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

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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

Hermen Jan Hupkes ^1, and Emmanuelle Augeraud-Véron ^2,

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

Abstract
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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.

Keywords: Functional differential equations, overlapping generations models, Functional differential equations, advanced and retarded arguments, initial value problems, indeterminacy..

Mathematics Subject Classification: Primary: 34K06, 34A12; Secondary: 91B6.

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 - A, 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. doi: 10.1016/j.jedc.2007.04.002.
[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. doi: 10.1016/j.jet.2004.02.006.
[3]	D. Cass and M. E. Yaari, Individual saving, aggregate capital accumulation and efficient growth,, in, (1967), 233.
[4]	H. d'Albis and E. Augeraud-Véron, In, preparation., ().
[5]	H. d'Albis and E. Augeraud-Véron, Competitive growth in a life-cycle model: Existence and dynamics,, International Economic Review, 50 (2009), 459. doi: 10.1111/j.1468-2354.2009.00537.x.
[6]	P. A. Diamond, National debt in a neoclassical growth model,, American Economic Review, 55 (1965), 1126.
[7]	O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H. O. Walther, "Delay Equations,", Springer-Verlag, (1995).
[8]	C. Edmond, An integral equation representation for overlapping generations in continuous time,, J. Economic Theory, 143 (2008), 596. doi: 10.1016/j.jet.2008.03.006.
[9]	D. Gale, Pure exchange equilibrium of dynamic economic models,, J. Economic Theory, 6 (1973), 12. doi: 10.1016/0022-0531(73)90041-0.
[10]	J. D. Geanakoplos and H. M. Polemarchakis, Walrasian indeterminacy and keynesian macroeconomics,, Rev. Economic Studies, 53 (1986), 755. doi: 10.2307/2297718.
[11]	J. Grandmont, On endogenous competitive business cycles,, Econometrica, 53 (1985), 995. doi: 10.2307/1911010.
[12]	J. K. Hale and S. M. Verduyn-Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).
[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. doi: 10.1512/iumj.2002.51.2188.
[14]	D. K. Hughes, Variational and optimal control problems with delayed argument,, J. Optimization Theory Appl., 2 (1968), 1. doi: 10.1007/BF00927159.
[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. doi: 10.1016/j.jde.2007.10.033.
[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. doi: 10.1007/s10884-006-9055-9.
[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. doi: 10.1512/iumj.2009.58.3661.
[18]	F. E. Kydland and E. C. Prescott, Time to build and aggregate fluctuations,, Econometrica, 50 (1982), 1345. doi: 10.2307/1913386.
[19]	B. J. Levin, "Distribution of Zeros of Entire Functions (translated from Russian),", American Mathematical Society, (1980).
[20]	T. Lloyd-Braga, C. Nourry and A. Venditti, Indeterminacy in dynamic models: when Diamond meets Ramsey,, J. Economic Theory, 134 (2007), 513. doi: 10.1016/j.jet.2005.12.005.
[21]	J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type,, J. Dyn. Diff. Eq., 11 (1999), 1. doi: 10.1023/A:1021889401235.
[22]	J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, J. Diff. Eq., ().
[23]	P. M. Romer, Increasing returns and long-run growth,, J. Political Economy, 94 (1986), 1002. doi: 10.1086/261420.
[24]	A. Rustichini, Functional differential equations of mixed type: the linear autonomous case,, J. Dyn. Diff. Eq., 11 (1989), 121. doi: 10.1007/BF01047828.
[25]	A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type,, J. Dyn. Diff. Eq., 11 (1989), 145. doi: 10.1007/BF01047829.
[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. doi: 10.1086/258100.
[27]	J. Tirole, Asset bubbles and overlapping generations,, Econometrica, 53 (1985), 1071. doi: i:10.2307/1911012.
[28]	A. Venditti and K. Nishimura, Indeterminacy in discrete-time infinite-horizon models,, in, 1 (2006), 273.

show all references

References:

[1]	M. Bambi, Endogenous growth and time-to-build: the AK case,, J. Economic Dynamics and Control, 32 (2008), 1015. doi: 10.1016/j.jedc.2007.04.002.
[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. doi: 10.1016/j.jet.2004.02.006.
[3]	D. Cass and M. E. Yaari, Individual saving, aggregate capital accumulation and efficient growth,, in, (1967), 233.
[4]	H. d'Albis and E. Augeraud-Véron, In, preparation., ().
[5]	H. d'Albis and E. Augeraud-Véron, Competitive growth in a life-cycle model: Existence and dynamics,, International Economic Review, 50 (2009), 459. doi: 10.1111/j.1468-2354.2009.00537.x.
[6]	P. A. Diamond, National debt in a neoclassical growth model,, American Economic Review, 55 (1965), 1126.
[7]	O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H. O. Walther, "Delay Equations,", Springer-Verlag, (1995).
[8]	C. Edmond, An integral equation representation for overlapping generations in continuous time,, J. Economic Theory, 143 (2008), 596. doi: 10.1016/j.jet.2008.03.006.
[9]	D. Gale, Pure exchange equilibrium of dynamic economic models,, J. Economic Theory, 6 (1973), 12. doi: 10.1016/0022-0531(73)90041-0.
[10]	J. D. Geanakoplos and H. M. Polemarchakis, Walrasian indeterminacy and keynesian macroeconomics,, Rev. Economic Studies, 53 (1986), 755. doi: 10.2307/2297718.
[11]	J. Grandmont, On endogenous competitive business cycles,, Econometrica, 53 (1985), 995. doi: 10.2307/1911010.
[12]	J. K. Hale and S. M. Verduyn-Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).
[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. doi: 10.1512/iumj.2002.51.2188.
[14]	D. K. Hughes, Variational and optimal control problems with delayed argument,, J. Optimization Theory Appl., 2 (1968), 1. doi: 10.1007/BF00927159.
[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. doi: 10.1016/j.jde.2007.10.033.
[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. doi: 10.1007/s10884-006-9055-9.
[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. doi: 10.1512/iumj.2009.58.3661.
[18]	F. E. Kydland and E. C. Prescott, Time to build and aggregate fluctuations,, Econometrica, 50 (1982), 1345. doi: 10.2307/1913386.
[19]	B. J. Levin, "Distribution of Zeros of Entire Functions (translated from Russian),", American Mathematical Society, (1980).
[20]	T. Lloyd-Braga, C. Nourry and A. Venditti, Indeterminacy in dynamic models: when Diamond meets Ramsey,, J. Economic Theory, 134 (2007), 513. doi: 10.1016/j.jet.2005.12.005.
[21]	J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type,, J. Dyn. Diff. Eq., 11 (1999), 1. doi: 10.1023/A:1021889401235.
[22]	J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, J. Diff. Eq., ().
[23]	P. M. Romer, