Convergence Of Solutions To The Integro-differential Equations

In this paper, the convergence of solutions to some integro-differential equations are stud-ied, implying with quasilinearization method and generalized quasilinearization method. The paper is composed of four chapters.In Chapter 1, the historical background and current situation of the integro-differential systems are introduced as well as the main work of the paper.In Chapter 2, the initial second-order integral differential equations with two integration op-erators on the normal cone are investigated by using the generalized quasilinearization method. Two monotone sequences which converge uniformly quadratically to the unique solution of the initial value problem are established.In Chapter 3, anti-periodic boundary value problems for first order integro-differential equations with the new definitions of upper and lower related solutions are discussed. By us-ing monotone iterative technique and quasilinearization method, two monotone sequences of iterates converging uniformly and quadratically to a solution of the problem are obtained.In Chapter 4, we use the quasilinearization method, Ascoli—Arzela theory and impulse inequation to the initial second-order impulsive integral differential equations on the normal cone and a class of first-order impulsive integro-differential equations with integral boundary conditions, then obtain monotone sequences which converge uniformly quadratically to the unique solution of the problem.In Chapter 5, a brief summary of this thesis and some advices about further development for the theory of integro-differential equations are advanced.