Positive Solutions Of Nonlinear Boundary Value Problems

Nonlinear functional analysis has been one of the most important branches of learning in modern mathematics at present. It provides an effect theoretical tool for studying many nonlinear problems. It plays an important role in dealing with nonlinear equations and differential equations arising in applied mathematics.The study of the nonlinear Sturm-Liouville boundary value problems has interested people for a long time and the research in this field is still very active. Recently, the existence of positive solutions of the nonlinear Sturm-Liouville boundary value problems attracts close attention. In much literature, the existence of positive solutions of the nonlinear Sturm-Liouville boundary value problems has been proved under the condition that the nonlinear term is nonnegative. Only a few authors have been proved the existence of positive solutions of the nonlinear Sturm-Liouville boundary value problems in the case that the nonlinear term is allowed to be negative. They mainly used the methods of the cone theory . Hence, for further discussion, we need to look for new methods to study the existence of positive solutions for nonlinear Sturm-Liouville boundary value problems.The boundary value problems of second order ordinary differential equations have been studied in much literature. Compared with boundary value problems of ordinary differential equations, a few authors have been studied the boundary value problems for systems of ordinary differential equations. For the demand of practical problems, the research on systems of ordinary differential equations has certain value.Measure chain theory is introduced by Stefan Hilger in [28] to unify the theory of continuous analysis and discrete analysis. With the development and improvement of measure chain theory, the study of the boundary value problems of dynamic equations on measure chains attracts close attention. How to use the tools of nonlinear functional analysis to study the boundary value problems of dynamic equations on measure chains has interested many authors.In this thesis we mainly study the global structure of positive solutions of nonlinear Sturm-Liouville boundary value problems, the existence of positive solutions of boundary value problems for systems of nonlinear ordinary differential equations and boundary valueproblems of nonlinear ordinary differential equations on measure chains. This thesis is composed of three chapters.In Chapter 1, we study the following nonlinear Sturm-Liouville boundary value problem-(p(z)u')' + ?(x)u = Xf(x, u), 0 < x < 1, /3btz'(0) = 0, (1) + 0iu'{l) = 0.Many authors have studied the existence of positive solutions for the nonlinear boundary value problem (1)( see[24,26,31-34,36,43-45,57-60] and the references therein) in the case that the nonlinear term / satisfies /(x, u) > 0(u > 0). But a few authors considered the existence of positive solutions for the nonlinear boundary value problem (1)( see[25,29,30,35]) in the case that the nonlinear term / does not satisfy f{x,u) > 0(ti > 0).In this chapter, we suppose that(Ho) a0 > 0, A) < 0, ax > 0, ft > 0, {al+0>){a\+0l) ± 0; p(x) € C% 1], q(x) € C[0,1], p(x) > 0, q(x) > 0, V x € [0,1]; and the homogenous equation with respect to (1) qu = 0, 0 < x < 1, aQu{0) + pou'(0) = 0, (2)has only the trivial solution.In the sublinear case, we obtain the following results.We assume that(Hx) f(x,u) = a(x)u + H(x,u), where a(x) e C[0,1], a(x) > 0(V x € [0,1]), H : [0,1] x R1 —? R1 is