Several Generalization Of Integral Inequalities

Although in only a very few special cases is it possible to obtain useful solu-tions to nonlinear differential equations via analytical calculations, the estimation ofthe solutions of nonlinear differential equations can be given by inequalities. Thisestimation can be applied to prove the existence, uniqueness, boundedness, stability,invariant manifolds and other qualitative properties of solutions of differential equa-tions. The inequalities of this type is called the integral inequalities. Since Gronwalland Bellman found the integral inequalities, which have epoch-making significance,Gronwall-Bellman's integral inequalities and theirs discrete versions have been devel-oped continuously.In 1957, in order to investigate the boundedness of the solutions of second orderdifferential equations, Ou-yang Liang generalized Gronwall-Bellman's integral inequal-ity and gave the inequality of which the left is the square of an unknown function.In 1979 while attempting to establish a connection between stability and the secondlaw of thermodynamics, Defermos improved the inequality of Ou-Yang to the integralinequality of which the integrand is the sum of one term including first power and sec-ond power of unknown function. Pachpatte extended the discrete version of Defermos'integral inequality to a sum-difference inequality. The sum in this inequality has twoterms, the one including first power of the unknown function, the other including thecompound function of the unknown function and nondecreasing function. In Chapter2 we generalize the Pachpatte's sum-difference inequality to a retarded form. In ourinequality we consider more nonlinear terms on unknown function but do not requiremonotonicity of given functions. We give the estimation of unknown function. Ourresult can be applied to giving boundedness and uniqueness for a initial value problemof a delay difference equation.On the other hand, in 1956 Bihari generalized first power of unknown functionof the integrand in Gronwall-Bellman's to the compound of unknown function andnondecreasing function. In 2000, Lipovan extended Bihari's integral inequalities by changing the upper and lower limits of integration into differentiable increasing func-tion, which leads to new integral inequality including delay. Furthermore, in 2005Agarwal et al improved the Lipovan's integral inequality to Gronwall-like integral in-equality with delay, where the constant out integration was changed to function andtwo integrations to more ones.In 2006 Cheung extended the Pachpatte's integral inequality in one variable andthe Lipovan's integral inequality in two variables to a delay integral inequality in twovariables. The left of this inequality is a power function of unknown function whilethe right is the sum of a constant and two integrations of which one includes a powerfunction of unknown function and the other contains a compound of unknown functionand nondecreasing function. On the bases of the works of Cheung and Agarwal, inSection 3.1, we establish a generalized retarded integral inequality of Gronwall-liketype in two variables. Comparing with Cheung's inequality in our inequality the con-stant out integration was changed to function in two variables and two integrations tomore ones. We do not require monotonicity of known functions. In order to overcomethe difficulties arising from the situation of without assumption of monotonicity, weemploy a technique of monotonization to construct a sequence of functions of whicheach possesses stronger monotonicity than previous one. For the purpose of decidingthe validity region of estimation of the unknown function. we have to consider theinclusion of more regions. By comparing the conditions of defining those regions, weconclude their inclusion relations. We obtain the estimation of unknown function inour inequality and apply this result to a boundary value problem of a partial differ-ential equation for boundedness, uniqueness and continuous dependence. Using ourresult we can estimate the bounds of the unknown function of the integral inequalitiesin both [Nonlinear Anal. TMA., 2006, 64, 2112-2128] and [Appl. Math. Comput.,2005, 165, 599-612]. In 2002 Pachpatte investigated a two variables inequality withtbur iterated integrals which only includes first power of unknown function. In Section3.2 we improve the Pachpatte's result by changing the first power of unknown functionin right of the inequality to two compound functions of nondecreasing function andunknown one. Moreover, we give an estimate for the unknown function. Our resultcan be applied to discussion on boundedness and uniqueness for a integro-differentialequation.In Section 4.1 we extend the inequality in [J. Math. Anal. Appl., 2006, 319,708-724] to a new sum-difference inequality which has a nonconstant out of sum sign and a compound function of unknown function and nonmonotonic function in sum sign.Moreover, we give an estimation for the unknown function. Our result can be appliedto discussion on boundedness, uniqueness and continuous dependence of solutionsof boundary value problem for difference equations in two variables. In Section 4.2we generalize the Pachpatte's sum-difference inequality with first power of unknownfunction only to a sum-difference inequality with four iterated sums and compound ofunknown function. Our result can be applied to study the boundedness and uniquenessfor difference equations with two iterated sums.Invariant curve (manifold) plays an important role in the theory of dynamicalsystems. Having an invariant curve (manifold), one can reduce a dynamical systemto a lower dimensional one by restricting the system to the curve (manifold). In 1997Ng and Zhang investigated invariant curves for a second order differential equationwith piecewise constant arguments. In 2001 Si and Zhang continued to discuss theanalyticity of the invariant curves in the cases that the eigenvalue at a fixed pointis off the unit circle and that the eigenvalue lies on the unit circle but satisfies theDiophantine condition. Another planar mapping is also studied in 2002 for analyticinvariant curves. Some recent efforts and results are also made beyond the restrictionof the Diophantine condition. In Chapter 5 we discuss the analytic invariant curves ofthe second order difference equation. We not only discuss the case of the eigenvalueoff the unit circle and the case on the unit circle with the Diophantine condition butalso consider the case of the eigenvalue at a root of the unity, which obviously violatesthe Diophantine condition.