Singular Nonlinear Boundary Value Solution And Its Application

Nonlinear functional analysis is an important branch of morderm analysis mathmatics, because it can explain all kinds of natural phenomenal, more and more mathematicans are devoting their time to it.Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. 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 classes of 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 arc published or accepted in important journals of the world or China, for example, << Nonlinear Functional Analysis and Applications>, << Journal of Qufu Normal University > , << Journal of Mathematical Study > , << Journal of Engineering Mathematics > etc. The paper is divided into charpt four according to contents.In the first Section of Charpt one, by using Schauder fixed point theorem we shall study a class of singular boundary value problems at nonres-where p, g, h satisfy:and there exist constants > 1, and constant 0 < M 1, for allt (0,1), u (0,+ ), such as(H3) h(t,u) C((0, 1) [0, + ),[0,+ )), and there exist constants 0 < 3 < 4 < 1, and constant 0 < M < 1 , for all t (0, 1), u (0, +00),such asWe obtain the following result:Theorem 2.1.1 Suppose that (H1),(H2),(H3) hold, and f01 t(1 -t)[g(t, 1) +h(t, 1)]dt < + , then problem (2.1.2) has at least a C[0,l] positive solution.Remark 2.1.4 If h(t,u) = 0, the terms (H3) can be removed, thus make Theorem of [10] become a special situation of Theorem 2.1.1 in this paper, so in fact this paper of theorem includes and popularises the relevant results of [10].Remark2.1.5 If g(l,u) = , h(t,u) = , p(t) satisfies (H1), then problem (2.1.1) has at least a C[0,1] positive solution, but we can't obtain it from [9],[10].Remark2.1.6 This results that are given out are new.In the second section , we exploit the fixed point theory of cone expansion and compression to study the positive solutions of a class second-order boundary value problems:(where > 0, / may be singular at t=0 and/or t=1.)A necessary condition and sufficient condition for the existence of C1[0, 1] positive solutions is obtained under the condition that f is superlinear (sub-linear), or f is sum of superlinear and sublinear, which improves the result of [25] in essence, and generalizes or includes some known results of [2,15,24,27]. And this paper is relatively great and different with methods. For the convenience, we make the following assumptions (H1) f(t,u) : (0,1) [0,+ ) - [0,+ ) is continuous, f(t,1) >0, 0 < t < 1, and there exists constants 1 2 >1, such that V t 6 (0,1), u [0, + ), with(H2) f(t, u) : (0,1) x [0, + ) - [0, +00) is continuous, f(t, 1) > 0, 0 < t < 1, and there exists constants 0 < 3 4 < 1, such that t (0, 1),u G [0, + ), withWo obtain the following result:Teorem 2.2.1 Assume (H1), (H3) or (H2), (H3) hold, then BVP(2.2.1) has a C1[0,1] positive solution of necessary condition and sufficient condition that is 0 < 01 f(s, G(s, s))ds < + .Theorem 2.2.2 If f(t,u) = g(t,u) + h(t,u), where g(t,u) satisfy (Hi), h(t,u) satisfy (H2), and f(t,u) such asthen (2.2.1) has a Cl[Q, 1] positive solution of necessary condition and sufficient condition that is 0 < J0 f(s,G(s,s))ds < +00.Remark2.2.4 This paper popularizes the result of paper [25] about function type f(t,u(t)) = p(t)i/(i)>' + q(t)u(t)rn, A > l,m > 1 under some boundary condition to getting with abstract nature of function type establish under general boundary condition, and announce inherent relation of existence quality of solution and Green function from it, so we pularize the result of [25] in essence. In addition, it makes results about Emden-Fowler equation become a ki...