Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

الملخص EN

Let T be a time scale and a,b∈T, a<ρ2(b).

We study the nonlinear fourth-order eigenvalue problem on T, uΔ4(t)=λh(t)f(u(t),uΔ2(t)), t∈[a,ρ2(b)]T, u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0 and obtain the existence and nonexistence of positive solutions when 0<λ≤λ* and λ>λ*, respectively, for some λ*.

The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.