Some Integral Inequalities with Maximum of the Unknown Functions

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1 Advances in Dynamical Systems and Applications ISSN , Volume 6, Number 1, pp Some Integral Inequalities with Maximum of the Unknown Functions Snezhana G. Hristova and Kremena V. Stefanova Plovdiv University Department of Applied Mathematics Plovdiv, Bulgaria Abstract This paper deals with a special type of delayed integral inequalities of Bihary type for scalar functions of two variables. The integrals involve the maximum of the unknown function over a past time interval. Several nonlinear types of integral inequalities are solved. The importance of these integral inequalities is based on their wide applications to the qualitative investigations of various properties of solutions of partial differential equations with maxima and it is illustrated by some direct applications. AMS Subject Classifications: 26D99, 34K38, 35A23. Keywords: Integral inequalities, maxima, scalar functions of two variables, partial differential equations with maxima. 1 Introduction In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in control theory correspond to the maximal deviation of the regulated quantity and they are adequately modeled by differential equations with maxima [13]. The qualitative investigation of properties of differential equations with maxima [2, 4, 6, 8 10] requires building of an appropriate mathematical apparatus. One of the main mathematical tools, employed successfully for studying existence, uniqueness, continuous dependence, comparison, perturbation, boundedness and stability of solutions of differential and integral equations, is the method of integral inequalities [1, 5, 14 17]. The Received October 1, 2010; Accepted November 29, 2010 Communicated by Eberhard Malkowsky

2 58 Snezhana G. Hristova and Kremena V. Stefanova development of the theory of partial differential equations with maxima [3, 7, 12] requires solving linear and nonlinear integral inequalities that involve the maximum of the unknown scalar function of two variables [11]. This paper deals with nonlinear integral inequalities which involve the maximum of the unknown scalar function of two variables. Various cases of nonlinear integral inequalities are solved. The form of the solution depends significantly on the type of nonlinear function in the integral. These results generalize the classical integral inequalities of Gronwall Belman and Bihari type. The importance of the solved integral inequalities is illustrated on some direct applications to partial differential equations with maxima. 2 Main Results Let h > 0 be a constant,,, X, Y be fixed points such that 0 < X and 0 < Y. Definition 2.1. We will say that the function α C 1 [, X, R is from the class F if it is a nondecreasing function and αx x for x [, X. Let the functions α i, β j F for i = 1, 2,..., n, j = 1, 2,..., m. Denote Consider the sets G, Ψ, Λ defined by J = min min α i, min β j. 1 i n 1 j m G = { x, y R 2 : x [, X, y [, Y }, Ψ = { x, y R 2 : x [J h, ], y [, Y }, Λ = { x, y R 2 : x [J h, X, y [, Y } = G Ψ. Theorem 2.2. Let the following conditions be fulfilled: 1. The functions α i, β j F for i = 1, 2,..., n, j = 1, 2,..., m. 2. The functions f i, g j C[J, X [, Y, R for i = 1, 2,..., n, j = 1, 2,..., m. 3. The function φ CΨ, [0, k] where k = const > The functions ω i, ω j CR, R are nondecreasing, ω i x > 0, ω j x > 0 for x > 0, i = 1, 2,..., n, j = 1, 2,..., m.

3 Some Integral Inequalities with Maximum of the Unknown Functions The function u CΛ, R and satisfies the inequalities ux, y k αi x f i s, tω i us, t dt ds g j s, t ω j max uξ, t dt ds for x, y G, i=1 α i βj x β j 2.1 Then for x, y G 1, the inequality ux, y φx, y, for x, y Ψ. 2.2 ux, y W 1 W k holds, where W 1 is the inverse function of G 1 = { r αi x i=1 α i βj x β j f i s, tdt ds g j s, tdt ds 2.3 W r = ds r 0 ωs <, 0 r 0 < k r, 2.4 ωt = max max it, max jt 1 i n 1 j m 2.5 x, y G : W k αi x i=1 α i βj x β j f i s, tdt ds g j s, tdt ds Dom } W 1. Proof. Define a function z : Λ R by the equalities αi x k f i s, tω i us, t dt ds i=1 α i zx, y = βj x x, y G, g j s, t ω j max uξ, t dt ds, β j k, x, y Ψ. The function zx, y is nondecreasing in its both arguments, z, y = k for y [, Y, and ux, y zx, y for x, y Λ. Note that max zξ, y = zs, y for

