Advertisement

Existence of solutions and existence of optimal solutions

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 979)

Keywords

Optimization Theory Fixed Point Theorem Existence Theorem Reflexive Banach Space Finite Dimensional Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. S. Angell, On controllability for nonlinear hereditary systems: a fixed point approach. Nonlinear Analysis 4, 1980, 529–546.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2] (a)
    H. Brezis, Equations et inequations nonlineaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier 18, 1968, 115–175.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [2] (b)
    , On some degenerate nonlinear parabolic equations. Nonlinear functional analysis. Proc. Symp. Pure Math., Amer. Math. Soc., 18, 1970, 28–38.MathSciNetCrossRefGoogle Scholar
  4. [2] (c)
    , Operateurs maximaux monotones, Amer. Elsevier, New York 1973.zbMATHGoogle Scholar
  5. [3]
    H. F. Bohnenblust and S. Karlin, On a theorem of Ville. Contributions to the Theory of Games (Kuhn and Tucker, eds.), Princeton Univ. Press 1950, vol. 1, pp. 155–160.MathSciNetzbMATHGoogle Scholar
  6. [4] (a)
    L. Cesari, Existence theorems for weak and usual solutions in Lagrange problems with unilateral constraints. Trans. Amer. Math. Soc. 124, 1966, 369–412, 413–429.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [4] (b)
    , Seminormality and upper semicontinuity in optimal control, Journ. Optimization Theory Appl. 6, 1970, 114–137.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [4] (c)
    , Closure, lower closure, and semicontinuity theorems in optimal control, SIAM J. Control 9, 1971, 287–315.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [4] (d)
    , Closure theorems for orientor fields and weak convergence. Archive Rat. Mech. Anal. 55, 1974, 332–356.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [4] (e)
    , Geometric and analytic views in existence theorems for optimal control in Banach spaces. I,II,III. Journ. Optimization Theory Appl. 14, 1974, 505–520; 15, 1975, 467–497; 19, 1976, 185–214.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [4] (f)
    , Functional analysis, nonlinear differential equations and the alternative method. Nonlinear Functional Analysis and Differential Equations. (L. Cesari, R. Kannan, J. D. Schuur, eds.), M. Dekker, New York 1976, pp. 1–197.Google Scholar
  12. [4] (g)
    , Existence theorems for optimal controls of the Mayer type. SIAM J. Control 6, 1968, 517–552.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [4] (h)
    Optimization Theory and Applications: Problems with Ordinary Differential Equations. Springer Verlag 1982.Google Scholar
  14. [5]
    L. Cesari and S. H. Hou, Existence of solutions and existence of optimal solutions. To appear.Google Scholar
  15. [6] (a)
    L. Cesari and R. Kannan, Functional analysis and nonlinear differential equations, Bull. Am. Math. Soc. 79, 1973, 1216–1219.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [6] (b)
    , An abstract existence theorem at resonance, Proc. Amer. Math. Soc. 63, 1977, 221–225.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [6] (c)
    An existence theorem for periodic solutions of nonlinear parabolic equations. Rend. Accad. Milano. To appear.Google Scholar
  18. [6] (d)
    Periodic solutions of nonlinear wave equations with damping. Rend. Circolo Mat. Palermo. To appear.Google Scholar
  19. [6] (e)
    Periodic solutions of nonlinear wave equations with nonlinear damping. Communications in partial differential equations. To appear.Google Scholar
  20. [6] (f)
    , Existence of solutions of nonlinear hyperbolic equations. Annali Scuola Norm. Sup. Pisa (4)6, 1979, 573–592; 7, 1980, p. 715.MathSciNetzbMATHGoogle Scholar
  21. [6] (g)
    Solutions of nonlinear hyperbolic equations at resonance. Nonlinear Analysis.Google Scholar
  22. [7] (a)
    L. Cesari and M. B. Suryanarayana Existence theorems for Pareto optimization. Multivalued and Banach space valued functions. Trans. Amer. Math. Soc. 244, 1978, 37–65.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [7] (b)
    An existence theorem for Pareto problems. Nonlinear Analysis, 2, 1978, 225–233.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [7] (c)
    Upper semicontinuity properties of set valued functions. Nonlinear Analysis, 4, 1980, 639–656.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [7] (d)
    On recent existence theorems in the theory of optimization. Journ. Optimization Theory Appl. 31, 1980, 397–415.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [8]
    J. P. Dauer, A controllability technique for nonlinear systems, J. Math. Analysis Appl. 37, 1972, 442–451.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [9]
    N. Dunford and J. T. Schwartz, Linear Operators I, Interscience 1958.Google Scholar
  28. [10]
    K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. U.S.A. 58, 1952, 121–126.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [11]
    I. Glicksberg, A further generalization of Kakutani fixed point theorem with application to Nash equilibrium points. Proc. Amer. Math. Soc. 3, 1952, 170–174.MathSciNetzbMATHGoogle Scholar
  30. [12]
    G. S. Goodman, The duality of convex functions and Cesari’s property (Q). J. Optimization Theory Appl. 19, 1976, 17–23.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [13]
    J. Henry, Étude de la controllabilité de certaines equations paraboliques non-lineaires, Proc. IFIP Working Conference on Distributed Parameter Systems, Rome 1976.Google Scholar
  32. [14] (a)
    S. H. Hou On property (Q) and other semicontinuity properties of multifunctions. Pacific J. Math. To appear.Google Scholar
  33. [14] (b)
    Implicit functions theorem in topological spaces, Applicable Mathematics. To appear.Google Scholar
  34. [14] (c)
    Controllability and feedback system. To appear.Google Scholar
  35. [14] (d)
    Existence theorems of optimal control problems in Banach spaces. Nonlinear Analysis. To appear.Google Scholar
  36. [15]
    S. Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. Journ. 8, 1941, 457–459.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [16] (a)
    R. Kannan, Une version stochastique du théoréme de Kakutani, C. R. Acad. Sc. Paris, t. 287, Ser. A, 551–552Google Scholar
  38. [16] (b)
    , Random correspondences and nonlinear equations. J. Multivariate Analysis 11, 1981, 230–243.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [17]
    N. Kenmochi, Existence theorems for certain nonlinear equations, Hiroshima Math. J. 1, 1971, 435–443.MathSciNetzbMATHGoogle Scholar
  40. [18]
    J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag 1971, xi+396.Google Scholar
  41. [19]
    J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, I,II,III, Springer Verlag 1972–73.Google Scholar
  42. [20] (a)
    G. J. Minty, On the maximal domain of a monotone function, Mich. Math. J. 8, 1961, 135–137.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [20] (b)
    , Monotone nonlinear operators in Hilbert space, Duke Math. J. 29, 1962, 341–346.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [20] (c)
    , On a monotonicity method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 50, 1963, 1038–1041.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [21] (a)
    M. B. Suryanarayana, Monotonicity and upper semicontinuity, Bull. Amer. Math. Soc. 82, 1976, 936–938MathSciNetCrossRefzbMATHGoogle Scholar
  46. [21] (b)
    Remarks on existence theorems for Pareto optimality, Dynamical Systems, a Univ. of Florida Intern. Symposium (Bednarek and Cesari, eds.), Academic Press 1977, pp. 335–347.Google Scholar
  47. [22]
    I. Tarnove, A controllability problem of nonlinear systems. Math. Theory of Control (Balakrishnan and Neustadt, eds.), Academic Press 1967, pp. 170–179.Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

There are no affiliations available

Personalised recommendations