Existence And Uniqueness Of Solutions For Singular Fourth-order Boundary Value Problems With Two Parameters

Singular boundary value problems for nonlinear ordinary differential equations,whicharise in a variety of areas such as mechanics,boundary layer theory,diffusion and reactionequations,biology,edt,become an impotent topic in ordinary equations fields.The paper present existence and uniqueness of solutions for singular fourth-orderboundary value problems(BVP) with two parameters.where f:[0,1]×(0,∞)→(0,∞)is continue,α,β∈R andβ<2π~2,α≥-β~2/4,α/π~4+β/π~2<1．This paper is composed of three parts.In the first chapter,we introduce the historicalbackground of the problems which will be investigated and the main results of this paper.In the second chapter,we present some lemmas and defines,which will be used in chapterthree.In the third chapter,by employing fixed point index in cones,in§3.1 it was provedthe existence of positive solutions for the singular boundary value problems;by employingmixed monotone operator,in§3.1 it was proved the uniqueness of positive solutions forthe singular boundary value problems.Owing to its importance in physics about elastic beam,the existence of solutions tothis nonsingular problem has been studied by many authors;see for example[1-17].In[14],the author used the fixed point index theorem to establish the existence ofpositive solutions,when f is nonsingular at u=0 and nonresonant;when f is singular atu=0 and nonresonant,in[15],the authors presented the existence of positive solutions ofsingular four-order boundary value problem by using fixed point index in cones;in[16],theauthors presented the uniqueness of positive solutions of singular four-order boundaryvalue problem with one parameter by using Mixed monotone operator.Recently,the singular boundary value problems have been studied extensively.However,there are only few works on existence and uniqueness of solutions for singular fourthorder boundary value problems with two parameters.As far as the author knows,theworks on existence and uniqueness of solutions about problem (BVP[1]-[2]) are quite rarelyseen. Motivated by the results mentioned above,In this paper we present some new existence and uniqueness results for(BVP[1]-[2]).