First- and Second-order Dynamic Equations with Impulse

We first briefly survey the recent results for existence of solutions to first-order problems with fixed-time impulses. Periodic boundary conditions using upper and lower solutions were considered in [19], using degree theory. A nonlinear alternative of Leray-Schauder type was used in [15] for initial conditions or periodic boundary conditions. The monotone iterative technique was employed in [14] for antiperiodic and nonlinear boundary conditions. Lower and upper solutions and periodic boundary conditions were studied in [20]. Semilinear damped initial value problems in a Banach space using fixed point theory were investigated in [6]. In [9], existence of solutions for the differential equation u′(t) = q(u(t))g(t,u(t)) subject to a general boundary condition is proven, in which g is Carathéodory and q ∈ L∞, and existence of lower and upper solutions is assumed. Schauder’s fixed point theorem was used there. This generalized an earlier result found in [18]. It appears that little has been done concerning dynamic equations with impulses on time scales (see [4, 5, 16] for earlier results). In Section 2, the present paper uses ideas from [9] to prove an existence result for discontinuous dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. The study of boundary value problems for nonlinear second-order differential equations with impulses has appeared in many papers (see [10, 11, 13] and the references therein). In Section 3, we use ideas from [12, 16] to prove an existence result for secondorder dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. Nonlinear boundary conditions cover, among others, the periodic and the Dirichlet conditions, and have been introduced for ordinary differential equations by Adje in [1]. Assuming the existence of a lower and an upper solution, we prove that the solution of the boundary value problem stays between them. In [2], it was shown that the upper and lower solution method will not work for first-order dynamic equations involving ∆-derivatives, unless restrictive assumptions are