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In this paper, we prove the stability results for measure differential equations, considering more general conditions under the Lyapunov functionals and concerning the functions f and g. Moreover, we prove these stability results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of (ω , Q) -periodic mild solutions for the following problem x ′ (t) = A x (t) + f (t , x (t)) , t ∈ R , on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable C 0 -semigroup and the nonlinearity f satisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of (ω , Q) -periodic mild solutions to the Hopfield-type neural network model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain converse Lyapunov theorems for measure functional differential equations which are known to encompass dynamic equations on time scales, impulsive differential equations and functional differential equations (see [7,8]). Our main tool is to consider the framework of generalized ordinary differential equations, introduced by Jaroslav Kurzweil, and to manage all other equations in this setting which has the advantage to handle well both highly oscillating functions and functions with many discontinuities. [ABSTRACT FROM AUTHOR]