Existence And Multiplicity Of Solutions To Boundary Value Problems Of Fourth-Order Ordinary Differential Equations

This thesis is a survey of the recent results in investigating the boundary value problems of fourth-order ordinary differential equations. We briefly overview the recent situation for studying this kind of problems and also survey the results with concentration on some im-portant boundary value problems, such as two-point boundary value problems, multi-point boundary value problems and periodic boundary value problems.We introduce the existence of (positive) solutions, the existence of multiple (positive) solutions to some fourth-order two-point boundary value problems; introduce the existence of solutions and the existence of multiple solutions to some fourth-order multiple-point boundary value problems; we also introduce some results to the existence, multiplicity and nonexistence of solutions to the fourth-order periodic boundary value problems.The thesis consists of 4 Chapters. In the first Chapter, by reviewing the theory of or-dinary differential equations and its development, we introduce the fourth-order boundary value problems with different boundary conditions, introduce the background of the prob-lems and outline the basic concepts and related theorems. We also introduce some basic definitions in nonlinear analysis and several important theorems including cone-stretching and cone-compressing fixed-point theorems, fixed-point index theorems of completely con-tinuous operators, Schauder fixed-point theorem and Arzela-Ascoli'lemma.In Chapter 2, we introduce the method of proving the existence and multiplicity of so-lutions to the fourth-order two-point boundary value problems with the use of fixed-point in-dex theories, upper and lower solutions methods and cone-stretching and cone-compressing fixed-point theorems. What we discuss includes both the case when the nonlinearity f de-pends on u" and the case when f does not depend on u".We first list some sufficient conditions to the multiplicity of solutions to the following boundary value problem with the use of fixed-point index theories. Under the following hypotheses:(H1)f:[0,1]×[0,∞)→[0,∞) is continuous;(H3) Set A= (∫01G(τ(s), s)ds)-1. There exists a p> 0 such that 0< f(x, y)< Ap for all x∈[0,1], y∈[0,p];(H5) Denote such that f(x, y)> Bp for all x∈[1/4,3/4],y∈[1/24 p,p].Theorem 1 Assume that f(x,y) satisfies (H1), (H2), (H3), then Problem (1) has at least two positive solutions u1, u2 satisfying 0< u1< p< u2.Theorem 2 Assume that f(x,y) satisfies (H1), (H4), (H5), then Problem (1) admits at least two positive solutions-u1, u2 satisfying 0 0 such thatTheorem 3 Assume that (H1')-(H4') hold and that r(L)> 1, then Problem (1) has at least two positive solutions.When f depends on u", one can get the existence result of the following problem with the use of the upper and lower solutions method. The main results are the following,Theorem 4 If there exist an upper solution a and a lower solutionβof Problem (2) such thatβ≤α,β"≥α", and i f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥0,β(t)≤u1≤u2≤a(t), v∈R, t∈[0,1], f(t, u, v2)-f(t, u,v1)≤0,α"(t)≤v1≤v2≤β"(t), u∈R,t∈[0,1].Then there exist two monotone sequences{αn}n=0∞and {β0}n=0∞, nonincreasing and nondecreas-ing, respectively, withα0=α,β0=β, which converge uniformly to the extremal solution to Problem(2) in[β,α].Theorem 5 If there exist an upper solution a and a lower solutionβof Problem (2) which satisfyβ≤α,β"+r(α-β)≥α", and if f:[0,1]×R×R→R is continuous and satisfies f(t, u2, v)-f(t, u1,v)≥-b(u2-u1), for allβ(t)≤u1≤u2≤α(t), v∈R, t∈[0,1]; f(t, u, v2)-f(t, u, v1)≤a(v2-v1), for v2+r(α-β)≥v1, a"+r(α-β)≤v1,v2≤β"+r(α-β), u∈R, t∈[0,1], where a,b≥0, a2-4b≥0, r1,2= (a±(?))