© Hindawi Publishing Corp. ON VOLTERRA INEQUALITIES AND THEIR APPLICATIONS

We present certain variants of two-dimensional and n-dimensional Volterra integral inequalities. In particular, generalizations of the Gronwall inequality are obtained. These results are applied in various problems for differential and integral equations.

The obtained results for integral inequalities in two variables are applied in various differential and integral problems.Some similar problems were considered in [4] for integral inequalities of the Volterra-Fredholm type: u(x, y) ≤ f (x,t)+ t 0 b a k(x, t; y,s)u(y,s)dy ds. (1.4) In this paper, we obtain better estimations than in [4], because in (1.3) double Volterra operator which plays there a dominant role arises.Moreover, integral inequalities in n independent variables are considered and applied to study boundedness and stability of solutions to n-dimensional nonlinear integral equation of the Volterra type.

Two-dimensional Volterra inequalities.
From the theory of Volterra linear integral equations, the following result follows.
Lemma 2.1.Let f and k be continuous functions in D and Ω, respectively.If k is nonnegative, and continuous function u satisfies inequality (1.3)   (2.7) Proof.We notice that inequality (2.5) is a special case of (1.3) with k(x, y, s, t) = a(x, y)b(s, t).In virtue of formula (2.4), we have (2.10) By induction, we obtain Then from (2.2), we get (2.12) Hence, using Lemma 2.1, the proof is finished.
The results were obtained by estimating the iterated kernels k i .We can get stronger inequality in the following case.where Theorem 2.12.Let u, a, and b be nonnegative continuous functions in D and let f be positive and continuous in D. If a/f is nonincreasing with respect to each variable, then inequality (2.5) implies (2.18).

Some applications of two-dimensional inequalities.
In this section, we present some applications of the given inequalities to study the boundedness, stability and uniqueness solutions of certain nonlinear integral equations, and value boundary problems for nonlinear hyperbolic partial differential equations.Moreover, the boundedness of solutions to a system of two-dimensional Volterra integral equations is studied.

3.1.
We consider two-dimensional Volterra nonlinear integral equation with the following assumptions: ( and satisfying the Lipschitz condition or (3) H is continuous in W and satisfies the condition where a and b are positive continuous functions in D, such that ab  Let the following assumptions be fulfilled: (1 ) g and p are continuous functions in D, (2 ) F is a continuous function satisfying one of the following conditions: where

Integral inequalities of n variables and their applications.
In this section, we establish n-independent variable generalizations of the integral inequalities established in Section 2. For this purpose, we introduce the following notations.
A point (x 1 ,...,x n ) in the n-dimensional Euclidean space R n is denoted by x and the origin of R n is 0 = (0,...,0).For x, y, s ∈ R n , we denote that x ≤ y x i ≤ y i for every i = 1, 2,...,n.
A function f is said to be nondecreasing if x ≤ y ⇒ f (x) ≤ f (y).
then (4.1) has a unique solution in (3.14) which depends continuously on f and h.
Proof.Let C(I) be normed by Abbreviating (4.1) as one has S : X → X continuous and, for v, w ∈ C(I), The integral is less than or equal to e ασ (x) α −n , and one obtains, after division by e ασ (x) , We choose α such that Lα −n = 1/2.When u is the solution, in particular, u α ≤ 2 S(0) α which leads to The theorem now follows.This also answers the existence and uniqueness in the infinite case, where the "quadrant" Introduce the notation Lemma 4.2 (see [6]).The solution of is given by where the resolvent kernel r is defined by with iterated kernels k n constructed by formulas where r is a resolvent kernel (4.11).
Proof.Abbreviate the right-hand side of (4.9) as f +Ku = Su.Then S is a monotone increasing operator (v ≤ w ⇒ Sv ≤ Sw).According to (4.9), u is a subsolution, u ≤ Su, hence the sequence u n = S n u obtained by successive approximation is increasing and converges to the solution ϕ = Sϕ.Since ϕ is the right-hand side of inequality (4.17), the theorem is proved.Proof.We assume first that f (x) = 1.With the same notation (but with k(x, s) = b(s)), we have u ≤ Su and we show that w = e B(x) satisfies s ≥ Sw, that is, B(s) ds, (4.21) according to Lemma 4.3.The sequence u n = S n u is increasing and w n = S n w is decreasing and both have the same limit ϕ = solution of ϕ = Sϕ.It follows that u ≤ w.
The theorem holds for f (x) = 1 and hence for f (x) = const = c ≥ 0. Now we fix x 0 > 0, put c = f (x 0 ), and consider the inequality in I = [0,x 0 ].We get u(x) ≤ ce B(x)  in I, in particular at x 0 .Since x 0 is arbitrary, this theorem follows.Proof.We notice that DM(x, s) m ≥ mb(s)M(x, s) m−1 .Using Lemmas 4.2, 4.3, and Theorem 4.5 and proceeding similarly as in the case of a two-dimensional inequality, we get (4.23).
We get the optimal estimation in the case of a special kernel.That is important since it allows to give the kernel explicitly and hence gives a much better bound. where Introducing the function for a positive function a, and

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

1 .
Introduction.Many authors have considered integral inequalities in two variables of the form u(x, y) ≤ f (x,y)+ x 0 y 0 b(s, t)u(s, t)ds dt (1.1)

Theorem 1 . 1 .
If b and f are nonnegative continuous functions in D and f is nondecreasing with respect to each variable, then inequality (1.1) implies the following Gronwall inequality: u(x, y) ≤ f (x,y)exp x 0 y 0 b(s, t)ds dt .

