General Lidstone Problems: Multiplicity and Symmetry of Solutions

For the 2mth order Lidstone boundary value problem, y 2m t = f y t y t y 2i t y 2 m−1 t t∈ 0 1 y 2i 0 = y 2i 1 = 0 0 ≤ i ≤ m − 1 where −1 mf m → 0 ∞ is continuous, growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions. This generalizes earlier papers which have applied Avery’s generalization of the Leggett–Williams theorem to Lidstone problems.W e then prove the analogous result for difference equations.