39 Clearly, z(x, y is a nonnegative, continuous and nondecreasing function in x. Treating y, y I 2 fixed in (39, a suitable application of Lemma 1 to (39 we get u(x, y z(x, ye 1 (x, y, (4 for (x, y Δ, where E 1 (x, y is defined as in (8. By (38 and (4, we obtain z(x, y c(x, y b j (x, te p 1(x, tz p (x, tdt. (41 Keeping x fixed in (41, an estimation of z(x, y can be obtained by a suitable application of Lemma 1 (part ii to (41, after that, we obtain z(x, y c(x, yq 1 (x, y, (42 for (x, y Δ, where Q 1 (x, y is defined as in (9. Finally, substituting the last inequality into (4, the desired inequality (7 follows immediately. Proof. Theorem 3 (B2. We define a function z(x, y by z(x, y =c(x, y by substituting (43 in (14, we get u q (x, y z(x, y a i (s, yu p (s, yds, (43 b j (x, tu p (x, tdt. (44

12 1972 H. Khellaf, M. Smakdji and M. Denche Clearly, z(x, y is a nonnegative, continuous and nondecreasing function in y. Treating x fixed in (44, a suitable application of Lemma 1 to (44 we get u(x, y z(x, y 1/q Q 4 (x, y, (45 for (x, y Δ,where Q 4 (x, y is defined as in (18. By (45 and (43, we obtain z(x, y c(x, y a i (s, yq p 4(s, yz p/q (s, yds. (46 Keeping y fixed in (46, an estimation of z(x, y can be obtained by a suitable application of Lemma 1 (part ii to (46, after that, we obtain z(x, y c(x, ye q 4 (x, y, for (x, y Δ,where E 4 (x, y is defined as in (19. Finally, substituting the last inequality into (45, the desired inequality (17 follows immediately. Proof. Theorem 12( B2 Setting N 2 (x, y =c(x, y d k (x, y, s, tw(u(s, tdsdt, (47 the inequality (31 can be restated as u q (x, y N 2 (x, y a i (s, yu p (s, yds b j (x, tu p (x, tdt, (48 Clearly, N 2 (x, y is nonnegative and nondecreasing function in each in x and y. Now a suitable application of the inequality (17 in Theorem 3 to (48, yields u(x, y N 1/q 2 (x, yẽ4(x, y Q 4 (x, y, (49 where Ẽ4(x, y, Q4 (x, y are defined in (37 and (36. From (47 and (49 and by using the fact that w is submultiplicative, we have N 2 (x, y c(x, y d k (x, y, s, tw (Ẽ4 (s, t Q 4 (s, t w(n 1/q 1 (s, tdsdt, (5 for (x, y Δ. By following the same steps from (??-(?? in (5, we get ] y N 2 (x, y H H(c(x, 1 y d k (x, y, s, tw (Ẽ4 (s, t w( Q 4 (s, tdsdt, for all x x 41, y y 4. Finally, substituting the last inequality into (49, the desired inequality (34 follows immediately.

13 Some new generalized nonlinear integral inequalities An APPLICATION In this section we present an immediate application of our results (Theorem 12 to study the boundless of the solution of partial differential equation. We consider the following nonlinear partial differential equation in R 2 : D 1 D 2 u p (x, y =h(x, y, u(x, y, D 1 g 1 (x, y, u(x, y D 2 g 2 (x, y, u(x, y, for all (x, y R 2. u(x, = σ 1 (x, u(,y=σ 2 (y, u(, = k, (51 Where h, g 1,g 2 C(R 3, R and σ 1,σ 2 C(R, R and k, p > are a constants. Assume that those functions are defined and continuous on their respective domains of definition. Theorem 16 Suppose that h(x, y, u d 1 (x, yu r, (52 g 1 (x, y, u b 1 (x, yu p, (53 g 2 (x, y, u a 1 (x, yu p, (54 e 1 (xe 2 (y k c, c (constant. (55 Where p>r>, b 1 (x, y,a 1 (x, y are as in Theorem 12 and d 1 (x, y C(R 2, R be nondecreasing function, with c 1 (x = σ 1 (x c 2 (y = σ 2 (y If u(x, y is any solution of (51, then g 2 (s,,σ 1 (sds, g 1 (,t,σ 2 (tdt. u(x, y (ẽ(x, y q(x, y 1/p c (p r/p p r d 1 (s, t(ẽ(x, y q(x, y r/p p r dsdt, p (56 for (x, y R 2, where ( ẽ(x, y = exp a 1 (s, yds, (57

