By using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.
Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family of continuous linear set-valued functions is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function is a solution of the problem , Φ(0,x) = x, for x ∈ K and t ≥ 0, where denotes the Hukuhara derivative of Φ(t,x) with respect to t and for x ∈ K.
We discuss the Hyers-Ulam stability of the nonlinear iterative equation . By constructing uniformly convergent sequence of functions we prove that this equation has a unique solution near its approximate solution.