Solutions For Elastic Beam Equation Boundary Value Problems

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. In this paper, by using fixed point theorem and lower and upper method,we get the existence of solutions for several classes of elastic beam equation boundary value problems. The dissertation contains four chapters:In chapter l,we investigates a class of four order nonlinear singular boundary value problems where f∈C((0,1) x (0,+∞) x (0,+∞) x (-∞,0),[0,+∞)), and nonlinear term f may be singular in t=0,1and u=0. By using Schauder’s fixed point theorem and lower and upper method and structing lower and upper solutions on equation, we get a necessary and sufficient codition for the existence of positive solutions for the above boundary value problems.In chapter2, we consider elastic beam equation with nonlinear boundary value condition of the boundary value problem where f∈C([0,1]×[0,+∞)×[0,+∞)×(-∞,0],[0,+∞))and g∈C(R).By using general comcave operator fixed point theorem,at least one monotone positive solutions are obtained.In chapter3,we study a class of elastic beam equation boundary value prob-where.f∈C([0,1]×R+â†'R1.By using fixed point theorem and the partial order method,the first is the existence of a λ*，and there is a unique positive solution when λ∈(0,λ*).Equation has no positive solution,and the existence of solution of the variable number is also discussed when λ>λ*.It also discussed the existence of multiple positive solutions when λ=1.In chapter4,we study a class of singular boundary value problems with p-Lapacian operatorwhereφ(t)=(|t|p-2t),p>1,0<α,η<1is a constant,α,β,γ,δ≥0,α2+β2≠0,γ2+δ2≠0,f∈C[(0,1)×(0,+∞),[0,+∞)],may be singular in the t=1,t=0,oru=0.By using the theory of fixed point theorem,we give equation at least two positive solution existence condition,improve and generalize some known results.