Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

Abstract

. We show that the solution of the third order parameter dependant dynamic boundary value problem y ΔΔΔ = f (cid:2) t , y , y Δ , y ΔΔ , λ (cid:3) , y ( t 1 ) = y 1 , y ( t 2 ) = y 2 , y ( t 3 ) = y 3 on a general time scale may be (delta) differentiated with respect to y 1 , y 2 , y 3 , t 1 , t 2 , t 3 , and λ . We show that the (delta) derivative of the solution solves the third order boundary value problem consisting of either the variational equation (in the dense case), the dynamic analogue (in the scattered case), or a modi fi ed variational equation in the parameter case with interesting boundary conditions in all cases.