Unique Tarski Fixed Points

Published Online:https://doi.org/10.1287/moor.2018.0959

References

  • [1] Abian S, Brown AB (1961) A theorem on partially ordered sets with applications to fixed point theorems. Canadian J. Math. 13:78–82.CrossrefGoogle Scholar
  • [2] Amann H (1972) On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal. 11(3):346–384.CrossrefGoogle Scholar
  • [3] Amann H (1976) Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18(4):620–709.CrossrefGoogle Scholar
  • [4] Amann H (1977) Order structures and fixed points. Technical report, Ruhr-Universität, Bochum, Germany.Google Scholar
  • [5] Baiocchi C, Capelo AC (1984) Variational and Quasivariational Inequalities: Applications to Free Boundary Problems (John Wiley & Sons, New York).Google Scholar
  • [6] Balbus L (2016) On non-negative recursive utilities in dynamic programming with nonlinear aggregator and CES. Working paper, University of Zielona Góra, Zielona Góra, Poland.Google Scholar
  • [7] Becker RA, Rincon-Zapatero JP (2017) Recursive utility and Thompson aggregators. CAEPR Working Paper Series 2018-006, Indiana University, Bloomington.Google Scholar
  • [8] Bertsekas DP, Shreve SE (1978) Stochastic Optimal Control: The Discrete-Time Case (Athena Scientific, Belmont, MA).Google Scholar
  • [9] Bloise G, Vailakis Y (2018) Convex dynamic programming with (bounded) recursive utilities. J. Econom. Theory 173:118–141.CrossrefGoogle Scholar
  • [10] Borovička J, Stachurski J (2017) Necessary and sufficient conditions for existence and uniqueness of recursive utilities. National Bureau of Economic Research Working Paper 24162, National Bureau of Economic Research, Cambridge, MA.Google Scholar
  • [11] Borwein JM, Dempster AH (1989) The linear order complementary problem. Math. Oper. Res. 14(3):534–558.LinkGoogle Scholar
  • [12] Coleman WJ (1991) Equilibrium in a production economy with an income tax. Econometrica 59(4):1091–1104.CrossrefGoogle Scholar
  • [13] Datta M, Mirman LJ, Morand OF, Reffett KL (2002) Monotone methods for Markovian equilibria in dynamic economies. Ann. Oper. Res. 114(1/4):117–144.CrossrefGoogle Scholar
  • [14] Davey BA, Priestley HA (2002) Introduction to Lattices and Order (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • [15] Deutsch F (2001) Best Approximation in Inner Product Spaces (Springer, New York).CrossrefGoogle Scholar
  • [16] Fujimoto T (1984) An extension of Tarski’s fixed point theorem and its application to isotone complementary problems. Math. Programming 28(1):116–118.CrossrefGoogle Scholar
  • [17] Gnana Bhaskar T, Lakshmikantham V (2006) Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65(7):1379–1393.CrossrefGoogle Scholar
  • [18] Guo J, He XD (2017) Recursive utility with investment gains and losses: Existence, uniqueness, and convergence. Working paper, Columbia University, New York.Google Scholar
  • [19] Hansen LP, Scheinkman JA (2012) Recursive utility in a Markov environment with stochastic growth. Proc. Natl. Acad. Sci. USA 109(30):11967–11972.CrossrefGoogle Scholar
  • [20] Kantorovich L (1939) The method of successive approximations for functional equations. Acta Math. 71(0):63–97.CrossrefGoogle Scholar
  • [21] Koopmans TC (1960) Stationary ordinal utility and impatience. Econometrica 28(2):287–309.CrossrefGoogle Scholar
  • [22] Krasnoselskii MA (1964) Positive Solutions of Operator Equations (Noordhoff, Groningen, Netherlands).Google Scholar
  • [23] Le Van C, Morhaim L, Vailakis Y (2008) Monotone concave operators: An application to the existence and uniqueness of solutions to the Bellman equation. University of Exeter Discussion Paper 0803, University of Exeter, Exeter, UK. Google Scholar
  • [24] Li J, Ok EA (2012) Optimal solutions of variational inequalities on Banach lattices. J. Math. Anal. Appl. 388(2):1157–1165.CrossrefGoogle Scholar
  • [25] Liang ZD, Wang WX, Li SJ (2006) On concave operators. Acta Math. Sinica 22(2):577–582.CrossrefGoogle Scholar
  • [26] Lucas RE, Stokey NL (1984) Optimal growth with many consumers. J. Econom. Theory 32(1):139–171.CrossrefGoogle Scholar
  • [27] Luxemburg WAJ, Zaanen AC (1971) Riesz Spaces I (North-Holland, Amsterdam).Google Scholar
  • [28] Marinacci M, Montrucchio L (2005) Ultramodular functions. Math. Oper. Res. 30(2):311–332.LinkGoogle Scholar
  • [29] Marinacci M, Montrucchio L (2010) Unique solutions for stochastic recursive utilities. J. Econom. Theory 145(5):1776–1804.CrossrefGoogle Scholar
  • [30] Markowsky G (1976) Chain-complete posets and directed sets with applications. Algebra Universalis 6(1):53–68.CrossrefGoogle Scholar
  • [31] Montrucchio L (1998) Thompson metric, contraction property and differentiability of policy functions. J. Econom. Behav. Organ. 33(3-4):449–466.CrossrefGoogle Scholar
  • [32] Nieto J, Rodríguez-Lopez R (2007) Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 23(12):2205–2212.CrossrefGoogle Scholar
  • [33] Nishimura H, Ok EA (2012) Solvability of variational inequalities on Hilbert lattices. Math. Oper. Res. 37(4):608–625.LinkGoogle Scholar
  • [34] Nussbaum RD (1988) Hilbert’s Projective Metric and Iterated Nonlinear Maps, Memoirs of the American Mathematical Society (American Mathematical Society, Providence, RI), 391.Google Scholar
  • [35] Ok EA (2019) Order Theory and its Applications (Mimeo, New York). Forthcoming.Google Scholar
  • [36] Puterman ML, Brumelle SL (1979) On the convergence of policy iteration in stationary dynamic programming. Math. Oper. Res. 4(1):60–69.LinkGoogle Scholar
  • [37] Ran ACM, Reurings MCB (2004) A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132(5):1435–1443.CrossrefGoogle Scholar
  • [38] Riddell R (1981) Equivalence of nonlinear complementary problems and least element problems in Banach lattices. Math. Oper. Res. 6(3):462–474.LinkGoogle Scholar
  • [39] Stokey NL, Lucas RE (1989) Recursive Methods in Economic Dynamics (Harvard University Press, Cambridge, MA).CrossrefGoogle Scholar
  • [40] Tarski A (1955) A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5(2):285–309.CrossrefGoogle Scholar
  • [41] Thompson AC (1963) On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14(3):438–443.Google Scholar
  • [42] Topkis DM (1998) Supermodularity and Complementarity (Princeton University Press, Princeton, NJ).CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.