Anti-periodic Boundary Value Problem For The Second-order Impulsive Integro-differential Equations

This paper have discussed the second-order integro-di?erential equations withimpulses and anti-periodic boundary conditionsIt use monotone iterative techniques and Banach fixed point theorem and the lowerand supper solutionsα0,β0 of anti-periodic boundary conditions to prove the exis-tence of extreme solutions of this equations.Firstly, it establish a new comparison principle about second-order impulsiveintegro-differential equations with anti-periodic boundary conditions x(t)≤0 ,which is proved by the method of contrary.Secondly, the definition of upper and lower solutions to the first-order andanti-periodic impulsive integro-di?erential equations and the second-order impulsiveintegro-differential equations with periodic and nonlinear boundary conditions havebeen not applied to solve problem (1), so an important definition of upper and lowersolutions for anti-periodic boundary problem in this paper are given.Thirdly, the linear equal equation of this class of anti-periodic boundary con-ditions second-order impulsive integro-di?erential equations briefly introduced andthe exist and unique of solution of it worked out, and the necessary conditions(A1) - (A3) are given.Lastly, an important theorem in this section is obtained, which under someconditions, the extreme solution of problem (1)is between upper and lower solutions,in the process of the proof comparison principle, monotone iterative techniques andlower and upper solutions are proved to prove the conclusion.