Lemma 4 . 3 .0Theorem 4 . 4 .
If b ∈ C(I) is nonnegative in (3.14), then 1 + x 0 b(s)e B(s) ds ≤ e B(x) , We notice that D i B = B i and D ij B = B ij are nonnegative and DB(x) = b(x), De B(x) ≥ b(x)e B(x) , x De B(s) ds = e B(x) − 1, x 0 b(s)e B(s) ds ≤ e B(s) − 1.(4.15)In the case n = 3, we haveDe B = e B B 1 B 2 B 3 + B 1 B 23 + B 2 B 13 + B 3 B 12 + b .(4.16)If f ∈ C(I) and a nonnegative function k ∈ C(J), then u ∈ C(I) satisfying the inequality u(x) ≤ f (x

Theorem 4 . 5 .
Let f be continuous, nonnegative, and nonincreasing function in I and let b ∈ C(I) be nonnegative.Then for u ∈ C(I), the inequality u(x) ≤ f (x)+ x 0 b(s)u(s)ds (4.19) implies u(x) ≤ f (x)e B(x) , B(x) =

First
Round of ReviewsMarch 1, 2009 b 1 , b 2 are positive, then this inequality is strict).

Corollary 2.10. If the assumptions of Theorem 2.9 are satisfied and f /a is nonde- creasing in D, then inequality (2.5) implies (2.18). Remark 2.11. If a = 1, we get the Gronwall inequality in two variables
(1.2).
Results of this paper can be extended on the class L 2 .
.25) Using Remark 2.8, we get u(x, y) f (x,y) ≤ exp x 0 y 0 a(s, t) f (s,t) b(s, t)f (s, t)ds dt .(2.26) Hence (2.21) follows.Theorem 2.13.If f , a, and b are nonnegative continuous functions in D, then In virtue of Theorem 1.1, we have u(x, y) W (x,y) ≤ exp x 0 y 0 W (s, t)b(s, t)ds dt.(2.31)That finishes the proof.Corollary 2.14.If the assumptions of Theorem 2.13 are fulfilled, then u(x, y) ≤ a(x, y) exp x 0 y 0 a(s, t)b(s, t)ds dt for a(x, y) ≤ f (x,y), u(x, y) ≤ f (x,y)exp x 0 y 0 f (s, t)b(s, t)ds dt for a(x, y) ≥ f (x,y).(2.32) Theorem 2.15.Let f be nonnegative continuous function in D and nondecreasing with respect to each variable, and let k be nonnegative function in Ω such that k(x, y, s, t) ≤ k(s, t, s, t) (2.33) for 0 ≤ s ≤ x, 0 ≤ t ≤ y.If the nonnegative and continuous function u satisfies inequality (1.3), then u(x, y) ≤ f (x,y)exp u(x, y) ≤ f (x,y)+ x 0 y 0 k(s, t, s, t)u(s, t)ds dt.(2.35)The proof follows using Theorem 1.1.Theorem 2.16.Let f be nonnegative continuous function in D and let k be positive function in Ω such that k(x, y, s, t) ≤ k(x, y, x, y) (2.36) for 0 ≤ s ≤ x, 0 ≤ t ≤ y.If f (x,y)/k(x,y,x,y) is nondecreasing with respect to variables x and y, then inequality (1.3) implies (2.34) for nonnegative and continuous function u in D. Theorem 2.17.Let f be positive continuous function in D and let k be nonnegative continuous function in Ω such that k(x, y, s, t) ≤ f (x,y).If the continuous and nonnegative function u satisfies inequality (1.3), then u(x, y) ≤ f (x,y)exp x 0 y 0 f (s, t)ds dt.(2.38) Proof.We notice that inequality (1.3) leads to u(x, y) ≤ f (x,y) 1 + Theorem 2.18.Suppose that f is a positive continuous function in D and K/f u(x, y) ≤ f (x,y)exp x 0 y 0 K(s, t)ds dt, (2.43) for nonnegative and continuous function u in D.
.32) We can use the presented theory of n-dimensional inequalities to study boundedness and stability of solutions for n-dimensional integral equation (4.1).Stability in a finite interval has been discussed earlier.In Q = R n + , one has to define what it means.One possibility is to require that u is an approximate solution that satisfies a bound.In the linear case, one can consider two equations with coefficients f , k and f , k and require |f − f |, |k − k| to derive a bound for |u − u|.