August  2011, 30(3): 737-765. doi: 10.3934/dcds.2011.30.737

Well-posedness of initial value problems for functional differential and algebraic equations of mixed type

1. 

Division of Applied Mathematics - Brown University, 182 George Street, Providence, RI 02912, United States

2. 

Laboratory of Mathematics, Images and Applications - University of La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle, France

Received  January 2010 Revised  January 2011 Published  March 2011

We study the well-posedness of initial value problems for scalar functional algebraic and differential functional equations of mixed type. We provide a practical way to determine whether such problems admit unique solutions that grow at a specified rate. In particular, we exploit the fact that the answer to such questions is encoded in an integer n#. We show how this number can be tracked as a problem is transformed to a reference problem for which a Wiener-Hopf splitting can be computed. Once such a splitting is available, results due to Mallet-Paret and Verduyn-Lunel can be used to compute n#. We illustrate our techniques by analytically studying the well-posedness of two macro-economic overlapping generations models for which Wiener-Hopf splittings are not readily available.
Citation: Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
References:
[1]

M. Bambi, Endogenous growth and time-to-build: the AK case, J. Economic Dynamics and Control, 32 (2008), 1015-1040. doi: 10.1016/j.jedc.2007.04.002.  Google Scholar

[2]

R. Boucekkine, O. Licandro, L. A. Puch and F. D. Rio, Vintage capital and the dynamics of the AK model, J. Economic Theory, 120 (2005), 39-72. doi: 10.1016/j.jet.2004.02.006.  Google Scholar

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D. Cass and M. E. Yaari, Individual saving, aggregate capital accumulation and efficient growth, in "Essays on the Theory of Optimal Growth," K. Shell (ed.), MIT Press, Cambridge, MA, (1967), 233-268. Google Scholar

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H. d'Albis and E. Augeraud-Véron, In, preparation., ().   Google Scholar

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H. d'Albis and E. Augeraud-Véron, Competitive growth in a life-cycle model: Existence and dynamics, International Economic Review, 50 (2009), 459-484. doi: 10.1111/j.1468-2354.2009.00537.x.  Google Scholar

[6]

P. A. Diamond, National debt in a neoclassical growth model, American Economic Review, 55 (1965), 1126-1150. Google Scholar

[7]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H. O. Walther, "Delay Equations," Springer-Verlag, New York, 1995.  Google Scholar

[8]

C. Edmond, An integral equation representation for overlapping generations in continuous time, J. Economic Theory, 143 (2008), 596-609. doi: 10.1016/j.jet.2008.03.006.  Google Scholar

[9]

D. Gale, Pure exchange equilibrium of dynamic economic models, J. Economic Theory, 6 (1973), 12-36. doi: 10.1016/0022-0531(73)90041-0.  Google Scholar

[10]

J. D. Geanakoplos and H. M. Polemarchakis, Walrasian indeterminacy and keynesian macroeconomics, Rev. Economic Studies, 53 (1986), 755-779. doi: 10.2307/2297718.  Google Scholar

[11]

J. Grandmont, On endogenous competitive business cycles, Econometrica, 53 (1985), 995-1045. doi: 10.2307/1911010.  Google Scholar

[12]

J. K. Hale and S. M. Verduyn-Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.  Google Scholar

[13]

J. Härterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.  Google Scholar

[14]

D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optimization Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.  Google Scholar

[15]

H. J. Hupkes, E. Augeraud-Veron and S. M. Verduyn-Lunel, Center projections for smooth difference equations of mixed type, J. Diff. Eqn., 244 (2008), 803-835. doi: 10.1016/j.jde.2007.10.033.  Google Scholar

[16]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dyn. Diff. Eq., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.  Google Scholar

[17]

H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487. doi: 10.1512/iumj.2009.58.3661.  Google Scholar

[18]

F. E. Kydland and E. C. Prescott, Time to build and aggregate fluctuations, Econometrica, 50 (1982), 1345-1370. doi: 10.2307/1913386.  Google Scholar

[19]

B. J. Levin, "Distribution of Zeros of Entire Functions (translated from Russian)," American Mathematical Society, Providence, 1980.  Google Scholar

[20]

T. Lloyd-Braga, C. Nourry and A. Venditti, Indeterminacy in dynamic models: when Diamond meets Ramsey, J. Economic Theory, 134 (2007), 513-536. doi: 10.1016/j.jet.2005.12.005.  Google Scholar

[21]

J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1999), 1-48. doi: 10.1023/A:1021889401235.  Google Scholar

[22]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, J. Diff. Eq., ().   Google Scholar

[23]

P. M. Romer, Increasing returns and long-run growth, J. Political Economy, 94 (1986), 1002-1037. doi: 10.1086/261420.  Google Scholar

[24]

A. Rustichini, Functional differential equations of mixed type: the linear autonomous case, J. Dyn. Diff. Eq., 11 (1989), 121-143. doi: 10.1007/BF01047828.  Google Scholar

[25]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar

[26]

P. A. Samuelson, An exact consumption-loan model of interest with or without the social contrivance of money, J. Political Economy, 66 (1958), 467-482. doi: 10.1086/258100.  Google Scholar

[27]

