Conical Coercivity Conditions and Global Minimization on Cones. An Existence Result

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 7)


We introduce in this paper some conical coercivity conditions, which are applied to the study of the global minimum on a convex cone in an infinite dimensional Banach space


Banach Space Variational Inequality Complementarity Problem Convex Cone Topological Vector Space 
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  1. 1.
    Ang D.D., Schmitt K. and Vy L. K. (1990), Noncoercive variational inequalities: some applications,Nonlin. Anal Theory, Meth. Appl.15 Nr. 6 497–512.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Auchmuly G . (1989), Duality algorithms for nonconvex variational principles,Num. Fund Anal Opt.,10 Nr. (3&4) 211–264.CrossRefGoogle Scholar
  3. 3.
    Bahya A. O . (1989) Ensembles côniquement bornés et cônes nucléaires dans les espaces localement convexe séparés,Thèse 3-ème cycle, Ecole Normale Supérieure, Takaddoum, Rabat, Maroc Google Scholar
  4. 4.
    Baiocchi C., Gastaldi F. et Tomarelli F. (1984) Inéquations variationnelles non-coercives,C. R. Acad. Se. Paris, T. 299, Sériel, Nr. 14, 647–650.MathSciNetGoogle Scholar
  5. 5.
    Barbu V. and Seidman T. I. (1987) Existence for minimization in Banach space with some applications,J. Math. Anal Appl 121, Nr. 1, 96–108.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berestycki H. and Lions P. L. (1983) Nonlinear scalar field equations, (I) Existence of a ground state,Archive Rat. Mech. Anal, 82, 313–345.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Berestycki H. and Lions P. L. (1983) Nonlinear scalar field equations, (II) Existence of infinity many solutions,Archive Rat. Mech. Anal, 82, 348–375.Google Scholar
  8. 8.
    Blot J . (1991) Calcul des variations pour les trajectoires presque-périodiques,Thèse (Docteur es Sciences) Université Paris DC, Dauphine Google Scholar
  9. 9.
    Bourgin R. D . (1973) Conically bounded sets in Banach spaces,Pacific J. Math.44, Nr. 2, 411–419.MathSciNetGoogle Scholar
  10. 10.
    Brezis H and Browder F. E. (1975) Maximal monotone operators in Nonrefiexive Banach spaces and nonlinear integral equations of Hammerstein type,Bull Amer. Math. Soc., 31 Nr. l, 82–88.Google Scholar
  11. 11.
    Browder F. E. (1964) Nonlinear eigenvalue problems and Galerkin approximations,Bull Amer. Math. Soc.,74, 651–656.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Browder F. E . (1970) Existence theorems for nonlinear partial differential equations,Proc. Sympos. Pure Math., Nr. 16AMS, Providence R. I. 1–60.Google Scholar
  13. 13.
    Browder F. E . (1973) Existence theory for boundary value problems for quasilinear elliptic systems with strongly nonlinear lower order terms,Proc. Sympos. Pure Math., Nr. 23 AMS, Providence R. I., 269–286.Google Scholar
  14. 14.
    Browder F. E. (1983) Fixed point theory and nonlinear problems,Bull. Amer. Math. Soc. 1 1–39.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caligaris O. and Oliva P. (1993) Compactness of level sets for integral functionate in infinite dimensional spaces,Boll. UMI (7), 7-B, 399–411.Google Scholar
  16. 16.
    Cimetière A . (1980) Un problème de fiambement unilateral en théorie des plaques,J. Mécanique, 19, Nr. 1, 183–203.Google Scholar
  17. 17.
    Cioranescu I . (1990) Geometry of Banach spaces, duality mappings and nonlinear problems,Kluwer Academic Publishers.zbMATHCrossRefGoogle Scholar
  18. 18.
    Costa D. G. and Gonçalves J. V. (1990) A. Critical point theory for nondifferentiable functional and applications,J. Math. Anal. Appt. 153, 470–485.zbMATHCrossRefGoogle Scholar
  19. 19.
    Costa D. G. and E Silva E. A. B. (1991) The Palais-Smale condition versus cœrcivity,Nonlinear Anal. Theory Meth. (amp)Appl., 16 Nr. 4, 371–381.Google Scholar
  20. 20.
    Dedieu J. P. (1979) Cônes asymptotes d’ensembles. Application à l’optimization.Bull. Soc. Math. France, Mémoire 60, 31–44.MathSciNetGoogle Scholar
  21. 21.
    Dietrich H. (1990) An application of the duality theory of Toland to a problem of optimal control,J. Math. Anal. Appl. 153, 301–308.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ekeland I. (1979) Nonconvex minimization problems,Bull. Amer. Math. Soc. 1, 443–474.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Goeleven D . (1993) Inequations variationnelles non-coercives,These, Université de Limoges, (France).Google Scholar
  24. 24.
    Goossens P. (1984) Asymptotically compact sets, asymptotic cone and closed conical hull,Bull. Soc. Royale Sciences, Liège, 53e-année, Nr. 1, 57–67.MathSciNetGoogle Scholar
  25. 25.
