Positive solutions to semi-positone second-order three-point problems on time scales

Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scalesuΔ∇(t)+λf(t,u(t))=0,u(a)=0,αu(η)=u(T).Here t∈[a,T]Tt∈[a,T]T, where TT is a time scale, α>0,η∈(a,ρ(T))T,α(η-a)0λ>0 belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and q-difference equations.