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We introduce new properties of Riemann-Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder's fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y 1 , Y 2 and ϖ 1 , ϖ 2 , which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay. [ABSTRACT FROM AUTHOR]

In this paper, we will establish some oscillation criteria for the fourthorder hybrid nonlinear functional dynamic equations with damping. The authors present new oscillation criteria to check whether all solutions of an equation, in this class, oscillate. This study aims to present some new sufficient conditions for the oscillatory of solutions to a class of fourth-order hybrid nonlinear functional dynamic equations by use of Riccati technique and other method. Illustrative examples are also provided. [ABSTRACT FROM AUTHOR]