On Some Dynamic (ΔΔ)^∇- Gronwall–Bellman–Pachpatte-Type Inequalities on Time Scales and Its Applications

Abstract

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Keywords

• Gronwall’s inequality
• dynamic inequality
• time scales
• Leibniz integral rule on time scales