Existence Of Solutions To Boundary Value Problems Of Second-Order Ordinnary Differential Equations

This paper is a survey on the studies about boundary value problems for second-order ordinary differential equations. We review the researchs done in the recent decade by using a masterstroke of several basilic boundary value problems. To semi-linear equations, we discuss the general second-order equations and Sturm-Liouville operator equations, mostly discourse upon the existence of positive solutions to the nonsingular and singular Sturm-Liouville boundary value problems. To quasi-linear equations, we mostly discourse upon the existence of positive solutions to the boundary value problems of p-Laplacian equation under the linear and nonlinear boundary value conditions.Moreover, we put forward some disquisitive problems to the boundary value problems of second-order ordinary differential equations.In this paper,we use the following notations : J := (0,1);I := [0,1];R⁺ := [0, +∞);R₀⁺ := (0, +∞) and;G(t, s) is the Green's function corresponding to the boundary value problems, λ₁ is the first eigenvalue corresponding to the boundary value problems. The following is our main results.I Semi-linear Equstions(i) Nonsingular Boundary Value ProblemsIn this case,we consider the following second order boundary value problem first:' u" + F{t,u) = 0, teJ,R^u) = cm(0) - pu'(0) = 0, (3.1)R2(u) = 7?(1) + 8u'(l) = 0.Let :° := hm max, /0 := hm mm o+te[oi], /0 u u->o+te[o,i] u, /oo := lim minHote[0lU:= Hm maxu-++oo t6[0,l]V = (/0X G(?, ajda)"1,/* = (/^ G(|, s)ds)-\ a = miWe adopt the following hypotheses:Hi: FeC(IxR+;R+).H2 : a, (3,7,5 > 0, and p := 7/3 + cry + a<5 > 0.H3 : There is a positive constant p\ such thatF{t,u) Hp2, for (t,u) e [-, -] X [CTP2.P2]-Then,the following theorem holdsTheorem4.1 Suppose one of the following conditions are satisfied.Then,the boundary value problem(3.1)has at least one positive solution,If one of the conditions(2)-(4) is satisfied,the boundary value problem(3.1) has at least two positive solutions (Theorem 3.1-3.6).(1) Hi,H2> and/0 = 0,700 = 00;(2) Hx.Ha, and/°° = 0,/o = oo;(3) Hi,H2, H3, and f0 = foo = oo;(4) Hi,H2,H4> and/o = /°° = O;(5) Hi -H4, and 0< a < 1;(6) Hi,H2,(7) Hi,H2,(8) Hi, H2, and /0, /?, 6 (ï¿¡, oo), 0 < a < 1, F(t, u) < VP * ?In order to use eigenvalue to describe the existence of positive solutions,we introduce the following notationsFq := limsup max u_>o+ *e[oi]:= limsup maxte[oi]F(t,u)uF(t,u) te[o,i] uand we assume that:Hi : Fq > Ai, and F^ > Ax. Hj : FQ < Ai, and F^ < Ai. H3 : there exist a constant p > 0 such that, Fn := hmmf min ----------,u->o+ te[o,i] m, Fr^ '■— hmint min ---------.te[o,i] uF(t,u) Ai and F^ < Ax;(5) FTO > Ai and Fo < Ax.(ii) Nonsingular Sturm-Liouville Bounday Value Problems We consider the following boundary value problem:Lu = F(t,u), t e J,Rx(u) := cm(0) -/?u'(0) = 0, (3.9)R2{u) := 7u(l) + 6u'(l) = 0,where — Lu = (p(t)u')' — q(t)u is Sturm-Liouville operator. We adopt the following hypotheses:o >Ai;U : liminf ^^ > Al5u->0+ U u-ioo uSo : limsup < Ai;Soo : limsup < At.u->0+ u u->oo UH3 : There is a constant p\ > 0 such thatF(t,u) 0 such that1 3F(t, u) > np2, for {t, u) e [-, -] x [ap2,p2].To this kind of boundary problems,the