The problem considered is of the form
x″ ( t ) + f ( x ( t ) ) sgn ( x′ ( t ) ) | x′( t ) | m + g ( t, x( t ) ) = s( t ),
for 0 < t < 2&pgr;, subject to the periodic boundary conditions x (0) = x ( 2&pgr; ), x′ (0) = x′ ( 2&pgr; ). The author uses results from cited literature to establish sufficient conditions for this boundary value problem to have at least three solutions. The first set of conditions includes 0 < m ≤ 2; g ( t, x ) → - ∞ as x → ∞ and g ( t, x ) → ∞ as x → - ∞, uniformly in t; and the existence of constants x1 and x2, with x1 < x2 and g ( t, x1 ) < g ( t, x2, for 0 ≤ t ≤ 2 &pgr;. The second set of conditions includes m ≤ 1; g ( t, x) → - ∞ as x → - ∞ and g ( t, x ) → ∞ as x → ∞, uniformly in t; and the existence of constants x1 and x2, with x1 < x2 and g ( t, x1 ) > g ( t, x2 ), for 0 ≤ t ≤ 2 &pgr;. Each theorem is illustrated by a simple example. To appreciate the significance of these results, the reader must be familiar with the cited literature.
