The Sign-changing Solutions For Nonlinear Third Order Differential Equation Boundary Value Problems

In this thesis, the existence and multiplicity of sign-changing solutions forthe nonlinear third-order differential equation and system are discussed by usingtopological degree method of nonlinear functional analysis. The new results areobtained.This thesis is composed of two chapters.In chapter I, at first, a new fixed point theorem is presented by usingLeray-Schauder topological degree theorem, namelyTheorem 1.1.1 Let A:E→E be completely continuous, and u₀∈E. Su-ppose that there exists a constant r_u0＞0 such that S={u∈E|u-u₀=λ(Au-u₀),0＜λ＜1}(?).Then A has at least one fixed point in the closed ball B(?).Theorem 1.1. 1 is a generalization of Leray-Schauder fixed point theorem.Making use of theorem 1.1.1 and schauder fixed point theorem, we obtain twosign-changing fixed point theorems as follow.Theorem 1.2.1 Let A:E→E be completely continuous, and u₀∈E. Ifd=dist(u₀,-P∪P)＞0 and there exists a constant r_u0∈(0,d) such that {u∈E|u-u₀=λ(Au-u₀),0＜λ＜1}(?).Then A has at least one sign-changing fixed point in E. Theorem 1.2.2 Let A:E→E be completely continuous, and u₀∈E. Ifd=dist(u₀,-P∪P)＞0 and there exists a constant r_u0∈(0,d) such thatAu∈(?) for u∈(?). Then A has at least one sign-changing fixedpoint in E.Secondly, the existence of sign-changing solution for third-order equationboundary value problemsis considered by using theorem 1.2.1 and theorem 1.2.2, where f:[t₁,t₃]×R¹Ã—R¹Ã—R¹â†’R¹ is continuous, -∞＜t₁＜t₂＜t₃＜+∞.Suppose that (H₁) f(t,x₀,x₁,x₂)=g(t, X₀,X₁,x₂)+a(t)ï¼›(H₂) f(t,x₀,x₁,X₂)=b(t)h(t,x₀,x₁,X₂).We obtained the results as follows:Theorem 1.3.1 Suppose that f satisfy, the condition (H₁) and the follo-wing requirements are satisfied:(1) there exists a constantμsuch that d_μ=dist(u_μ,-P∪P)＞0;(2) there exist nonnegative functionsγ,δ₀,δ₁,δ₂∈C[t₁, t₃] such that |g(t,x₀,x₁,x₂)-μ|≤γ(t)+sum from i=0 to 2δ₁(t)|x₁|, for all t∈[t₁,t₃], x₀,x₁,x₂∈R¹ and 0＜(‖γ‖₀+sum from i=0 to 2‖δ₁‖₀‖u_μ¹â€–₀)c/1-sum from i=0 to 2 c₁‖δ₁‖₀＜d_μ. Then the problem (1.3.1) has at least one sign-changing solution in C³[t₁,t₃].Theorem 1.3.2 Suppose that f satisfy the condition (H₂) and the follo-wing requirements are satisfied: (1) there exists a constantμ≠0 such that d_μ=dist(ν_μ, -P∪P)＞0;(2) there exist nonnegative functionsγ,δ₀,δ₁,δ₂∈C[t₁,t₃] such that |h(t, x₀,x₁,x₂)-μ|≤1/‖b‖₀(γ(t)+sum from t=0 to 2δ₁(t)|x₁|),for all t∈[t₁,t₃],x₀,x₁,x₂∈R¹ and 0＜(‖γ‖₀+sum from t=0 to 2‖δ_i‖₀‖ν_μ^(t)‖₀)c/1-sum from t=0 to 2 c₁‖δ₁‖₀＜d_μ.Then the problem (1.3.1) has at least one sign-changing solution in C³[t₁,t₃].Theorem 1.3.3 Suppose that f satisfy the condition (H,) and the following requirements are satisfied:(1) there exists a constantμsuch that d_μ=dist(u_μ,-P∪P)＞0;(2) there exists constant r_μ∈(0,d_μ) such thatμ-r_μ/c≤g(t, x₀,x₁,x₂)≤μ+r_μ/c,for all t∈[t₁,t₃],|x₁|≤r_μ+‖u_μ‖(i=0,1,2). Then the problem (1.3.1) has at leastone sign-changing solution in C³[t₁,t₃].Theorem 1.3.4 Suppose that f satisfy the condition (H₂) and the following requirements are satisfied:(1) there exists a constantμ≠0 such that d_μ=dist(ν_μ,-P∪P)＞0;(2) there exists constant r_μ∈(0,d_μ) such thatμ-r_μ/c‖b‖₀≤h(t,x₀,x₁,x₂)≤μ+r_μ/c‖b‖₀,for all t∈[t₁,t₃],|x₁|≤r_μ+‖u_μ‖(i=0,1,2). Then the problem (1.3.1) has at leastone sign-changing solution in C³[t₁,t₃].Finally, the existence of sign-changing solution for third-order system bou- ndary value problemsis considered by using theorem 1.2.1 and theorem 1.2.2, where f,g:R¹Ã—R¹â†’R¹ are continuous, a,b:[0,1]→R¹ are continuous. We obtained the results asfollows:Theorem 1.4.1 Suppose that the following conditions are satisfied:(1) there exists a constantμ≠0 such that d_μ=dist((u_μ,ν_μ),-P∪P)＞0;(2) there exist nonnegative constantsγ,δ₀,δ₁ such that |f(x₀,x₁)-μ|≤1/‖a‖₀(γ+δ₀|x₀|+δ₁|x₁|), |g(x₀,x₁)-μ|≤1/‖b‖₀(γ+δ₀|x₀|+δ₁|x₁|),and 0＜‖γ‖₀+‖δ₀‖₀‖u_μ‖₀+‖δ₁‖₀‖ν_μ‖₀/12-‖δ₀‖₀-‖δ₁‖₀＜d_μ,for x₀,x₁∈R¹. Then the problem (1.4.1) has at least one sign-changing solutionin C³[0,1]×C³[0,1].Theorem 1.4.2 Suppose that the following conditions are satisfied:(1) there exists a constantμ≠0 such that d_μ=dist((u_μ,ν_μ),-P∪P)＞0;(2) there exists constant r_μ∈(0,d_μ) such thatμ-12r_μ/‖a‖₀≤f(x₀,x₁)≤μ+12r_μ/‖a‖₀,μ-12r_μ/‖b‖₀≤g(x₀,x₁)≤μ+12r_μ/‖b‖₀,for |x₀|≤r_μ+‖u_μ‖,|x₁|≤r_μ+‖ν_μ‖. Then the problem (1.4.1) has at least one sign-changing solution in C³[0,1]×C³[0,1].In chapterâ…¡, at first, the multiplicity of sign-changing solutions forthree-order equation boundary value problemsis considered by using topological degree and the fixed-point index theory ofcone, where f:R¹â†’R¹ is continuous.Assume that(H1) f(0)=0, yf(y)＞0 forall y∈R¹\{0};(H2) there exist positive integers n₀ and n₁ such thatλ_2n0＜β₀＜λ_2n0+1,λ_2n1＜β₁＜λ_2n1+1;(H3) there exists C₀＞0 such that |f(y)|＜4/3C₀, for all y with |y|≤C₀.We obtained the results as follows:Theorem 2.1.1 Suppose that (H1)-(H3) hold. Then boundary value problem(2.1.1) has at least two sign-changing solutions, two positive solutions and twonegative solutions.Corollary 2.1.1 Suppose that (H1)-(H3) hold, and f is odd function.Then boundary value problem (2.1.1) has at least four sign-changing solutions,two positive solutions and two negative solutions.Secondly, the multiplicity anti-symmetric sign-changing solutions forthree-order boundary value problemsis considered by using extending positive solution method and the fixed-pointindex theory, where f is continuous nonnegative and even function in R¹. Weobtained the results as follows: Theorem 2.2.1 Suppose that(1) f₀=f_∞=∞;(2) there existsρ＞0 such that f(y)＜216/3^1/2ρ, for 0≤y≤ρ.Then the problem (2.2.1) has at least two anti-symmetric sign-changing solutions in C³[0,1].Theorem 2.2.2 Suppose that(1) f₀=f_∞=0;(2) there existsρ＞0 and t₀∈(0,1/2) such that f(y)＞(integral from n=8 to 3 ⁸G(t₀,s)ds)^-1ρ,for 3/32ρ≤y≤ρ. Then the problem (2.2.1) has at least two anti-symmetricsign-changing solutions in C³[0,1].