following theorems are about the existence and multiplicity of positive solutions.Theorem4.3 Suppose H!,H2, and Ai are satisfied,and one of the following conditions holds:(1)IO and Soo,(2)1^ and So,(3)H3 and H4,(4)I0 and H'3,(5)S0 and H4,(6)Ioo and H3)(7)Soo and H4. Then,the boundary value problem(3.9)has at least one positive solution.Theorem4.4 Suppose H1;H2, and Ai are satisfied,and one of the following conditions holds:(l)Io,Ioo and H3,(2)80,800 and H4,(3)I0, H3 and H4 , with 0 < px < p2,(4)So, H'3 and H4 , with 0 < p2 < P\,(5)Ioo, H'3 and H4 , with 0 < p2 < Pi,(6)Soo, H3 and H4 , with 0 < pi 0, and p := y{3 + ccy + a5 > 0.H13 : For any t € J, F(t, u) is decreasing with respect to u.There exists a b > 0 such that,for any 0 < r < 1F(t, ru) < r^bF(t,?), V(*, u) G J x R+.H*3 : For any t ï¿¡ J, F(t, u) is increasing with respect to u. There is a function g(b) : [l,+oo) -> R+ such thatF(t, bu) < g(b)F(t, u), V(t, u) e J x R+,where g(b) < 6,and ^ is a measurable function defined in (1, +00).Theorem4.6 Suppose Ha - Hi3 or Hu - H*3 are satisfied.Then,the BVP(3.14) has a positive solution if and only if0< / G(s,s)F(s,l)ds< +00. ./oTheorem4.7 Suppose Hu - H13 or Hn - U*l3 are satisfied.Then.the BVP(3.14) has a CX{I) positive solution if and only if0< / F(s,G(s,s))ds< +00. Jo(iv) Singular Sturm-Liouville Singular Boundary Value ProblemsAs far as we know,for the singular boundary value problems(3.9),the existence of positive solutions has not been studied so much under general boundary conditions.However,for the eigenvalue problems with F — A/(i,u), there are some known results.We assume thatA'i: peC1(I',Rt),qeC(I;R+).H7 : For each M > 0 there exists a continuous function qm 'â–  J —> R+ so that f{t, z) < gM{t) for (t, z) ï¿¡ J x [0, M] and/ G(s,s)gM(s)ds < 00. JoH8 : For each t ï¿¡ J, f(t, u) is nondecreasing in u for u > 0. Hg : There is a function p\ 6 C(J;R+),Pi ^ 0 on any subinterval of (0, l),so that for each constant o > 0 there is an Ra > 0 withf(t, z) > azPl{t), for (t, z)eJx (0, Ra}.H10 : There exists a b > 0 and a continuous function p2 â–  J —> R+ such that f(t,z) > bzp2 for (t, z) ï¿¡ J x i?+,andG(s,s)p2(s)ds < 00.To this kind of problems,the following two theorems are about the existence of positive solutions.Theorem4.8 Assume Hu, H12, A[ and H7 - H9 hold.Then,there exists a Ao > 0 such that the BVP(3.9) has at least one positive solution for 0 < A < Ao.Theorem4.9 Assume H7 - Hi2, and A[ hold.Then,there exists a A* > 0 such that the BVP(3.9) has at least one positive solution for 0 < A < A* and has no solution for A > A*././oMoreover,if F(t, u) — h(t)f(u),we assume that H21 : hEC(J;R+),h^O, and /? G(s, s)h(s)ds < +00. H22: feC(R+;R+).Theorem4.10 Suppose conditions Ai,H2,H2i,H22 are satisfied,and if one of the following conditions holds:(1) /oo>A1,/°A1)/°°>a ^,II Quasi-linear(p-Laplace operator) Equations(i) Boundary Value Problems of Linear Boundary ConditionsWe consider the following two-point boundary value problem:ti), 0p{s) = \s\p²s,p > 1 , R^u) = au(0) - Pu'(Q);R2(u) = ju(l) + Su'(l), and a,P,j,5>0,p = a8 + aj + 'yp> 0.According to the article that we have found,under the general conditions,there are a few results about the existence of positive solutions.Most of them are