A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative u ( i υ ) ( t ) = f ( t , u ( t ) , u ′ ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′′ ( 0 ) = u ′′ ( 1 ) = 0 , where f : [ 0 , 1 ] × [ 0 , ∞ ) × R → [ 0 , ∞ ) is continuous.