Elastic Beam Equation Boundary Value Problems

The main purpose of this dissertation is to study the eigenvalue problems for two kinds of equations: simply supported beam equation and cantilever beam equation.Owing to the importance in both theory and in application, boundary value problems for ordinary differential equations have been attracted many researchers, and a large number of results have been obtained. In recent years, many works related to beam equations appeared in literature, most of them focused on the existence, multiplicity of solutions (or positive solutions). Some efficient tools such as topological degree theory, fixed point theory and lower and upper method have been applied.For the BVP (1), the case η = ζ = 0, λ = 1 and the case ( ζ= 0, λ = 1 have been considered, but for the case η, ζ ≠ 0, λ ≠ 1, the existence of solutions for (1) is unknown. When η, ζ≠ 0,λ = 1, Li [45] discussed the existence of positive solutions for (1) under some conditions, which shows that (η, ζ) is only in the small part of the (η, ζ) plane. In the other part of plane, the problems remain open. Under the nonresonance conditions, the eigenvalue lines of the homogenous problem corresponding to (1) play a very important role, they intersect each other, which leads to some difficulties to solve the above mentioned problems. In this thesis, we shall overcome these obstacles.First, under the conditions η < 2π~2, ζ ≥-(η~2)/4, ζ/(π~4)+η/(π~2) < 1,λ > 0, by the fixed theorems and semiorder method, we consider the existence, multiplicity and uniqueness of solutions for (1). Under the conditions that f{t,u) is positive, nondecreasing in u with some growth condition, we obtain that there exists a λ~* > 0 such that (1) has at least two, one positive solutions for 0 < λ < λ~*, λ = λ~*, respectively, and has no positive solutions for λ > λ~*. When the growth condition is replaced by the concave condition, by means of the existence theorem of minimal solutions for the concave operators in normal cones, we obtain that there exists λ~* > 0 such that (1) has a unique positive solution for 0 < λ < λ~* and has no positive solutions for λ ≥ λ*. In addition, we discussed the solutions' dependence on the parameter A in both of the sections. Second, we studied the existence of solutions of (1) when (η, ζ) is in many possible different parts on the (η, ζ) plane. In the first section, under the conditions that ζ = - (η~2)/4, η < 2π~2, λ > 0, by the decomposition of operators from[14] and critical point theory, we obtain that (1) has at least one, two, three and infinitely many solutions when A is in different intervals. In second section, using the variational structure and critical point theory, we obtain that if 77, ^ < 0 which implies that (17, ￡) is on the left side of the line ^ -f ^ = 1, then (1) has only trivial solution for A G (—00, 0), and has infinitely many solutions for A G (0, +00);if (n, ￡) is on the right side of the line ^ + ^ = 1, then (1) has only trivial solution or has at least n* distinct pairs of solutions, which depends on its different position further. Third, under local condition for the nonlinear term, we obtain the sufficient condition for the existence of solutions (or positive solutions) for (1) using the Leray-Schauder fixed point theorem. By means of the asymptotic derivate and the fixed point theory for differentiable operators, we obtain the existence of positive and sign-changing solutions for (2), and further, we show that (1) has sign-changing solutions. Finally, we are concerned with the cantilever problem, and obtain the existence, nonexistence and multiplicity of the problem. Parts of our results have been published or accepted for publication by the following journals: Journal of Mathematical Analysis and Applications, Dynamic Systems and Applications.