given on the condition that the nonlinear item is in some special forms or the boundary conditions are of special kinds.The following results for the existence of positive solutions are representatives.case(l) F = k(t)f(u), andp = 5 = 0We adopt the following hypotheses:G10 : f{u) is positive,right continuous,nonincreasing in (0, +00) , andGn : k(t) is a nonnegative measurable function defined in (0,1). Theorem4.11 Suppose conditions Gi0 — Gn are satisfied , then the boundary value problem(3.28) has a positive solution if and only if,.1/2 ,-1/2 pi ps0Gi6 : there exists a constant L > 0 such that,for any compact set e C (0,1), there is an e = ee > 0,such thatF(t,u) > L, for all (t, u) G e x (0,e]. G17 : for any r > 0,there is a hr G C(J;R+) with\F(t,u)\ r, and there exist 0 0 , then the solution is unique.case(3)F = k(t)f(u), a = 7 = 1, and (3 > 0,5 > 0Where we assume f(u) G C(R+;R+), k{t) G C(J;R+), and let/? = Mm M, /. = Urn M0+ XP"1 4fWe make the following hypothesesL2 : 0 < 0 < /01/2 <ï¿¡g( /s1/2 k(r)dr)ds + f^ cj>q{f°/2 k(r)dr)ds < +00.Theorem4.14 Suppose that L2 is satisfied , and in the case that either of the following hold:(i)/o = 0, /oo = +oo, (ii)/0 = +00, /oo = 0. Then,the boundary value problem(3.31 )has at least one positive solution.Theorem4.15 Suppose that L2 is satisfied , and the following conditions holdsL3 : fo = foo = +00,L4 : there a constant r > 0 and a constant Mi € (0,such that/(?) < (A/ir)"-1, for 0 < u < r.Then, the boundary value problem(3.31) has at least two positive solutions ui,u2 with 0 < ||?i|| < r < \\u2\\-Case(4) F = k(t)f(u)Now we introduce the following notations,set/o" - iim^y ^(u)Theorem4.16 Suppose f(u) € C{R+;R+),k(t) P(mi) < /o" < °°'where Mi, and mi are the same as in the article, then the boundary value problem(3.33) has at least one positive solution u G C(I).(ii) Boundary Value Problems of Nonlinear Boundary ConditionsAs far as we know, only a few mathematician has studied this problems with nonlinear boundary conditions. We consider now the boundary value problems0 0,0,7 > 0, F{t, u) G C(I x i?+;R+) , and let/oW = ,im ï¿¡M /coW =Choose 0 € (0,1/2), and let / = [ï¿¡=!(!^)iï¿¡T]-\m = [1 + make the following hypotheses.Ni: /o(<)<^p(m), t6[0,l],NJ: /oo(*)<^M, ?€ [0,1],n2: h{t)xi>P{iie), te[e,i-9],N3(A) : F(i,u) < 0p(mA), t G /,? G [0, A].N4(t7): F(t,u)>(j>p(lV), te[0,l-d],ue[8ri,TJ\.Theorem4.17 The boundary value problem(3.37) has at least one positive solutions in the case that one of the following hold:(1)(2)(3)Theorem4.18 The boundary value problem(3.37) has at least two positive solutions in the case that either of the following hold:(1) N3(A),N2 and N*2.(2) N4(t7),N1 and NJ.Theorem4.19 The boundary value problem(3.37) has at least three positive solutions in the case that either of the following hold:(1) N1,Nt,N2)N^N3(A2) and N^A^if 0 < X1 < A2.(2) Ni.N^N^N^N^AO and N4(A2),if 0 < X, < A2.Ill Problems Worthily StudyBy the results mentioned above, we think the following problems are worthy of further study: (l)The boundary value problems relevant with first order derivative;(2)The boundary value problems of nonlinear boundary conditions;(3)Necessary and sufficient conditions for the existence of positive solutions to various boundary problems.