Third-order Nonlocal Problems with Sign-changing Nonlinearity on Time Scales

We are concerned with the existence and form of positive solutions to a nonlinear third-order three-point nonlocal boundary-value problem on general time scales. Using Green’s functions, we prove the existence of at least one positive solution using the Guo-Krasnoselskii fixed point theorem. Due to the fact that the nonlinearity is allowed to change sign in our formulation, and the novelty of the boundary conditions, these results are new for discrete, continuous, quantum and arbitrary time scales. 1. statement of the problem We will develop an interval of λ values whereby a positive solution exists for the following nonlinear, third-order, three-point, nonlocal boundary value problem on arbitrary time scales (px∆∆)∇(t) = λf(t, x(t)), t ∈ [t1, t3]T, (1.1) αx(ρ(t1))− βx(ρ(t1)) = ∫ ξ2 ξ1 a(t)x(t)∇t, x(t2) = 0, (px)(t3) = ∫ η2 η1 b(t)(px∆∆)(t)∇t, (1.2) where: p is a left-dense continuous, real-valued function on T with p > 0; λ > 0 is a real scalar; (H1) the real scalars α, β > 0 and the three boundary points satisfy t1 < t2 < t3 ∈ T such that 0 < ∫ σ(t3)