Approximate controllability of time varying measure differential problem
of second order with state-dependent delay and non-instantaneous impulse
It is comprehended that the systems without any limitation on their Zeno
action are enthralled in a vast class of hybrid systems. This article is
influenced by a new category of non-autonomous second order measure
differential problems with state-dependent delay (SDD) and
non-instantaneous impulse (NII). Some new sufficient postulates are
created that guarantee solvability and approximate controllability. We
employ the fixed point strategy and theory of Lebesgue–Stieltjes
integral in the space of piecewise regulated functions. The measure of
non-compactness is applied to establish the existence of a solution.
Moreover, the measured differential equations generalize the ordinary
impulsive differential equations. Thus, our findings are more prevalent
than that encountered in the literature. At last, an example is
comprised that exhibits the significance of the developed theory.