Abstract
Abstract In this paper, some new Gronwall-Bellman-type integral inequalities in two independent variables on time scales are established, which can be used as a handy tool in the research of qualitative and quantitative properties of solutions of dynamic equations on time scales. The inequalities established unify some of the integral inequalities for continuous functions in (Meng and Li in Appl. Math. Comput. 148:381-392, 2004) and their discrete analysis in (Meng and Li in J. Comput. Appl. Math. 158:407-417, 2003). MSC:26E70, 26D15, 26D10.
Highlights
It is well known that Gronwall-Bellman inequality [, ] plays an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations
In the s, Hilger created the theory of time scales [ ] as a theory capable to contain both difference and differential calculus in a consistent way
Since many authors have expounded on various aspects of the theory of dynamic equations on time scales
Summary
It is well known that Gronwall-Bellman inequality [ , ] plays an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. We establish some new Gronwall-Bellman-type integral inequalities in two independent variables containing integration on infinite intervals on time scales, which unify some of the continuous inequalities in [ ] and the corresponding discrete analysis in [ ]. For some x ∈ Tκ , and a function f (x, y) ∈ (T × T, R), the partial delta derivative of f (x, y) with respect to x is denoted by (f (x, y))x and satisfies f σ (x), y – f (s, y) – f (x, y) x σ (x) – s ≤ ε σ (x) – s , ∀ε > , where s ∈ U, and U is a neighborhood of x.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