continuous, andU-.0 U(H2) there exists a 6 R1 such thatfix u) limsup ' < a, uniformly on x € [0,1].U—+00 USuppose that (Ho) ~ (H2) are satisfied, and if a in (H2) satisfies — > Ai , then foraC+ n ({A} x S+) ^ 0, VAiAj.Hence, for any A € (Ai,+oo), the boundary value problem (1) has at least a positive solution, (where Ai is the first eigenvalue of the linear operator corresponding to (1),fl Xb is the first eigenvalue of the linear operator Bu{x) — I k(x,y)u(y)dy, k(x,y) is theJo Green's function with respect to (2)).When f(x,u) = h(x)g(u), and h is allowed to be sigular at x = 0 or x — 1. We study the following nonlinear Sturm-Liouville boundary value problem-(p(x)u'Y + q(x)u = \h(x)g(u), 0 < x < 1,a0u{0) + pou'{0) = 0, (3)We assume that(H'x) g{u) = au + H(u), where a > 0. H : R1 ^ Rl is continuous andlim0;u->0 U(H'2) there exists a € Rl such thatg(u) hmsup----- < a;u~*+oo u(H3) /i : (0,1) —> [0, +00) is continuous, h(x) ^ 61, and/ h(y)dy < +oc. JoSuppose that (Ho)(H'1)(H'2)(H3) are satisfied, if 0 < a■ < a in (H^XI^), then there exists an unbounded connected component C in L"1" which meets (Ai.#), andCn({A}QThus, for any A € (Ai,-----), the boundary value problem (3) has at least a positivea solution; if a < 0 in (H'2), then there exists an unbounded connected component C in L+,containing (Ai,0), and0, VA>AlThus, for any A € (Ai,+oo), the boundary value problem (3) has at least a positive solution.In §1.2, we obtain the global structure of positive solutions in the case that f(x,u) 2 0(u > 0), and the existence of positive solutions is proved, which method is different fromthat of [25,29,30,35]. The cone theory is used in [25,29,30,35] to prove the existence of positive solutions. By approximation, we obtain the global structure of positive solutions of the singular boundary value problem (3) and the existence of positive solutions.Similar to the sublinear case, we obtain the global structure of positive solutions of the boundary value problems (1)(3) and the existence of positive solutions in the superlinear case(see§1.3).In Chapter 2, we study the existence of positive solutions of boundary value problem for the system of second order nonlinear differential equations—L\u = f(x,v,u), 0 < x < 1, -L2v = g(x, u), 0 < x < 1, + &iu'(0) = 0, ci?(l) + diu'(l) = 0,a2v{Q) + b2v'(0) = 0, C2?(l) + d2v'(l) = 0,where L{U = (pin')' + qiU.In the sublinear case, we obtain the following results.We assume that(Co) Pi(x)eC%l], 9i(i)€C[0,l], Pi(x)>0,qi(x)<0, Vi€[O,l], z = 1,2;Ot>0, 6i<0, Ci>0, di>0, {af + bf)(c^ + df)^O, i = 1,2;/ G C([0,1] x R+ x R+, R+), g € C([0,1] x i?+, ,R+) (R+ = [0,/(i,0,0) = 0, g{x,0) = 0; and the homogenous equation with respect to (4)f -LiU = 0, 0 < x < 1,\ aiU{0) + biu'{0) = 0, ciu(l) + diu'{\) = 0 Uhas only the trivial solution.(Ci) there exist t € (l,+oo) and continuous functions a\(x) > 0, a2{x) > 0, (3(x) > 0, 0i(x) > 0, fo(x) > 0, such thatf{x,u,v) 0, v > 0;(5(a:.?))' < <*2(x)u + A(x), V x G [0,1], u > 0.(C2) there exist s € (0, +00) and continuous functions t](x) > 0, 771 (x) > 0, 772(0:) > 0, such thatf(x,u,v) >77i(x)us+7](x)v, VxG [0,1], Q<% v T]2{x)u^ V x G [0,1], 0 < u < r.IVwhere r is a sufficiently small positive constant. LetB\u{x) = / / ki(x,y)ai{y)kt2(y,T)a2(T)u(T)d,Tdy+ ki{x,y)/3(y)u{y)dy, Jo Jo Jof1 ( f1 1 V flB2u{x)= I h(x,y)rii(y)[ k2(y,r)ri2(T)u~°(T)dT) dy + ki{x,y)r)(y)u(y)dy,Jo Vo J Jowhere ki(x,y) and k2(x,y) are the Green's functions with respect to (5).