/2. Then there exist two monotone sequences {an}a=0∞and {βn}n=0∞, nonincreasing and nondecreasing, respectively, with a0=α,β0=β, which converge uniformly to the extremal solution to Problem (2) in [β,α].In Chapter 3, we use the upper and lower solutions method and cone-stretching and cone-compressing fixed-point theorems to introduce the existence of solutions to the multi-point boundary value problem of fourth-order equations. The problems under consideration include the case f= f(t, u), the case f= f(t, u, u") and the case f= f(t, u, u'u", u'"). We consider the following four-point boundary value problem: where a, b, c, d≥0 and 0≤ξ1≤ξ2≤1.The main conclusion is the following theorem.Theorem 6 If the following conditions hold:(B1) a,b, c, d≥0,0≤ξ1≤ξ2≤1 and b-aξ1≥0, d-c+cξ2≥0,δ= ad+be+ ac(ξ2-ξ1)≠0;(B2) f(t,u)∈C([0,1] x [0,∞),R+) is nondecreasing relative to u, f(t,t(1-t))(?) 0 for t∈(ξ1,ξ2) and there exists a constant 0<μ< 1 such that kμf(t, u)≤f(t, ku) for any 0≤k≤1. Then Problem (3) has at least one positive solution.When the nonlinearity f= f(t, u, u"), denoteWith the use of cone-stretching and cone-compressing fixed-point theorems one can prove the following theorem.Theorem 7 Assume that f satisfies:(C1)f∈C([0,1] x [0,∞) x (-∞,0], [0,∞));(C2)f is sublinear, namely min f0=+∞, max f∞= 0.Then Problem (3) has at least one positive solution.Theorem 8 Assume (C1) and(C3) f is superliner, namely max f0= 0, min f∞=+∞.Then Problem (3) has at least one positive solution.One can prove the existence of multiple solutions to Problem (3) by modifying the above conditions.Assume that(C4) min f0= min f∞,=+∞;(C5) There exists an l1> 0 such that for all t∈[0,1], x∈[0,l1],-y∈[0,l1];(C6) max f0= max f∞,= 0;(C7) There exists an l2> 0 such that for any t∈[0,1], x∈[0,l2],-y∈[l2/4,l2].Theorem 9 Assume (C1), (C4) and (C5). Then Problem (3) admits at least two posi-tive solutions u1, u2 satisfying 0<||u1||2< l1< ||u2||2.Theorem 10 Assume (C1), (C6) and (C7). Then Problem (3) admits at least two positive solutions u1, u2 satisfying 0<||u1||2< l2< ||u2||2.As for the following problem where a, b, c, d≥0,ρ= ad+be+ac> 0,f:[0,1]×R4→R is continuous. One can get the existence and uniqueness of solutions with the use of the upper and lower solutions method and Schauder fixed point theorem.Theorem 11 Suppose that v,w are upper and lower solutions to Problem (4) such that w"(t)≥v"(t) and f satisfies the Nagumo condition with respect to w" and v". Then Problem (4) has at least one solution u such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].Theorem 12 Suppose that v, w are upper and lower solutions to Problem (4) and f satisfies the Nagumo condition with respect to w" and v". If f(t,x1,x2, x3, x4) is decreasing in x1,x2 and strictly increasing in x4, then Problem (4) has a unique solution u(t) such that w(t)≤u(t)≤v(t), w'(t)≤u'(t)≤v'(t), v"(t)≤u"(t)≤w"(t), t∈[0,1].In Chapter 4, we study the existence, multiplicity and nonexistence of solutions to the following periodic boundary value problemAssume that(D1) For any givenβ,α∈C[0,2π] withβ(t)<α(t), t∈[0,2π], there exist 0< A< B such that A(v2-v1)< f(t, u, v2)-(t, u,v1)< B(v2-v1) for a.e. t∈[0,2π] wheneverβ(t)< u< a(t), v1,v2∈R, v1 0 such that b2≥4a and f(t,u2,v2)-f(t,u1,v1)≥-a(u2-u1)+b(v2-v1); for a. e. t∈[0,2π] whenever B(t)≤u1≤u2≤α(t), v1, v2∈R, v1< v2.Then Problem (5) has a solution u∈W4,1 [0,2π] satisfyingβ(t)≤u(t)≤α(t). Theorem 15 Suppose that there exist a lower solutionβ(t)and an upper solutionα(t) such thatβ(t)≤α(t),t∈[0,2π],and f(t,u,v)satisfies the hypothesis(D4)There exist C,D>0 such that D<4C+1/4,D2>4C and f(t,u2,V1)-f(t,u1,v2)≥-C(u2-u1)-D(v1-v2) for a.e.t∈[0,2π]wheneverβ(t)≤u1≤u2≤α(t),v1,v2∈R,v1≤v2. Then Problem(5)has a solution u∈W4,1[0,2π]satisfyingβ(t)≤u(t)≤α(t).At the end of the thesis,we outline the results in studying the existence,multiplicity and nonexistence of solutions to the following problem under suitable conditions to f where f:[0,1]×R+→R+is continuous,a,b∈R such that 00, a/π4+b/π2+1>0.Denote (?)=(?) (?)=(?)Theorem 16 Assume that one of the following two conditions holds: (E1)f0a；(E2)f0>a,f∞a,then Problem(6)has at least one positive solution satisfying min{c,d}<||u||< max{c,d}.Hereφ(l)=max{f(t,c)/c:t∈[0,1],c∈[σ-l,l]},φ(l)=min{f(t,c)/c:t∈[0,1],c∈[σ-l,l]), Theorem 18 Suppose that there exist n+1 positive constants a1< a2<…a, l∈(0,+∞).Then Problem (6) has no positive solution u∈K.