4 60 Snezhana G. Hristova and Kremena V. Stefanova s [β j, β j X, j = 1, 2,..., m and y [, Y. Then from inequality 2.1 and the definition of the function ωt, we get for x, y G αi x zx, y k f i s, tω zs, t dt ds i=1 α i βj x 2.6 g j s, tω zs, t dt ds. β j Define a function K : G [k, by the equality αi x Kx, y = k f i s, tω zs, t dt ds i=1 α i βj x g j s, tω zs, t dt ds. β j 2.7 Note that Kx, y is an increasing function and the inequality zx, y Kx, y holds for x, y G. From 2.7 and condition 1 of Theorem 2.2, we obtain for the partial derivative K xx, y with respect to x K xx, y = f i αi x, t αi ω zα i x, t x dt i=1 g j βj x, t βj ω zβ j x, t x dt i=1 From inequality 2.8, we get f i αi x, t αi ω Kα i x, t x dt g j βj x, t βj ω Kβ j x, t x dt ω Kx, y K xx, y ω Kx, y i=1 f i α i x, tα i x dt g j β j x, tβ j x dt. αi x f i αi x, t dt i=1 βj x g j βj x, t dt

5 Some Integral Inequalities with Maximum of the Unknown Functions 61 Integrate inequality 2.9 with respect to x from to x, x [, X, change the variable η = α i s i = 1,..., n in the first sum of integrals, and η = β j s j = 1,..., m in the second sum of integrals, use the definition 2.4, the equalities x K x s, y ds Kx,y ω Kx, y = du k ω = W Kx, y W k u and the inequality zx, y Kx, y, and obtain for x, y G W Kx, y W k αi x i=1 α i βj x β j f i η, tdt dη g j η, tdt dη Since W 1 is increasing, from inequalities 2.10 and ux, y zx, y Kx, y we obtain the required inequality 2.3 for x, y G 1. Define the following class of functions. Definition 2.3. The function ω CR, R is said to be from the class Ω 1 if i ωx > 0 for x > 0 and it is a nondecreasing function; ii ωtx tωx for 0 t 1. In the case when the nonlinear functions in the integrals, additionally to the conditions of Theorem 2.2, are submultiplicative, then the constant k in inequality 2.1 may be substituted by an increasing function. Theorem 2.4. Let the following conditions be fulfilled: 1. The conditions 1 and 2 of Theorem 2.2 are satisfied. 2. The function k CG, [1, is nondecreasing in its both arguments. 3. The function φ CΨ, [0, k], where k = k,. 4. The functions ω i, ω j Ω 1, i = 1, 2,..., n, j = 1, 2,..., m. 5. The function u CΛ, R and satisfies the inequalities ux, y kx, y αi x f i s, tω i us, t dt ds g j s, t ω j max uξ, t dt ds for x, y G, i=1 α i βj x β j 2.11

6 62 Snezhana G. Hristova and Kremena V. Stefanova Then for x, y G 2, the inequality ux, y φx, y, for x, y Ψ ux, y kx, y W 1 W 1 αi x i=1 α i βj x β j f i s, tdtds 2.13 g j s, tdtds holds, where the functions W, ω are defined by 2.4 and 2.5, respectively, { αi x G 2 = x, y G : W 1 f i s, tdt ds i=1 α i βj x g j s, tdt ds Dom W 1 }. β j Proof. Define the continuous nondecreasing function K : Λ [1, by { kx, y, for x, y G, Kx, y = k, y, for x, y Ψ. From inequalities 2.11, 2.12, and obtain and ux, y kx, y 1 1 αi x i=1 α i βj x β j αi x i=1 α i βj x β j ux, y k, 1 kx, y , and condition 4 of Theorem 2.4, we us, t f i s, tω i dt ds kx, y max uξ, t g j s, t ω j kx, y us, t f i s, tω i dt ds Ks, t max uξ, t g j s, t ω j φx, y k, Ks, t dt ds, dt ds, x, y G , x, y Ψ For every j : 1 j m and t [, Y, s [β j, β j X, the inequalities max uξ, t ξ [s h,s] Ks, t = uξ 1, t Ks, t uξ 1, t Kξ 1, t max ξ [s h,s] uξ, t Kξ, t max Zξ, t 2.17 ξ [s h,s]

