© Hindawi Publishing Corp. A MORE GENERALIZED GRONWALL-LIKE INTEGRAL INEQUALITY WITH APPLICATIONS

This paper deals with a new Gronwall-like integral inequality which is a generalization of integral inequalities proved by Engler (1989) and Pachpatte (1992). The new Gronwall-like integral inequality can be used in various problems in the theory of certain class of ordinary and integral equations.


Introduction.
It is well known that integral inequalities play a very crucial role in the study of differential equations, integral equations, functionaldifferential equations, and integro-differential equations.Besides the famous Gronwall-Bellman inequality and its first nonlinear generalization by Bihari (see Bellman and Cooke [1]), there are several other very useful Gronwall-like inequalities.Haraux [3,Corollary 16, page 139] derived one Gronwall-like inequality and used it to prove the existence of solutions of wave equations with logarithmic nonlinearities.On the other hand, Engler [2] utilized the following slight variant of inequality due to Haraux [3, page 139] in the study of global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity.
Lemma 1.2.Let p i ∈ L 1 (I, R 0 = (0, ∞)) and p ∈ L 1 (I, R + ).Let g be a continuously differentiable function defined on R + , and g > 0 and g ≥ 0 on R 0 .If (1.3) where G −1 is the inverse of G and t 1 is chosen so that The aim of the present paper is to establish a new generalization of all the inequalities discussed in the above lemmas.One application example is also included.

Main results.
For convenience, we give some basic notations and definitions which will be used in our subsequent discussion.Let I = [0,T ], T > 0, be finite but can be arbitrarily large.
We denote by L 1 (I, R) the class of all measurable functions p(t) defined on the set I and with range in the set R with satisfying T 0 |p(t)| dt < ∞, and denote by C k (M, S) the class of all k-times continuously differentiable functions on the set M with range in the set S. We define the differential operators L i , 0 ≤ i ≤ n, by with p n (t) = 1, where x(t) and p i (t) > 0 are some functions defined on I.For t ∈ I and some functions q j (t) > 0, j = 1,...,n − 1 and q(t) ≥ 0 defined on I, we define A more generalized version of the inequality appearing in Lemma 1.2 is given in the following theorem. ) where and G −1 is the inverse function of G and t 1 is chosen so that Proof.Let ε > 0 be an arbitrary small constant and define on I a nondecreasing function (2.7) From (2.3) and (2.7), we have (2.9) Since ϕ and g are nondecreasing, by (2.8) and (2.9), we observe that (2.10) Using the fact that v ε (t) is positive, and L n v ε (t) ≥ 0 for t ∈ I, it follows from (2.10) that (2.12) that is, (2.13) Integrating (2.13) from 0 to t and using the fact that It also follows from (2.14) that which, upon integrating from 0 to t and using the fact that L n−2 v ε (0) = 0, leads to (2.16) Repeating the above argument successively, we obtain (2.17) For any invertible and continuously differentiable function a(t), by changing the variable η = a −1 (ξ), we have Using the above fact and integrating (2.17) from 0 to t, we obtain Define a function w(t) by and w(0 ( where G and G −1 are defined as in Theorem 2.1 and Letting ϕ = u k (k > 1 is a constant) in Theorem 2.1 leads to the following corollary.
Corollary 2.4.Suppose that the functions u, p, p 1 ,...,p n−1 and g are defined as in Theorem 2.1 and k > 1 is a constant, then the inequality where G and G −1 are defined as in Theorem 2.1 and , we arrive at Ou-Iang's integral inequality given in [5].
Remark 2.6.By choosing other suitable special functions to ϕ, we get other interesting inequalities which could not be derived from Lemma 1.2.

Application. Consider the differential equation
where k > 1 and C i−1 , 1 ≤ i ≤ n are constants, p and f ∈ C(I, R), and L n is defined as in Section 2. It is easy to observe that (3.1) is equivalent to the integral equation where where J(x) denotes the maximal existent interval of x(t).

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: