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FIRST- AND SECOND-ORDER DYNAMIC EQUATIONS WITH IMPULSE F. M. ATICI AND D. C. BILES Received 3

FIRST- AND SECOND-ORDER DYNAMIC EQUATIONS WITH IMPULSE F. M. ATICI AND D. C. BILES Received 3 December 2004 and in revised form 11 February 2005 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. | FIRST- AND SECOND-ORDER DYNAMIC EQUATIONS WITH IMPULSE F. M. ATICI AND D. C. BILES Received 3 December 2004 and in revised form 11 February 2005 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 Caratheodory and q e 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 second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. Nonlinear boundary conditions