7 Some Integral Inequalities with Maximum of the Unknown Functions 63 ux, y hold, where ξ 1 [s h, s] and Zx, y =. From inequality 2.17 it follows Kx, y that the inequalities 2.15, 2.16 may be written in the form Zx, y 1 αi x i=1 α i βj x β j f i s, tω i Zs, t dt ds g j s, t ω j max Zξ, t dt ds, x, y G, 2.18 Zx, y 1, x, y Ψ From inequalities 2.18 and 2.19 according to Theorem 2.2, we obtain for x, y G 2 Zx, y W W 1 1 αi x i=1 α i βj x β j f i s, tdtds g j s, tdtds Inequality 2.20 and definitions of the functions Zx, y, Kx, y imply the validity of inequality Define the following class of functions. Definition 2.5. We will say that the function ω CR, R is from the class Ω 2 if it satisfies the following conditions: i ωx > 0 for x > 0 and it is a nondecreasing function; ii ωtx tωx for 0 t 1; iii ωx ωy ωx y. Remark 2.6. For example, the functions ωx = x and ωx = x are from the class Ω 2. In the case when the nonlinear functions in the integrals are from the class Ω 2, the following result holds. Theorem 2.7. Let the following conditions be fulfilled: 1. The conditions 1 and 2 of Theorem 2.2 are satisfied. 2. The functions ω i, ω j Ω 2, i = 1, 2,..., n, j = 1, 2,..., m. 3. The function k CΛ, R.

8 64 Snezhana G. Hristova and Kremena V. Stefanova 4. The function µ CG, [1,. 5. The function u CΛ, R and satisfies the inequalities αi x ux, y kx, y µx, y f i s, tω i us, t dt ds i=1 α i βj x g j s, t ω j max uξ, t dt ds for x, y G, β j 2.21 ux, y kx, y for x, y Ψ Then for x, y G 3, the inequality ux, y kx, y Mx, yp x, yw W 1 1 Ax, y 2.23 holds, where the functions W, w are defined by 2.4 and 2.5, respectively, W 1 is the inverse of W, { G 3 = x, y G : W 1 Ax, y Dom W 1}, P x, y = 1 αi x i=1 α i βj x β j f i s, tω i ks, t dt ds g j s, t ω j αi x max kξ, t Ax, y = f i s, tms, tdt ds i=1 α i βj x g j s, tms, tdt ds, β j { µx, y for x, y G, Mx, y = µ, y for x, y Ψ. dt ds, x, y G, Proof. Define a function z : Λ R by the equalities αi x f i s, tω i us, t dt ds i=1 α i βj x zx, y = g j s, t ω j max uξ, t dt ds β j for x, y G, for x, y Ψ.

9 Some Integral Inequalities with Maximum of the Unknown Functions 65 From inequality 2.21 and the definition of the function zx, y, we obtain ux, y kx, y Mx, yzx, y, x, y Λ Since the function Mx, y is nondecreasing in its both arguments and for s [J, X, y [, Y we obtain max uξ, y max kξ, y Ms, y max zξ, y From inequality 2.28, the definition of the function zx, y and conditions 1 and 2 we get for x, y G and αi x α i αi x α i αi x α i αi x βj x f i s, tω i us, t dt ds f i s, tω i ks, t Ms, tzs, t dt ds f i s, tω i ks, t dt ds f i s, tms, tω i zs, t dt ds α i β j βj x g j s, t ω j max uξ, t dt ds g j s, t ω j max kξ, t dt ds β j βj x β j g j s, tms, t ω j max zξ, t dt ds From the definition of the function zx, y and inequalities 2.28, 2.30, 2.31, it follows that αi x zx, y P x, y f i s, tms, tω i zs, t dt ds i=1 α i βj x 2.32 g j s, tms, t ω j max zξ, t dt ds, for x, y G, β j zx, y 0, for x, y Ψ From inequalities 2.32, 2.33, according to Theorem 2.4, we obtain the inequality 2.23.

10 66 Snezhana G. Hristova and Kremena V. Stefanova 3 Applications Consider the scalar partial differential equation with maxima u x y = F x, y, ux, y, max us, y for x, y G 3.1 s [σx, τx] with the initial conditions u, y = ϕ 1 y for y [, Y, ux, = ϕ 2 x for x [, X, ux, y = ψx, y for x, y [τ h, ] [, Y, 3.2 where u R, ϕ 1 : [, Y R, ϕ 2 : [, X R, ψ : [τ h, ] [, Y R, F : [, X [, Y R R R. Theorem 3.1 Upper bound. Let the following conditions be fulfilled: 1. The functions τ F, σ C[, X, R and there exists a constant h > 0 : 0 τx σx h for x [, X. 2. The function F C[, X [, Y R R, R and satisfies for x, y G and γ, ν R, the condition F x, y, γ, ν Qx, y γ Rx, y ν, where Q, R CG, R. 3. The function ψ C[τ h, ] [, Y, R. 4. The functions ϕ 1 C[, Y, R, ϕ 2 C[, X, R and ϕ 1 = ϕ 2, ϕ 1 y = ψ, y for y [, Y. 5. The function ux, y is a solution of the initial value problem 3.1, 3.2 which is defined for x, y [τ h, X [, Y. Then the solution ux, y of the partial differential equation with maxima 3.1, 3.2 satisfies for x, y G the inequality ux, y Kx, y P1 x, y 1 1 x [ ] 2 Qs, t Rs, t dt ds, where Kx, y = { ϕ 1 y ϕ 2 x ϕ 2, x, y G ψx, y, x, y [τx0 h, ] [, Y, P 1 x, y = 1 x x 0 x Qs, t Ks, t dt ds Rs, t max Kξ, t dt ds. ξ [σs, τs]

