Boundary Value Problems For Second Order Singular Differential Equations

In recent years, the existence of positive solutions for singular boundary value prob-lems have been widely studied. It arises from many practical subjects, such as nuclear, newt-onian fluid mechanics, gas dynamics and so on. In 1927, L.H.Tomas and E.Fermi induced second order ordinary differential equations singular boundary value problems by studying electrmotive force of atom. Since the second boundary value problems of singular type have widely applying background, the study of them has important theo-retical and practical meaning.The study of second order singular boundary problems was earlier studied by S.Talia ferro. Since then, many scholars studied the second order singular boundary value problems. For instance, Guang Chong Yang and Ge Weigao, R.P.Agarwal, D.O'Regan and Stank have obtained many results under different conditions. There are many results under the hypothesis of the nonlinearity f＞0, however, the study of singular boundary problems with sign changing is very fare. In this thesis, we discuss the existence of positive solutions of second order boundary value problems for singular differential equations (1) as follows:In [1] R.P.Agarwal and D.O'Regan studied existence of positive solutions of second order singular boundary value problems [24], Guang Chong Yang studied existence of positive solutions of second order singular boundary value problems where the nonlinearity f＞0, f(t,x,x') may be singular at x=0, x'=0. When f may change sign but is independent on x', in [26], Guang Chong Yang proved the existence of positive solutions of second order singular boundary value problems Recently, Chen Yun, Zhang Lili, Baoqiang Yang, in [36], proved the existence of positive solutions for BVP (1). Under the conditions that f(t,x,z) may change sign and may be singular at x=0 or(and) z=0,Based on the ideas above, this thesis obtains the existence of positive solutions to equation (1) when the nonlinearity f(t,x,z) related to x'changes sign and may be sin-gular at t=1, x=0 or(and) z=1.In chapter one, we give some preparatory knowledge and discuss the positive so-lutions to equation (1). The nonlinearity f(t,x,z) changes sign, without singularity;∫01 1/(p(r))dr = +∞, Because of∫01 1/(p(r))dr = +∞, solutions to equation (1) may be unbounded. We define a special Banach space to solve the problem.In chapter two, using the theory of fixed point index on a cone, first, we present the existence of positive solutions to equation (1) when f is singular at x=0 but not at z=0. We use f(t,x,z)≥ω(t)> 0, whenz∈[-δ,0), to overcome the difficulties caused by sign changing. Similarly, we obtain positive solutions to equation (1) when f(t,x,z) is singular at z=0, but not at x=0 and is singular at x=0, z=0. In-spired by the idea of [24], by transforming the second order equation into first order, in the fourth part, we obtain the positive solutions to equation (1) without considering Under some conditions, we prove the solutions are unbounded. In the fifth part,using f(t, x, z)≥α(t, z)> 0,0< x(t)< b(t) to overcome the difficulties caused by sign changing, we present the existence of positive solutions to equation (1).