Levinson Theorem for Differential Equations with Piecewise Constant Argument Generalized

Abstract: In this work, it is presented an adaptation of an asymptotic theorem of N. Levinson of 1948, to differential equation with piecewise constant argument generalized, which were introduced by M. Akhmet in 2007. By simplicity and without loss of generality, the case where the argument is delayed is considered. The N. Levinson's theorem which is adapted is that dealt by M. S. P. Eastham in his work which is present in this bibliography. The more relevant hypotheses of this theorem are highlighted an it is established a version of this theorem with these hypotheses for ordinary differential equations. Such a version is that which is adapted to differential equation with piecewise constant argument generalized. The adaptation is proved by mean the Banach fixed point where contractive operator is built form a suitable version of the constant variation formula.