The Study Of Existence Of Solutions For Nonresonant Impulsive Functional Boundary Value Problems

In the nature, the law of development of many things relies on not only the state of the time, but also at some state that has gone or will come, and there usually are impulses. The mathematical model of these phenomenons can be described by functional differential systems([1]-[14]) with impulses. Nonresonant differential systems have widely practical background in many fields, such as physics, biology, medicine and other areas and there were many results on this aspect in late years([14]-[18]). Nonlinear Functional analysis became an important mathematical branch from 1930s, and it provides useful tools for nonlinear problems coming from science and technology field([1]-[9]). In this paper, we study existence of solutions for boundary value problems of nonresonant impulsive functional differential equations using the theory of nonlinear functional analysis.In chapter one, we consider the following nonresonant impulsive functional equationsusing fixed point index.In this chapter, using upper and lower technique and fixed point index theory, we first prove that there is a λ^* > 0 and λ^** > 0 such that the above problem has at least two positive solutions if 0 < λ < λ^* and there are no solutuons if λ > λ^**. Our results improve the conclusions in [14]In chapter two, we present the existance of solutions for the following non-resonant impulsive functional boundary value problems' -u" + f3u{t) = f(t, u{t), ut), t e (0,1), u(t) = (t), t€[-r,0];Aw|t=tl = Lu(ti);Au'\t=tl = L*u'{tx);(2)The key difference between this chapter and chapter one is that the right end is not f(t,u(iu(t))) but f(t,u(t)^ut). Namely, the nature of the system at the moment of t associated not only with the state of w(t), but also the overall state from t — r to t. Monotone iterative technique has broad applications in periodical boundary value problems([19]-[22]), but it is rarely used to discuss impulsive functional boundary value problems([23j). In this chapter, we first establish the corresponding comparison theorem, and then we get the existence of solutions of (2). At last, an example is worked out to indicate that our conditions is reasonable.