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1999 | 71 | 3 | 253-271
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Existence of solutions for a multivalued boundary value problem with non-convex and unbounded right-hand side

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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).
Opis fizyczny
  • Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Facoltà di Ingegneria, Università di Reggio Calabria, Via Graziella (Feo di Vito), 89100 Reggio Calabria, Italy
  • Dipartimento di Matematica ed Applicazioni, Facoltà di Ingegneria, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy
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