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1999 | 71 | 3 | 253-271
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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).
Rocznik
Tom
71
Numer
3
Strony
253-271
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-02-04
poprawiono
1998-06-08
Twórcy
autor
  • 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
Bibliografia
  • [1] J. Appell, E. De Pascale, H. T. Nguyê n and P. P. Zabreĭko, Multi-valued superpositions, Dissertationes Math. 345 (1995).
  • [2] Z. Artstein and K. Prikry, Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl. 127 (1987), 540-547.
  • [3] D. Averna, Lusin type theorems for multifunctions, Scorza Dragoni's property and Carathéodory selections, Boll. Un. Mat. Ital. (7) 8-A (1994), 193-202.
  • [4] G. Bonanno, Two theorems on the Scorza Dragoni property for multifunctions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 83 (1989), 51-56.
  • [5] G. Bonanno, Differential inclusions with nonconvex right hand side and applications to implicit integral and differential equations, Rend. Accad. Naz. Sci. (detta dei XL) 20 (1996), 193-203.
  • [6] A. Bressan, Upper and lower semicontinuous differential inclusions: A unified approach, in: Controllability and Optimal Control, H. Sussmann (ed.), Dekker, New York, 1989, 21-31.
  • [7] C. Castaing, A propos de l'existence des sections séparément mesurables et séparément continues d'une multiapplication séparément mesurable et séparément semi-continue inférieurement, Sém. Analyse Convexe, Montpellier 1976, Exp. no. 6.
  • [8] F. S. De Blasi and G. Pianigiani, Solution sets of boundary value problems for nonconvex differential inclusions, Topol. Methods Nonlinear Anal. 1 (1993), 303-313.
  • [9] K. Deimling, Multivalued Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. 1, de Gruyter, Berlin, 1992.
  • [10] C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.
  • [11] J. B. Hiriart-Urruty, Images of connected sets by semicontinuous multifunctions, J. Math. Anal. Appl. 111 (1985), 407-422.
  • [12] A. Kucia, Scorza Dragoni type theorems, Fund. Math. 138 (1991), 197-203.
  • [13] S. A. Marano, Existence theorems for a multivalued boundary value problem, Bull. Austral. Math. Soc. 45 (1992), 249-260.
  • [14] S. A. Marano, On a boundary value problem for the differential equation f(t,x,x',x'') = 0, J. Math. Anal. Appl. 182 (1994), 309-319.
  • [15] O. Naselli Ricceri and B. Ricceri, An existence theorem for inclusions of the type Ψ(u)(t) ∈ F(t,Φ(u)(t)) and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270.
  • [16] J. Oxtoby, Measure and Category, Springer, New York, 1971.
  • [17] B. Ricceri, Applications de théorèmes de semi-continuité inférieure, C. R. Acad. Sci. Paris Sér. I 295 (1982), 75-78.
  • [18] B. Ricceri, On multifunctions of one real variable, J. Math. Anal. Appl. 295 (1987), 225-236.
  • [19] B. Ricceri and A. Villani, Openness properties of continuous real functions on connected spaces, Rend. Mat. 2 (1982), 679-687.
Typ dokumentu
Bibliografia
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