| The singular ordinary differential equations arise in the fields of nuclear physics, Newtonion fluid mechanics, the theory of boundary layer, nonlinear optics and so on. The existence of positive solutions for singular differential equations have been considered extensively.In 11.1, we considered the following singular fourth order boundary value problem:Where Until now, fourth order differential equations has been studied widely. But most of these papers only considered the nonsingular case for the nonlinear term. In this paper, we will establish the existence results of positive solutions for the BVP(l. 1.1. )(1.1. 2) by means of fixed point theorem, Arzela ?Ascoli theorem and prior bounds of solution. Different from all known results, we permit the nonlinear term / can be singular at t = 0, 1 and = 0, h may be singular at t = 0.For convenience, we list the following assumptions: is bounded for t (0, 1) . (H4) For any H > 0, there exists y € (7(0,1] satisfies y(t) > 0 t 6 [0, 1] and such that VIWe have the following main result:Theorem 4.1 Suppose the conditions (Hi) - (H4) be satisfied. Then problem (1.1.1),(1.1.2) has a positive solution.In 11.2 , consider the following fourth order differential equation:Where A > 0 is a real positive parameter, , can be singular at t = 0.1.When the nonlinear term / is independent on , The problems has been studied widely, see [14-19] and their reference. In [13], Ma Ruyun showed the existence of positive solutions to equation u;'4'(i) = /(2,u)(f), and boundary condition (2.1.2) when / is either superlinear or sublinear with respect to the variable ".Under weaker conditions in this section, we obtained some existence results of (72[0, 1] positive solutions to problem (2.1.1)(2.1.2). The methods here are same as [14]. Incited by [15], we still get a multiple solutions result. Different from the above results, the conditions that / is satisfied here depended on u".Theorem 2.1.1 Suppose h and / satisfied the following assumptions. continuous, nondecreasing for , nonincreasing for p ;(H2) h : (0, 1) - [0, oo) continuous, h 0 in (0, 1) and satisfied the following inequality.Then(I) There exists > 0 such that 5VP(2.1.1)(2.1.2) has at least one positive solution when ;(II) Fourthermore / satisfied the following conditions:(Hz) There exists constant d > 0 such that when u > 0 then . Then there exists A* > 0 such that WP(2.1.1)(2.1.2) has at least one positive solution when ; there is no positive solution when .Moreover when / satisfied some conditions depended on u>", we can get the multiple solutions result. For convenience, we give the following assumptions.In addition condition (H2) still hold. If / satisfied continuous and we get the following main result.Theorem 2.3.1 Assume that (Ai) - (A3) and (H1*), (H2) hold. Then for an arbitrary given real number M > 0, there exists a a > 0 such that the B-VP(2.1.1)(2.1.2) has at least two nonnegative solutions a,where and max In the second chapter, consider the Emden-Fowler singular boundary value problem:where In [26], Mao give the sufficient and necessary conditions for problem (3.1.3)(3.1.4) when < 0, m < 0. . Other results see [19],[21]-[23]. Here we will consider the case of > 0, m > 0. Now the nonlinear term have singularity. We extended some results.For convenience we give the hypothesis as follows:.Now we state the main results.Theorem 1.1 Suppose condition (H) is satisfied. Then. Then problem (3.1.3), (3.1.4) has a C"[0,l] positive solution only and only ifVIII(II) If . Then problem (3.1.3), (3.1.4) has a C^O, 1] positive solution only and only if(III) If ft / 0, S = 0. Then problem (3.1.3), (3.1.4) has a Ca[0, 1) positive solution only and only if(IV) If . Then problem (3.1.3), (3.1.4) has a C[0, 1] positive solution only and only ifTheorem 1.2 Suppose that (H) is satisfied. Th... |
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