Abstract Solution Of Nonlinear Impulsive Equation And Its Application In The Space

Along with the development of science and technology, various nonlinear problemshave resulted from mathematics, cybernetics, physics, biology, medicine, economics, en-gineering and so on. During the process of solving these problems, nonlinear functionalanalysis has been one of the most important research felds in modern mathematics. Thenonlinear functional analysis is based upon nonlinear problems of maths and science,constructs general theories and methods and ofers efective theoretic tools for theseproblems arising from many applied mathematics. Many well known domestic and for-eign mathematicians had proud works in various felds of nonlinear functional analysis.The nonlinear analysis has become one of the most important research directions inmodern mathematics. In addition, nonlinear impulsive diferential equations describeprocesses which experience a sudden change of their state at certain moments. Be-cause it can explain well various the natural phenomenon, these problems have arousedpeople’s widespread attention day by day. It is one of the most active domains ofnonlinear functional analysis at present. The boundary value problem for nonlinearimpulsive diferential equation in Banach space, especially, the boundary value prob-lem with integral boundary conditions is also the hot spot which has been discussed inrecent years. It becomes a very important domain of diferential equation research atpresent.The present paper is mainly concerned with existence of solutions for severalclasses of nonlinear impulsive equations in abstract spaces by using cone theory, fxedtheorem. It consists of three chapters and the main contents are as follows:In Chapter1, we investigate the following boundary value problems with integralboundary conditions for second-order nonlinear impulsive integro-diferential equationsof mixed type in Banach space k) represent the right-hand limit and left-hand limit of x(t) at t=tk,Δx|t=tkhas a similar meaning for x (t). By the use of the Mo¨nch fxed point theorem,we obtain the existence of solutions for boundary value problem (1.1.1) under weakerconditions. The results obtained in this paper improve and generalize the results in[8](see Remarks1.3.1-1.3.2).In Chapter2, we talk about the solvability for a class of nonlinear singular impul-sive Volterra integral equtions in Banach spacesWe use the Mo¨nch fxed point theorem and the method of estimate step by step, thesolvability of the nonlinear singular impulsive Volterra integral equtions is investigatedunder weaker conditions. We also give some applications to initial value problemsfor second-order nonlinear singular impulsive diferential equations of mixed type inBanach spaces. The results generalize and improve the results in [1,3,24](see Remarks2.4.1-2.4.4).In Chapter3, we use the Schauder fxed point theorem and concern the existenceof positive solutions for second-order nonlinear impulsive singular BVPs with integral boundary conditions in Banach spacesk) represent the right-hand limit andleft-hand limit of u(t) at t=tk, Δu|t=tkhas a similar meaning for u (t).