ON THE EXISTENCE OF POSITIVE SOLUTIONS TO SECOND ORDER PERIODIC BVPS
Abstract The existence of positive solutions to second-order periodic B V P s - u ″ + M u = f ( t , u ) , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) a n d u ″ + M u = f ( t , u ) , u ( 0 ) = u ( 2 π ) ⋅ u ′ ( 0 ) = u ′ ( 2 π ) is proved by a simple application of a Fixed Point Theorem in cones due to Krasnoselskii.
We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported into the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given ::: threshold. ::: ::: We prove that it is Pi-2-p-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions.