Initial And Boundary Value Problems For Singular Functional Differential Equations On Half-line

Initial and boundary value problems are generated from many natrural sciences, for example., lemology. nuclear physics, control theory and so on. Many phenomena can be depicted in functional differential equations, so reseach work about these is of great value both in theory and application. In 1970s, there was some fundamental theory about functional differential equations. Subsequently, many reseach workers contiuned to study and develop the theory of these equations. Alao there were some existence results under some conditions. e.g. Jiang Daqing, Weng Peixuan in China and Ravi. Agarval, CH. G. Philos, Donal. O'Regan, L.H. Erbe abroad have done a lot work. In these reseach work, some nonlinearities have singularities, some have not. Most methods they applied are fixed point theorem, fixed point index theory on cone, lower and upper solution method and so on.The dissertation contains two chapters. We will overcome the difficulties which lie in sign changing and singularities using the fixed point theory on cone and approximation technique. Thus we can get the existence of positive solutions for singuar initial and boundary value problems.In chapter one. we consider singular initial value problem on half-linewhere f(t, (?), y) may be sign-changing and singular at (?) = 0 and y = 0. In [3], R.P. Agarwal, Ch. G. Philos and P.Ch. Tsamatos considered the following functional differential equationwhere f(t,(?),y) may be singular at t = 0. In the third section of this chapter, we main discuss the case p(t)φ(t) = 1,t∈[0,∞),i.e, we generalize the above equation to the singularities at (?) = 0 and(or) y = 0. Using fixed point index theory, we obtain the solution sequence of approximate equations, then we get the solution to (1) by ArzelaAscolitheorem. When (?) is positive and singular aty=0. we get the existence of at least one positive solution in section 4. The conditonsare less strict than these in section 3.In the second chapter, we consider singular boundary value problemwhere f(t,(?),y) may be sign-changing and singular at (?) = 0 and y = 0. In[33]. Ch.G.Philos considered the following equationwhereτ= - (?)(t - T_j(t)). In section 2. we mainly consider the case p(t)φ(t)≡1, t∈[0, +∞). The difference between the above equation and the one we consedered is that f(t,(?),y) is derivative independent and sigular at (?) = 0 and y = 0 . Constructing a cone, we get the existence of at least one positive solution to the operator sequence by fixed point theory, then by Arzela-Ascoli theorem, the existence of at least one positive solution to (2) are obtained. In section 3, f(t,(?), y) is positive on [0,∞)×(X~+ -0)×(0,∞).