J. Tirole, Asset bubbles and overlapping generations, Econometrica, 53 (1985), 1071-1100. doi: i:10.2307/1911012.  Google Scholar

[28]

A. Venditti and K. Nishimura, Indeterminacy in discrete-time infinite-horizon models, in "Handbook of Optimal Growth: Vol. 1: The Discrete Time Horizon," R.A. Dana, C. Le Van, T. Mitra and K. Nishimura (eds.), Kluwer, (2006), 273-296. Google Scholar

show all references

References:
[1]

M. Bambi, Endogenous growth and time-to-build: the AK case, J. Economic Dynamics and Control, 32 (2008), 1015-1040. doi: 10.1016/j.jedc.2007.04.002.  Google Scholar

[2]

R. Boucekkine, O. Licandro, L. A. Puch and F. D. Rio, Vintage capital and the dynamics of the AK model, J. Economic Theory, 120 (2005), 39-72. doi: 10.1016/j.jet.2004.02.006.  Google Scholar

[3]

D. Cass and M. E. Yaari, Individual saving, aggregate capital accumulation and efficient growth, in "Essays on the Theory of Optimal Growth," K. Shell (ed.), MIT Press, Cambridge, MA, (1967), 233-268. Google Scholar

[4]

H. d'Albis and E. Augeraud-Véron, In, preparation., ().   Google Scholar

[5]

H. d'Albis and E. Augeraud-Véron, Competitive growth in a life-cycle model: Existence and dynamics, International Economic Review, 50 (2009), 459-484. doi: 10.1111/j.1468-2354.2009.00537.x.  Google Scholar

[6]

P. A. Diamond, National debt in a neoclassical growth model, American Economic Review, 55 (1965), 1126-1150. Google Scholar

[7]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H. O. Walther, "Delay Equations," Springer-Verlag, New York, 1995.  Google Scholar

[8]

C. Edmond, An integral equation representation for overlapping generations in continuous time, J. Economic Theory, 143 (2008), 596-609. doi: 10.1016/j.jet.2008.03.006.  Google Scholar

[9]

D. Gale, Pure exchange equilibrium of dynamic economic models, J. Economic Theory, 6 (1973), 12-36. doi: 10.1016/0022-0531(73)90041-0.  Google Scholar

[10]

J. D. Geanakoplos and H. M. Polemarchakis, Walrasian indeterminacy and keynesian macroeconomics, Rev. Economic Studies, 53 (1986), 755-779. doi: 10.2307/2297718.  Google Scholar

[11]

J. Grandmont, On endogenous competitive business cycles, Econometrica, 53 (1985), 995-1045. doi: 10.2307/1911010.  Google Scholar

[12]

J. K. Hale and S. M. Verduyn-Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.  Google Scholar

[13]

J. Härterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.  Google Scholar

[14]

D. K. Hughes, Variational and optimal control problems with delayed argument, J. Optimization Theory Appl., 2 (1968), 1-14. doi: 10.1007/BF00927159.  Google Scholar

[15]

H. J. Hupkes, E. Augeraud-Veron and S. M. Verduyn-Lunel, Center projections for smooth difference equations of mixed type, J. Diff. Eqn., 244 (2008), 803-835. doi: 10.1016/j.jde.2007.10.033.  Google Scholar

[16]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dyn. Diff. Eq., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.  Google Scholar

[17]

H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487. doi: 10.1512/iumj.2009.58.3661.  Google Scholar

[18]

F. E. Kydland and E. C. Prescott, Time to build and aggregate fluctuations, Econometrica, 50 (1982), 1345-1370. doi: 10.2307/1913386.  Google Scholar

[19]

B. J. Levin, "Distribution of Zeros of Entire Functions (translated from Russian)," American Mathematical Society, Providence, 1980.  Google Scholar

[20]

T. Lloyd-Braga, C. Nourry and A. Venditti, Indeterminacy in dynamic models: when Diamond meets Ramsey, J. Economic Theory, 134 (2007), 513-536. doi: 10.1016/j.jet.2005.12.005.  Google Scholar

[21]

J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1999), 1-48. doi: 10.1023/A:1021889401235.  Google Scholar

[22]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, J. Diff. Eq., ().   Google Scholar

[23]

P. M. Romer, Increasing returns and long-run growth, J. Political Economy, 94 (1986), 1002-1037. doi: 10.1086/261420.  Google Scholar

[24]

A. Rustichini, Functional differential equations of mixed type: the linear autonomous case, J. Dyn. Diff. Eq., 11 (1989), 121-143. doi: 10.1007/BF01047828.  Google Scholar

[25]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar

[26]

P. A. Samuelson, An exact consumption-loan model of interest with or without the social contrivance of money, J. Political Economy, 66 (1958), 467-482. doi: 10.1086/258100.  Google Scholar

[27]

J. Tirole, Asset bubbles and overlapping generations, Econometrica, 53 (1985), 1071-1100. doi: i:10.2307/1911012.  Google Scholar

[28]

A. Venditti and K. Nishimura, Indeterminacy in discrete-time infinite-horizon models, in "Handbook of Optimal Growth: Vol. 1: The Discrete Time Horizon," R.A. Dana, C. Le Van, T. Mitra and K. Nishimura (eds.), Kluwer, (2006), 273-296. Google Scholar

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