Eigenvalue problem for p-Laplacian three-point boundary value problems on time scales

Let TT be a time scale such that 0,T∈T0,T∈T, β,γ⩾0β,γ⩾0 and 0<η<ρ(T)0<η<ρ(T). We consider the following p-Laplacian three-point boundary problem on time scales(φp(uΔ(t)))∇+λh(t)f(u(t))=0,t∈(0,T),u(0)−βuΔ(0)=γuΔ(η),uΔ(T)=0, where p>1p>1, λ>0λ>0, h∈Cld((0,T),[0,∞))h∈Cld((0,T),[0,∞)) and f∈C([0,∞),(0,∞))f∈C([0,∞),(0,∞)). Some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. In doing so the usual restriction that f0=limu→0+f(u)φp(u) and f∞=limu→∞f(u)φp(u) exist is removed. An example is also given to illustrate the main results.