Positive Solutions Of Some Classes Of Differential Equation Boundary Value Problems

With the great development of science and technology, all sorts of nonlinear problems have resulted from mathematics, physics, chemistry, biology, and so on. These nonlinear problems have attracted wide attention. However, it is a difficult and interesting aspect to study nonlinear ordinary differential equation singular boundary value problems. It is well known that the ordinary differential equations singular boundary value problems aries in the fields of gas dynamics, newtonian fluid mechanics, the theory of boundary layer, epidemic problems and so on, and has been considered extensively. O'Regan, the Ireland mathematician, dealed with the singular theory detailedly and systematically in literature [6]. On the one hand, many nonlinear ordinary differential equations singular boundary value problems come forth all sorts of applied subject. This forces many people to study them.On the other hand, nonlinear functional analysis has made great progress. Its powerful and fruitful theoretical tools and its advanced methods have been ripeness gradually. Thus, by using many advanced analysis tool of nonlinear analysis in recent years, to study differential equations singular boundary value problems is a subject which is much more interesting and may gain much more important fruitful new results.The present paper employs the cone theory, fixed point index theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several order differential equations singular boundary value problem. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. Most results of this paper are published or to appear in important journals of the world or China, for example,results of this paper are published or to appear in important journals of the world or China, for example, <>,<>, << Journal of Qufu Normal University>>,etc. The paper is divided into four sections.In the first Section, by using fixed point index theory, we present the existence of positive solutions for the singular boundary value problem where a is a continuous function and p(i),g(i) may be singular at t = 0 and/or t = 1. We adopt the following assumptions:(H1) For any and there exist a, b ?(0, 1) such that 0 < / G(t, t)p(t)g(t)dt < +00. JoWe obtain the following results:Theorem 1.3.1 Assume that the conditions (H\) and (H^) are satisfied. In addition, assume that(H3) 0 < F?= limsupx_^0+ maxt€[0il] ^^ < L~l,F(t x) 0 < l~l < FOO = liminf min v ' ' < +00.x-t+oo te[a,b] XThen the boundary value problem (1.1.1) has at least one positive solution in K for anyAe(_L_i_),Theorem 1.3.2 Suppose that the conditions (Hi) and (H2) hold. In addition, assume thatThen the boundary value problem (1.1.1) has at least one positive solution for eachwhere L-1,l-1 of Theorem 1.3.1 and Theorem 1.3.2 are defined as following Note that 0 < L-1 < l-1 < +00.Remark 1.3.1 Prom Theorem 1.3.1, we can see that F(t,u) need not be superlinear or sublinear. So our conclusion extend and improve the corresponding results in [8, 14 - 16]. In fact, Theorem 1.3.1 still holds if one of the following conditions holds:Remark 1.3.3 It seems to be difficult to prove our results by using the norm-type expansion and compression theorem used in [14] and [15]. From Examples 1.3.1 and 1.3.2, we can show not only the existence of positive solutions of the BVP (1.1.1), but also the A interval , which is different from the previous papers (see [14], [15]).Remark 1.3.4 Note that, if F is superlinear, i.e., F?= 0 and F = +00 or sublinear, i.e., F0 = +00 and F = 0, for any A € (0, +00), the BVP (1.1.1) has at least one positive solution. In particular, if p(t) = 1 and g(t)F(t,u) = a(t)f(u), the conclusions of Theorems 1.3.1 and 1.3.2 hold. Thus we generalize the main results of Ma [15]. Our results still hold for the non-singular cases...