Instability and existence results to a non-autonomous two point boundary value problem

We study the existence results and stability properties to the boundary value problem where λ > 0 is a parameter and f is a smooth function such that lim uniformly in x, f (x, −u)= − f (x, u) for x ∈ (−1, 1) and u > 0, and We use the method of sub-super solutions to establish our existence results and then we prove stability and instability of nontrivial nonnegative solutions under various choices of f.