14 1974 H. Khellaf, M. Smakdji and M. Denche For all (x, y R 2. ( q(x, y = exp b 1 (x, tẽ(x, tdt. (58 Proof. It is easy to see that, the solution u(x, y of the problem (51 satisfies the equivalent integral equation u p (x, y = σ 1 (xσ 2 (y k g 2 (s, y, u(s, yds From (59 and with (52-(55, we have u p (x, y c a 1 (s, yu p (s, yds h(s, t, u(s, tdsdt g 2 (s,,σ 1 (sds b 1 (x, tu p (x, tdt g 1 (x, t, u(x, tdt g 1 (,t,σ 2 (tdt. (59 d 1 (s, tu r (s, tdsdt, (6 for (x, y R 2, with p>r>. Now, a suitable application of Theorem 12 ( B1 to (6, with w(s =s r, d 1 (x, y, s, t =d 1 (s, t, n 1 = n 2 = n 3 = 1 and = =, yields u(x, y (ẽ(x, y q(x, y 1/p c (p r/p p r d 1 (s, t(ẽ(x, y q(x, y r/p p r dsdt p for all (x, y R 2, where ẽ(x, y and q(x, y are defined in (57 and (58. Remark 17 In special case (p = r> in the boundary value problem (51 with (52-(55 in Theorem 16, we can obtain the following estimation of u(x, y ( u(x, y c (ẽ(x, y q(x, y 1/p exp d 1 (s, tẽ(s, t q(s, tdsdt, for all (x, y R 2, where ẽ(x, y and q(x, y are defined in (57 and (58. finally, we note that under some suitable conditions, the uniqueness and continuous dependence of the solutions of (51, can also be discussed using our results. OPEN PROBLEM. I think, our results can be generalized to some new nonlinear integral inequalities for function of n independent variables. The problem here is, under some suitable conditions, how to give an estimation

15 Some new generalized nonlinear integral inequalities 1975 and some applications of the following integral inequality in R n (and its nonlinear variants : u(x c(x n n k (x i i a i (x 1,..., x i 1,s i,x i1,..., x n u(x 1,..., x i 1,s i,x i1,..., x n ds i d k (x, sw(u(sds, for x = (x 1,x 2,..., x n Δ R n, s = (s 1,s 2,..., s n Δ and = ( 1, 2,..., n Δ, and we note by ds = n...ds x 1 x 2 x n n...ds 2 ds 1. ACKNOWLEDGEMENT. This research is supported by ANDRU, Algeria (Project PNR nos. V25/R25. The author would like to thank Prof. M. Denche (From Univ. of Constantine 1, Algeria for his assistance. References 1] D. Bainov and P. Simeonov; Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, ( ] M. Denche, H. Khellaf, Integral inequalities similar to Gronwall inequality, Electron. J. Diff. Eqns.,, Vol27 (27, ] H. Khellaf; On integral inequalities for functions of several independent variables, Electron. J. Diff. Eqns.,, Vol. 23 (23, No.123, pp ] Y.H. Kim, On some new integral inequalities for functions in one and two variables, Acta Math. Sin. (Engl. Ser., 21 ( ] B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 ( ] B. G. Pachpatte; Bounds on Certain Integral Inequalities, J. Ineq. Pure and Appl. Math., 3(3 (22, Article No ] B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, ] Zareen A. Khan, On Certain New Gronwall-Bellman Type Integral Inequalities of Two Independent Variables, Int. Journal of Math. Analysis, Vol. 6, 212, no. 1,

16 1976 H. Khellaf, M. Smakdji and M. Denche 9] C. C. Yeh and M.-H. Shim, The Gronwall-Bellman Inequality in Several Variables, J. Math.Anal. Appl., 87 (1982, ] C. C. Yeh, Bellman-Bihari Integral Inequality in n Independent variables, J. Math. Anal. Appl., 87 (1982, Received: April 12, 213