Generalizations of Gronwall-Bihari Inequalities on Time Scales

We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.Comment: This is a preprint of an article accepted (16/May/2008) for publication in the "Journal of Difference Equations and Applications"; J. Difference Equ. Appl. is available online at http://www.informaworld.co

Note on some nonlinear integral inequalities in two independent variables on time scales and applications

Abstract Our aim in this note is to investigate some nonlinear integral inequalities in two independent variables on time scales.The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales

Some new dynamic Inequality on time scales in three variables

In this paper we obtain the estimates on some dynamic integral inequalities in three variables which can be used to study certain dynamic equations. We give some applications to convey the importance of our result

Holder's and Hardy's Two Dimensional Diamond-alpha Inequalities on Time Scales

We prove a two dimensional Holder and reverse-Holder inequality on time scales via the diamond-alpha integral. Other integral inequalities are established as well, which have as corollaries some recent proved Hardy-type inequalities on time scales.Comment: Accepted for publication (October 21, 2009) in the journal "Annals of the University of Craiova, Mathematics and Computer Science Series" (http://inf.ucv.ro/~ami

Halanay type inequalities on time scales with applications

This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that beside generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalites, which are the most important particular cases of dynamic inequalities on time scales

Qualitative analysis of dynamic equations on time scales

In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz condition is not necessary for uniqueness. The existence of epsilon-approximate solution is established under suitable assumptions. Moreover, we study the approximate solution of the dynamic equation with delay by studying the solution of the corresponding dynamic equation with piecewise constant argument. We show that the exponential stability is preserved in such approximations.Comment: 13 page

Inequalities and majorisations for the Riemann-Stieltjes integral on time scales

We prove dynamic inequalities of majorisation type for functions on time scales. The results are obtained using the notion of Riemann-Stieltjes delta integral and give a generalization of [App. Math. Let. 22 (2009), no. 3, 416--421] to time scales.Comment: Submitted 30-Apr-2009; revised 15-Feb-2010; accepted 24-Mar-2010; for publication in Math. Inequal. App