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

Diego Averna¹ ; Gabriele Bonanno¹

¹

Annales Polonici Mathematici, Tome 71 (1999) no. 3, pp. 253-271

Résumé

Let $F:[a,b] × ℝ^n × ℝ^n → 2^{ℝ^n}$ 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 ($P_F$)    {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 ($P_F$) with another one ($P_G$), with a suitable convex right-hand side G, such that every generalized solution of ($P_G$) is also a generalized solution of ($P_F$) (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).

Détail

DOI : 10.4064/ap-71-3-253-271

Keywords: multivalued differential inclusions, boundary value problems, non-convex and unbounded right-hand side, directional continuous selections, implicit equations