The Solution Of Nonlinear Boundary Value And Its Applications

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,Leggett-williams fixed point theorem and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several classes of differential equations singular m-point boundary value problem. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. The paper is divided into three sections according to contents.In the first Section, we shall study a class of m-point boundary value problemsThe following conditions hold:(H₁) q(t) ∈ L¹[0,1], q(t) ≥ 0,(?) t∈ [0,1];(H₂)f∈C([0,1]×[0,+∞),[0,+∞));(H₃) a_i ≥ 0, b_i ≥ 0, ∑_i=1^m-2a_i < 1,∑_i=1^m-2b_i < 1, p(t) ∈ C[0,1],and p(t)>0,(?)t∈[0,1] ;(H₄) there exist nonnegative functions a_i(t), g_i(y)(i = 1,2), t ∈ [0,1], y ∈ [0, +∞), satisfy ∫₀¹q(t)a_i(t)dt > 0,and (?)t∈ [0,1], (?)r > 0,if 0 ≤ y ≤ r then a₁(t)g₁(y) ≤ f(t,y); if R>0,y≥R then f(t,y) ≤ a2(t)g₂{y);(H₄^*) there exist nonnegative functions a_i(t),g_i(y)(i = 1,2), t ∈ [0,1], y∈ [0, +∞), satisfy ∫₀¹ q(t)a_i(t)dt > 0,and a₁(t)g₁(y) ≤ f(t, y) ≤ a₂(t)g₂(y).We obtain the following result: Theorem 1 If (H4) hold,andwhere λ_i(i = 1,2) are the first eigenvalue of T_i defined by (2.1.6), then BVP (2.1.2)has at least a positive solution. Theorem 2 If (H₄^?) hold,andwhere λ₂ is the first eigenvalue of T₂ defined by (2.1.6).then BVP (2.1.2)has at least a positive solution.Remark 1 If p(t) ≡ 1,then t/(p(t))≥∫₀^t1/(p(s))ds, (?)t ∈ [0,1].Remark 2 When p(t) = 1,q(t) = 1,f(t,y) = a(t)f(y), we obtain the same results with [11] under weaker condition,so in fact this paper of theorem includes and popularizes the relevant results of [11].By making use of Leggett-williams fixed point theorem,we obtain the exitence of two positive solutions of the m-point BVPThe following conditions hold:(#1) di > 0, bt > 0, 0 < Yh=i ai < !? EI^2 &i < 1; (F2)/€C([0,+oo),[0,+oo));(H3) a(t) E La[0,1], a(t) > 0, V t e [0, l],and 0 < /on o(t)rfi < +oo; (tf4) p(t) € C[0,l],p(t) > 0,^ > /o^yds.V t 6 [0,l],and if -72 6i ^ 0 thendx 1 ^ f1 dx \ /?* rfa;We obtain the following result:Theorem 1 If there exist constantO < a < b < c, such that / satisfy:(#5) :/M > f, c < W < ^c;(Jf7) : /(w) > 2, 0 p{u'))' + a(t)f(t, u) = 0, 0 < t < 1,?Z? 3.1.4cm(0) - pu'(0) = 0, u(l) = } at1=1where 4>p{s) is p — Laplacian function,^(s) is inverse function to 0p(s),i.e 0P = |s|p-2s, p > 1, J + i = 1, 0 < -ei < ... < U-2 < 1, a > 0, P > 0, a2 + (32 > 0.The following conditions hold:(^2) nonnegative function a(i) 6 C(0,1), 0 < Jo a(s)ds < +00, and there exists Xq G (^m-2,1) such that a(x0) > 0;(H3) B = a{l- ZZ'i2 ?*&) + P(l- Etf 0.We obtain the following result:Theorem 1 Suppose one of the following hold:(H4) : 3 pi,p2 ￡ (0, +oo), and px < 7p2, such that Z^1 < $p(m),(H5) : 3 pi,pi € (0, +00), and px < p2, such that f￡2 < <&p(m),then BVP (3.1.4)has at least a positive solution. Theorem 2 Suppose one of the following hold: (H6) : 3 pi, p2, P3 e (0, +oo), and pi < 7p2, p2 < p3)such that(H7) : 3 pi, p2, P3 G (0, +oo), and pi < p2 < 7p3,such that f￡2 < $p(m), f%px > $p(^7)>u $" Au, V u G d KP2, f!*3 > 3>P(M7). then BVP (3.1.4)has at least two positive solutions.Remark2 Whenp = 2, m — 3,under conditions hmu_>0+ ^^, limu₊00+ ^ one equal 0 the other equal +00,[20] obtained the existence of a positive solution of BVP (3.1); When a = 1,(3 = 0,we improve [22] to singular conditions.In the third section, by making use of the fixed point theorem of cone expansion or compression type , we set up the existence of positive solutions for third-order three-point boundary value problem with semipositonenonlinearity' u'"(t) - \f(t, u,u') = 0, 0 0 such thatf(t,u,v)>-M(t)...