Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u″(t)+au′(t)+bu(t)=f(t,u(t))−u″(t)+au′(t)+bu(t)=f(t,u(t)) for all t∈[0,1]t∈[0,1] with u(0)=u(1)=0u(0)=u(1)=0, where a,b∈R1a,b∈R1, f∈C1([0,1]×R1,R1)f∈C1([0,1]×R1,R1). By using the Morse theory, we impose certain conditions on ff which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.