(C3) r(Bi) < 1, where r(B\) is the spectral radius of the linear operator B\\ and there exists u*(x) € C[0,1], tt*(x) > 0, V x € [0,1], u*(x) ^ 0, such that B2u* > u*. And f?2 is a homogenous operator.Suppose that (Co) ~ (C3) are satisfied. Then the boundary value problem for the system of second order ordinary differential equations (4) has at least a positive solution.we study the existence of positive solutions of boundary value problem for the system of singular second order nonlinear differential equations-Liu = hi{x)f(v,u), 0 < x < 1,-L2v = h2{x)g{u), 0 < x < 1,aiu(0) + 6iu'(0)=0, cxu(l) + d1u/(l) = 0,a2i>(0) + b2v'(0) = 0, c2u(l) + d2v'(l) = 0,where L{U = (piU1)' + qiu. We assume that (C'o) Pi(x) € Cl[Q, 1], qi(x) E C[0,1], Pi(x) > 0,qi(x) < 0, V x e [0,1], i = 1, 2;ai > 0, bi < 0, a > 0, d% > 0, (a? + 62)(C2 + df) ^ 0, i = 1, 2; / 6 CCR+ x i?+,i?+), g e C(R+,R+)(R+ = [0,+co)); /(0,0) = 0, 5(0) = 0;(Cj) there exist i € [1, +00) and constants qi > 0, a2 > 0, /3 > 0, /?i > 0, /32 > 0 such thatf(u,v) < aiu* +(3v + j3i, V u > 0, 1; > 0;(C2) there exist s € (0, +00) and constants 77 > 0, r/i > 0, 772 > 0 such that f(u, v) > T]ius + rjV, V 0 < u, v < r; g(u) > t]2U~> , V 0 < u < r.where r > 0 is a sufficient small constant.Letki(x,y)hi(y)k^(ytT)h^(T)u(r)dTdy + 0 / ki(x,y)hi{y)u{y)dy; JoB2u(x) = T)iV2 h(x,y)hi{y)( k2{y,T)h2{T)ui{r)dT)dy + r) I h{x,y)hi{y)u(y)dy. Jo Wo ' Jowhere ki(x,y) and fc2(x,y) are the Green's functions with respect to (5).(C3) r(B\) < 1, where r{B\) is the spectral radius of the linear operator B\; and there exists u?(x) G C[0,1], u,{x) > 0 such that B2U* > ut with u+(x) ^ 0, V x G [0,1]. B2 is a homogenous operator.f1 (C4) /it(0,1) —> [0,+00) is continuous, hi(x) ^ 6, (i = 1,2), / /ii(x)dx < +00;1 / h2{x)dx < +00.Suppose that (C'o) ~ (0^(04) are satisfied. Then the boundary value problem for the system of singular second order ordinary differential equations (6) has at least a positive solution.We use topological degree methods to study the boundary value problem (4). The conditions in §2.2 extend and improve those of [20,21,37]. And we use the invariance of small perturbation of the fixed point index to obtain the existence of positive solution for the system of singular ordinary differential equations. We have not seen the paper for systems of singular ordinary differential equations so far.Similarly, we can obtain the existence of positive solutions of the boundary value problems (4) and (6) in the superlinear case (see§2.3).In Chapter 3, we apply topological methods to study the boundary value problem of the second order nonlinear differential equation on a measure chainLx(t) = -\r(t)xA(t))A = f(t,x(a(t))), t G [a,6],ax(a)-/?zA(a) = 0, (7)We assume(Fx) a > 0, p > 0, 7 > 0, 6 > 0, (a2 + /?2)(72 + S2) ± 0, r(t) > 0, t E and r is A-differential;(F2) / : \a,a(b)] x R+ -> R+ continuous. Suppose that (F1XF2) are satisfied. If1; pu—>0+ U u->+oowhere Ai is the first eigenvalue of the relevant linear operator, then the boundary value problem (7) has at least a positive solution.We also study the following boundary value problem of second order nonlinear differential equation tET, \ s(0) = x(l) = 0.We obtain the global structure of positive solutions of the boundary value problem (8). And the existence of positive solution is proved.The conditions in §3.2 and §3.3 are weaker than those of [27,46-48,50-52]. The cone theory was used to prove the existence of positive solutions in [55]. The global structure theory is used to prove the existence of positive solutions in the case that the nonlinear term may be negative in §3.4, which is different from that of [55].