First- and second-order dynamic equations with impulse
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We present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. 1. Introduction 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 Carath´eodory 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 Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 119–132 DOI: 10.1155/ADE.2005.119
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Dynamic equations with impulse
made. Hence, in Section 2, we work with the ∇-derivative. In Section 3, we can use the more conventional ∆-derivative. The monographs [17, 21] are good general references on impulsive differential equations—discussion of applications may be found in these books. Applications of the results given in this paper could involve those typically modelled on time scales which are subjected to sudden major influences, for example, an insect population sprayed with an insecticide or a financial market affected by a major terrorist attack. For our purposes, we let T be a
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