Existence of solutions for a multivalued boundary value problem with non-convex and unbounded right-hand side

Let ${\displaystyle F:[a,b]\times {ℝ}^n\times {ℝ}^n\to 2^{{ℝ}^n}}$ be a multifunction with possibly non-convex and unbounded values. The main result of this paper (Theorem 1) asserts that, given the multivalued boundary value problem (${\displaystyle P_F}$)    {u'' ∈ F(t,u,u'), u(a) = u(b) = ϑ_{ℝ^n}, if an appropriate restriction of the multifunction F has non-empty and closed values and satisfies the lower Scorza Dragoni property and a weak integrable boundedness type condition, then we can substitute the problem (${\displaystyle P_F}$) with another one (${\displaystyle P_G}$), with a suitable convex right-hand side G, such that every generalized solution of (${\displaystyle P_G}$) is also a generalized solution of (${\displaystyle P_F}$) (see also Remark 1 and Corollary 1). As a consequence of our results, in conjunction with those in [13] and [18], some existence theorems for multivalued boundary value problems are then presented (see Theorem 2, Corollary 2 and Theorem 3). Finally, some applications are given to the existence of generalized solutions for two implicit boundary value problems (Theorems 4-6).