11 Some Integral Inequalities with Maximum of the Unknown Functions 67 Proof. For the function ux, y we obtain ux, y ϕ1 y ϕ 2 x ϕ 2 x s, F t, us, t, max ϕ1 y ϕ 2 x ϕ 2 x x ξ [σs, τs] uξ, t dt ds Qs, t us, t dt ds 3.6 Rs, t max uξ, t dt ds for x, y G, ξ [σs, τs] ux, y = ψx, y, for x, y [τx0 h, ] [, Y. 3.7 Denote ux, y = Ux, y for x, y [τx0 h, X [, Y. From 3.6 and 3.7, we have Ux, y ϕ1 y ϕ 2 x ϕ 2 x x Rs, t max ξ [σs, τs] Qs, t Us, t dt ds Uξ, t dt ds, x, y G, 3.8 Ux, y = ψx, y = Kx, y, for x, y [τx0 h, ] [, Y. 3.9 max ξ [σx, τx] max ξ [τx h, τx] Change the variable s = τ 1 η in the second integral of 3.8, use the inequality Uξ, y Uξ, y for y [, Y and x [, X that follows from condition 1 of Theorem 3.1 and obtain x Ux, y Kx, y Qs, t Us, tdtds τx Rτ 1 η, tτ 1 η max Uξ, t dtdη, x, y G. τ ξ [η h, η] 3.10 Note that the conditions of Theorem 2.7 are satisfied for n = m = 1, αx x, βx τx, kx, y Kx, y, where the function Kx, y is defined by 3.4, fx, y Qx, y for x, y G, gx, y Rτ 1 x, yτ 1 x for x [τ, X, y [, Y, ωv = ωv = v, W v = 2 v, W 1 v = 1 4 v2 for v R, and DomW 1 = R. According to Theorem 2.7, from inequalities 3.10, 3.9, we obtain for x, y G the bound Ux, y Kx, y P 1 x, y 1 x [ ] 2 2 Qs, t Rs, t dt ds, where the function P 1 x, y is defined by equality 3.5. Inequality 3.11 and the definition of the function Ux, y imply the validity of the required inequality 3.3.

12 68 Snezhana G. Hristova and Kremena V. Stefanova In the case when the initial conditions in problem 3.1, 3.2 are constants and Lipschitz functions for the function F are constants, we may apply Theorem 2.2 instead of Theorem 2.7 and we obtain a better bound for the solution. Theorem 3.2. Let the conditions of Theorem 3.1 be satisfied, where ϕ 1 y C, ϕ 2 x C, ψx, y C, Qx, y Q, Rx, y R, C, Q, R are constants. Then the solution ux, y of the partial differential equation with maxima 3.1, 3.2 satisfies for x, y G the inequality References C 2 ux, y 1 2 Q Rx y [1] R. P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput no. 3, [2] D. D. Bainov, S. G. Hristova, Monotone-iterative techniques of Lakshmikantham for a boundary value problem for systems of differential equations with maxima, J. Math. Anal. Appl., 190, no. 2, 1995, [3] D. Bainov and E. Minchev, Forced oscillations of solutions of hyperbolic equations of neutral type with maxima, Applicable Anal., 70, , [4] D. D. Bainov, V. A. Petrov and V. S. Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second-order neutral differential equations with maxima, J. Comput. Appl. Math., 83, no. 2, 1997, [5] D. D. Bainov, P. S. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht [6] T. Donchev, S. G. Hristova, N. Markova, Asymptotic and oscillatory behavior of n-th order forced differential equations with maxima, PanAmer. Math. J., , No. 2, [7] T. Donchev, N. Kitanov, D. Kolev, Stability for the solutions of parabolic equations with maxima, PanAmer. Math. J., 20, no. 2, 2010, [8] S. G. Hristova, Practical stability and cone valued Lyapunov functions for differential equations with maxima, Intern. J. Pure Appl. Math., 57, no , [9] S. G. Hristova, Stability on a cone in terms of two measures for impulsive differential equations with supremum, Appl. Math. Lett., ,

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