We consider positive solutions of the Dirichlet problem u(x) + λf(u(x)) = 0 on (−1, 1), u(−1 ) =u(1) = 0. depending on a positive parameter λ. Each solution u(x) is an even function, and hence it is uniquely identified by α = u(0). We present a formul . . .

We present exact multiplicity results for the boundary value problems of the type (1.1) u’’+ λf(x, u) = 0 for − L < x < L, u(−L) = u(L) = 0, with the nonlinearity f behaving like a cubic polynomial in u.

We use bifurcation theory to show the existence of infinitely many solutions at the first eigenvalue for a class of Dirichlet problems in one dimension.

For very general two-point boundary value problems we show that any positive solution satisfies a certain integral relation. As a consequence we obtain some new uniqueness and multiplicity results.

We consider sign-changing solutions of the Dirichlet problem u +λ f(u )= 0, 0 < x < 1, u(0 )= u(1 )= 0, with n 0 interior roots. We give a necessary and sufficient condition that a turn occurs at the solution (λ,u(x)), depending only on the maximum va . . .

We present an algorithm for computing the direction of pitchfork bifurcation for two-point boundary value problems. The formula is rather involved, but its computational evaluation is quite feasible. As an application, we obtain a multiplicity result. . . .

We revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in 161. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approa . . .

For a class of two-point boundary value problems we prove exactness of an S-shaped bifurcation curve. Our result applies to a problem from combustion theory, which involves nonlinearities like for [formula].