Upper and lower pointwise bounds are presented for the solution of certain nonlinear Sturm-Liouville problems. Our method requires an inclusion domain for the solution and an integral equation formulation in terms of a nonlinear Hammerstein equation with associated contraction operator. Several different sets of bounds are discussed, including a simple first order set and two new second order sets. The Barnsley-Robinson embedding procedure is used as a basis for the second order bounds. The relative merits and efficiency of the various bounding results are examined in calculations for a problem arising in the theory of electric current flow in excitable cells, in which estimates are obtained for the current and the potential at a point.
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