Nonlinear Ordinary Differential Equation Boundary Value Problems

Nonlinear functional analysis is an important branch of modern analysis mathmatics, because it can explain many kinds of natural phenomena, more and more mathematicians are devoting their time to it. As a branch of Nonlinear functional analysis ,the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, and it is one of the most active fields that are studied in analysis mathematics at present. Singular boundary value problems are important in applied mathematics, which model a wide spectrum of nonlinear phenomena, such as gas diffusion through porous media,catalysts theory,thermal self-ignition of a chemicalac-tive mixture of gases and so on.In this paper we investigate the existence of positive solutions of several classes of differential equations boundary value problems by using the cone theorem, fixed point index theorem, and the fixed point theorem of cone expansion and compression and so on. 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 be appeared in important journals of China, for example, 《 Journal of Qufu Normal University 》 , 《 Journal of Engineering Mathematics 》 etc. The paper is divided into three chapters according to the contents.In the first section of the first chapter, we study some second-order eigenvalue boundary value problem:when A is in some range,the postive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression under the extensive condition, and the results of this paper partially improve and extend the former corresponding work. where: λ > 0, α, β, γ ,δ ≥ 0, ρ = αγ +βδ + γβ > 0, a∈C((0,1), (0, +∞)), f is singular at t = 0,1, u = 0, and f, asatisfy:{Hi) f : (0,1) x (0, +oo) -? [0, +oo) is continuous, f(t, u) < p(t)q(u), where p : (0,1) -> [0, +oo), q : [0, +oo) -? [0, +oo) is continuous;(H2) (i) lim sup ^ < M, (ii) lim inf inf ^^ > M, whereu_>0+ u-?+ooO M;u>|oo u->0+ 0+o(F5) lim inf inf ^ > M lim inf inf IMl > M;u-$0+0H6 There exists i?i > 0, such that 0 <|| u ||< i?x implies f(t,u) > RiMMg;(H7) There exists R2 > 0, such that 0 <|| u ||< i?2 implies max{^(u) : 0 <|| u ||< i?2} < i?2M.We obtain the following results:Theorem 1.1.1 Suppose (Hi), (H2) or (Hi), (H3) hold, then for V A :MeMjg^eG(^s)a(s)ds < ^the boundary value problem (1.1.1) has at least a positive solution. Theorem 1.1.2 Suppose (i*i), (H4), (H6) hold, then for V A :X1the boundary value problem(1.1.1) has at least a positive solution. Theorem 1.1.3 Suppose (Hi), (H5), (H7) hold, then for V A :------—i_---------9G(\,s)a(s)dsthe boundary value problem (1.1.1) has at least two positive solutions.Remark 1.1.1 If / does not include variable t, the paper [2,6] studied the boundary value problem (l.l.l)under the Dirichlet boundary value problem, but / is singular in this section.Remark 1.1.2 The paper [3] considered the boundary value problem (1.1.1) under the condition that A = l,p(t) = 1. The condition of this section is obviously weaker than the paper [4].In the second section of the first chapter, we study the positive solutions of the following boundary value problem:(p(t)u')' + a(t)f(t,u) = O, 0 < ￡ < 1,(1.2.1) u'(0) = u(l) = 0where:/ ￡ C((0,1) x [0, +oo), [0, +oo)), p G C([0,1], (0, oo)).Concering the boundary value problem (1.2.1), we make the following assumptions:(Hi) iimu_y0+ inf ^^- > M,]imu_>+0o inf ^^ > M,Vt G (0,1), where0 < M^x = I ' -4- I drds < +oo.Jo P(s)JoTheorem 1.2.1 Suppose (Hi) hold, and there exists H, such that ||u|| < H implies f{t,u) < NH,V t G (0, l),where1 = /" — fJo P(s)JfS0 < N'1 = I -fr / drds < +oo.then the boundary value problem (1.2.1) has at least two positive solutions. In the second chapter, by means of the fixed point index theorem we consider the existence of the positive solution for fourth-order singular semi-positone boundary value problems:u(4) = Xf{t, u) + g(t, u), 0 < t < 1, u(0) = u'(0) = u(l) = u'(l) = 0. The documents in the past seldem involve in the researches of the semi-positone boundary value problems, most of which make very strong monotone assumption to function /, and the condition is quite complicated. This paper is under a kind of new condition, by utilizing one kind of new method we get the sufficient condition the existence of positive solutionof the semi-positone boundary value problem, furthermore we simplified the proof of this issue. Few people study singular semi-positone boundary value problem of fourth order differential equations, so the results of singular semi-positone boundary value problem of fourth order has seldom been seen until now. The identification of this section theorem is new, and the result is new too, and we give the applications of this theorem also. The result of this paper improves and extends many recent results. In this paper, the existence of the positive solution of the boundary value problenv (2.1.1)is obtained on a special cone under the condition that g is semi-positone,and one example is presented to illustrate the application of the obtained results. Concering the boundary value problem(2.1.1), we make the following assumptions:(Hi) f : (0,1) x [0, oo) -$ R+ is continuous', g : (0,1) x (0, oo) ->? R is continuous, g(t,u) > -q(t), V(i,u) € (0,1) x (0,oo), where q(t) : (0,1) ->? (0, oo) is continuous, 0 < JQ s2(l — s)2q(s)ds < oo;(H2) f(t,u) < pi(t)qi(u), g(t,u) < p2(t)q2(u), where Pl : (0,1) ->■ [0, oo) is continuous, p2 : (0,1) —> R is continuous, q\ : [0, oo) -> [0, oo) is continuous, q2 : [0, oo) —>■ R is continuous;(H3) 0 < /o1 s2(l - s)2(XPl(s) +p2(s) + q(s))ds < oo ;(Hi) There exists r : r > 2 fQ q(s)ds, such that0M, where M^l = max f^6 f G(t, s)ds, 9 eWe obtain the following result:Theorem 2.1.1 Suppose (Hi) — (H$) hold, then the boundary value problem (2.1.1) has at least a positive solution.The following example is presented to illustrate the application of theobtained result. T)^i (212)(0) = u'(0) = u(l) = u'(l) = 0 where,It is clear that {Hi), (H2), (H3), (H5) hold. Let r = 14, then f1 , A1 27 >-------------------------->0 i^l) > -2 , 1 , so we can not find a constant M > 0, such that g(t,u) > —M, thus we can not obtain the existence of the positive solution of the boundary value problem (2.1.2) by the means of the theorem of [7]. The theorems of [33,35] are the special case of theorem 2.1.1 of this paper, for g is singular at u = 0 and g is semi-positone in this paper.Remark 2.1.2 By the same way, we can consider the following second-order singular semi-positone boundary value problem:u" + f{t, u) + g(t, u) = 0,0 < t < 1,(2.1.3) cm(0) - 0u'(O) = 0,7u(l) + 6u'(l) = 0.where: a, 0,^,6 > 0. /is singular at t = 0,1, g is singular at t — 0,1, u = 0, and g is semi-positone.In the third chapter we shall consider the existence of the positive solutions for some system boundary value problems. In the first section, weshall consider the existence of the positive solutions for some second-order system eigenvalue boundary value problems:(p(t)u')f + Xa(t)f(u(t),v(t) = 0,0 < t < 1,(p(t)v'y + Xb(t)g(u(t),v(t) = 0,0 < t < 1, (3.1.1)u'(0) = u(l) = v(l) = v'(l) = 0.We obtain two positive solutions by means of the fixed point theorem of cone expansion and compression, and the result of this paper improves and extends many recent results. Where /, g, a, b satisfy:(#0 /, g G C[(R+ x R+), [0, +oo)], a, b,p € C((0,1), (0, oo)),(H2) There exists H > 0, such that 0 < ||(w,f)|| < H implies (f(u,v),g(u,v)) ||< N\\(u,v)||, where = max{f1 If1 fl 1 /"* { / —— / a(s)dsdt, / —^ / b(s)dsdt} < +oo.7o P(*J 7o Jo P{t) JoWe obtain the following results:Theorem 3.1.1 Suppose (Hi) — (H2) hold, when A satisfies:_<sub><sub><sub><sub><sub><sub>M_<sub><sub><sub><sub><sub><sub>min{/ , &,, / ,g } ^wheref1 1 /"* A1 1 /"*M = mini / —t-t- / a(s)dsdt, / —-— / b(s)dsdt)} < +oo,Jo P(*) Jo Jo P{t) Jothen the second-order system boundary value problem(3.1.1) has at least two positive solutions xx,x2, and 0 < ||xi|| < H < ||x2||.In the second section,we shall consider the existence of the positive solutions for some fourth-order system boundary value problems by means of the fixed point index theorem :Concering the boundary value problem(3.2.1), we make the following assumption:(Hi) f,g : [0,oo) x [0,oo) —>■ [0,oo) is continuous, a, b : [0,1] —> [0, oo)is continuous and a, b € L(0,1).Let7 ,. f{u,v) v ? e f(u'v)/o = hm sup-^—777,/ = hm mf||()||0 K||(ut/)||>io ||()| TTT---------rrr. \\(U,V)\\ = hm sup .,. . 1,, / = hmWe obtain the following results: Theorem 3.2.1 Suppose (Hx) holds, if(!) Zoo > a(^oo > a)' f° a(gi)>a)also holds, then the boundary value problem(3.2.1) have at least one positive solution.Theorem 3.2.2 Suppose (HJ, f^ > a(g^ > a), ￡, > a(g^ > a) and there exists Mi, such that 0 < \\u, v\\ < Mi implies\\(f(u,v),g(u,v))\\ 1, then the boundary value problem(3.2.1) have at least one positive solution.Theorem 3.2.3 Suppose (Hi), fQ < ft, g0 < 0, /? < P, 9oo < P and there exists M2, such that 0 <\\(u,v)\\ < M2 implies\\(f(u,v),g(u,v))\\>5M2where S = ?& min{f*j* G(r,r)a(r)dT, f* G(r,T)b(r)dr}, then the boundary value problem(3.2.1) have at least two positive solutions.Remark 3.2.1 Suppose (Hi) holds, ifi) /o = 0, g0 = 0, /a, = 00 (&O = 00);ii) /oo = 0, goo = 0, /o = 00 (p0 = 00)also hold, then the boundary value problem(3.2.1) has at least one positive solution.