    Gwinner J . (1989) Convergence and error analysis for variational inequalities and unilateral boundary value problems,Habilitationsschrift TDH-Preprint 1257, Tehchnische Hochschule Darmstadt Google Scholar
  26. 26.
    Hess P. (1972) On nonlinear mappings of monotone type homotopic to odd operators,J. Funct. Anal. 11, 138–167.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Hogbe-Nlend H . (1971) Théorie de homologies et applications,Lecture Notes in Math. Springer- Verlag, 213.Google Scholar
  28. 28.
    Hu S. (1949) Boundedness in a topological space,J. Math. Pures Appl. 28, 287–320.MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ionescu I. R. and Rosea I. (1990) Fridrichs extensions for nonconvex variational problems,Nonlin. Anal. Theory, Meth. Appl. 14 Nr. 11, 905–914.CrossRefGoogle Scholar
  30. 30.
    Isac G. (1985) Nonlinear complementarity problem and Galerkin method,J. Math. Anal. Appl. 108, Nr.2, 563–574.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Isac G. (1989) Conically bounded sets and optimization,Arch. Math. 53, 288–299.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Isac G. On an Altman type fixed point theorem on convex cones,Rocky Mountain J. Math. (In printing).Google Scholar
  33. 33.
    Isac G . (1992) Complementarity problems,Lecture Notes in Math. Nr.1528, Springer-VerlagGoogle Scholar
  34. 34.
    Isac G. and Théra M. (1988) Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate,J. Optim. theory Appl. 58 Nr. 2, 241–257.CrossRefGoogle Scholar
  35. 35.
    Isac G. and Gowda M. S. (1993), Operators of class (S)+,Altaian’s condition and the complementarity problem,J. Fac. Sci. Univ. Tokyo Sec. IA, 40 Nr. 1, 1–16.MathSciNetGoogle Scholar
  36. 36.
    Klee V. L. Jr. (1956) Iteration of the lin operation for convex sets,Math. Scan. 4, 231–238.MathSciNetzbMATHGoogle Scholar
  37. 37.
    Krasnoselskii M. A. and Zabreiko P.P. (1984). Geometrical methods of nonlinear analysis,Berlin- Heidelberg-New York Google Scholar
  38. 38.
    Lezanski T. (1980), Sur le minimum de fonctionnelles dans les espaces de Banach,Studia Math.68, 49–66.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mel’nik V. S. (1992) On noncoercive extremal problems in Banach space,Soviet Math. Dokl., 44 Nr. 3, 824–829.MathSciNetGoogle Scholar
  40. 40.
    More J. J . (1990) A collection of nonlinear model problems,Lectures in Applied Math. Vol. 26 AMS, 723–762.Google Scholar
  41. 41.
    Nikodym O. M . (1953), (1954) On transfinite iterations of the weak linear closure of convex sets in linear spaces,Part A & BRend. Circ. Mat. Palermo (2),285–105; 35–.Google Scholar
  42. 42.
    Nirenberg L. Variational methods in nonlinear problems,In:Topics in Calculus of Variations (Ed. M.Giaquinta), Lecture Notes in Math. Nr. 1365, Springer-Verlag.Google Scholar
  43. 43.
    Petryshyn W. V. (1971) Antipodes theorem for A-proper mappings and its applications to mappings of the modified type (S) or (S)+ and to mappings with the pm property,J. Funct. Anal. 7, 165–221.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Seidman T. I. (1983) Existence and regularity of extrema,J. Math. Anal. Appl. 94 Nr.2, 470–478.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Seidman T. I. and Wolf P. (1988) Equilibrium states of an elastic conducting rod in a magnetic field,Arch. Rat. Mech. Anal. 102, Nr. 4, 307–329.Google Scholar
  46. 46.
    Stampacchia G. and Lions J. L. (1967) Variational inequalities,Comm. Appl. Math. 20, 493–519.MathSciNetzbMATHGoogle Scholar
  47. 47.
    Szulkin A. (1990) A relative category and applications to critical point theory for strongly indefinite functional,Nonlin. Anal. Theory, Meth., Appl. 15 Nr. 8, 725–739.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Vahlberg M. M . (1964) Variational methods for the study of nonlinear operators,Holden-Day Inc. San Francisco, London, Amsterdam.Google Scholar
  49. 49.
    Weber H . (1984) φ-asymptotisches spectrum und suijektivitätssätze vom Fredholm type für nichtlineare Operatoren mit anwendungen,Math. Nachr.117,1–35.Google Scholar
  50. 50.
    Wysocki K . (1990) Critical points in the presence of order structure,In:Nonlinear Functional Analysis (Ed P.S. Milojevic’) Lecture Notes in Pure and Appl. Math. Vol. 121AMS.Google Scholar
  51. 51.
    Zalinescu C . (1989) Stability for a class of nonlinear optimization problems and applications,In: Nonsmooth optimization and Related Topics (Eds. F. H. Clark, V. F. Dem ’ynov and F. Giannessi), 437–458.Google Scholar
  52. 52.
    Zelati V. C., Ekeland I. Nd Séré E. (1990) A variational approach to homoclinic orbits in Hamiltonian Systems,Math. Annalen, 288 nr. 1, 133–160.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • G. Isac
    • 1
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada

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