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

This paper is an comprehensive survey of the recent results obtained from the study of the boundary value problems of 4th order ordinary differential equations. Classified by the boundary conditions, the author outlines the results for two point and/or three point boundary value problems. For the two point boundary value problem, the author mainly studies the equation for plastic beams with 2 ends fixed and supported and states the existence of positive solutions with nonsingular or singular boundary value problems. For the three point boundary value problem, the author mainly discusses the existence of positive solutions with linear or nonlinear boundary values. At the end, the author proposes some problems that should be further solved.The main results are the following:I. Two point boundary value problems1. Beams supporting at the two ends1) Nonsingular boundary value problemsThe following two point boundary value problem is considered first. We make the following assumptions: to the corresponding boundary value problem. H₅: There is a positive numberηsuch thatThen the following theorems hold.Theorem 1 For the case whenα,β=0 in (1), if one of the conditions(1),(2)is satisfied, the boundary value problem(1) has at least one positive solutions; forβ=0,if one of(3),(4)holds, the problem has at least one positive solution; for general cases, if(5)holds, the problems (1) has at least one positive solution and if contitions(6)(7) hold, the problem (1) has at least 2 positive solutions. (1) H₁ , H₂,f0′=0,f∞′=∞; (2) H₁ , H₂,f0′=∞,f∞′=0; (3) H₁ , H-2,H₃,f0′=0,f∞′=∞; (4) H₁ , H₂,H-3,f0′=∞,f∞′=0; (5) H₄,H5; (6) H₄ , H₅,A₁,A₂;The paper also summarized the case when f depends on y′, y′′. Under suitable conditions, the existence of positive solutions is guaranteed. Some results for the existence of multiple solutions are also discussed.2) Singular boundary value problems The following boundary value problem is studied.One assumes that: H₆ : f∈C(J×R₀⁺ ,R⁺),f(t,1)≡/0,t∈Jand there is constantsλ,μ,N,M, ( -∞<λ≤0≤μ<1,0λf(t,y), if 0≤c≤N. cλf(t ,y)≤f(t,y)≤c^μf(t,y), if c≥M.H₆^*: There are constantsλ,μ,N ,(1<λ≤μ,00⁺. H₇ : f∈C(J×R₀⁺ ,R⁺),f(t,1)≡/0, t∈J.One obtained the following result. Theorem 2 Let ( H 6* ),(H7) hold, and s (1 s)f(s,1)ds.Then the boundary value problem(2)has a C 2[0,1] solution. Asuume H 6 andα(μ?λ)/(1-?μ)<1, Then the singular boundary value problem(2)has a C 2 (I) positive solution if and only if the following inequalities hold: sfsssds . Let ( H 6* ),(H7) hold and f (t ,1)≤f(t,s),t,s∈J. Then the singular boundary value problem (2)has a C 2[0,1] positive solution if and only ifIn the paper, we also mentioned some other necessary and sufficient conditions for the existence of C³[0,1] positive solutions to the singular boundary value problem(2).2. Beams with fixed 2 ends1) Nonsingular boundary value problems In this part, we mainly study the existence of positive solutions of the following equation in 2 cases. The problem is We haveTheorem 3 Assume that f is continuous and satisfies somemonotonic conditions (we omit the details here). Then if one of the following conditions holds, problem(3)has at least one positive solution; furthermore, if(4)and(5)hold, (3)has at least2 positive solutions.(1) There are 2 positive numbersλ,ηsuch that Some authors also studied the existence of 3 positive solutions to some special boundary value problems.2) Singular boundary value problemsWe consider the following singular boundary value problems:We make the following assumptions:Theorem 4 Problem (4) has at least one positive solution if one of the following conditions hold.People also studied the problem when there is a parameterλin the nonlinear term. Some results of nonexistence, existence of positive solutions and the multiplicity of the solutions to the caseλ>0 are proven.II. Three point boundary value problems1. Linear boundary conditionsThere are not many results of 3 point boundary value problem to the 4th order equations. We find out some of the results according to the references we get. The problem is the following:The existence of solution is mentioned as Theorem 5 Assume that there exist 2 positive numbers a, b such thatφ( a )≤aA,φ(b)≥bB,then BVP (5) has at least one positive solution y *∈Kwith min{a ,b}≤y*≤max{a,b}, whereThe existence of n positive solutions are also obtained, which is :Theorem 6 If there exist n +1 positive numbers a1 < a2<2 ):The solution to the initial value problem of equation (6) may be extended to [ a ,c] or the maximal existence interval is unbounded and the following hold:( A₂ ):there is a supersolutionφ(t) and a lowersolutionψ(t) on [ a ,c]such that( A₃ ):g1(x,y,z,w) is continuous in R⁴ and nonincreasing in x, w, nondecreasing in y for fixed z , also(A₄ ):h1(x,y) is continuous in R 2, nondecreasing in y for fixed x, furthermore h1 (φ(b),φ′′(b))≤0≤h1 (ψ(b),ψ′′(b));(A₅ ):g₂(x,y,z,w) is continuous in R⁴, nonincreasing in x, y, nondecreasing in w for fixed z , andFurthermore, if we assume that ( A₁^* ):f(t,y,z,ω,η) is nonincreasing in y on [ a ,c] for fixed t ,Z,ω,η. (A₆ ):h₂(x,y)is continuous in R², nonincreasing in y for fixed x,and h₂ (φ′( b),φ′′(b))≤0≤h₂(ψ′(b),ψ′′(b)).Then we haveTheorem 8 Assume (Y 1 ),(Y2),(A1*),(A2)? (A6), then (6) has solution under the following boundary conditions:In the context, we also mentioned some other related results.Remark: We may notice from this paper that the study to 4th order differential equations is still very limited, especially, people are mainly studying the case when the function f depends only on the unknown function but not on the derivatives of the unknowns. Even in the case when this function depends on the derivatives, most of the results are still concentrated on the case when the function depends only the 2nd derivatives since in this case some equations may be changed to be a 2nd order equation so that the results of 2nd order equation may be applied. Hence, we have to find some new method and perform careful analysis in studying the general 4th order equations which need our further efforts.