An Evolutionary Property of the Bifurcation Curves for a Positone Problem with Cubic Nonlinearity

We study an evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity {u ''(x)+lambda f(u) = 0, -1 < x <1, u(-1) = u(1) = 0, f(u) = -epsilon u(3) + sigma u(2) + tau u + rho, where lambda > 0 is a bifurcation parameters, epsilon > 0 is an evolution parameter, and sigma, rho > 0, tau >= 0 are constants. In addition, we improve lower and upper bounds of the critical bifurcation values (epsilon) over tilde of the problem.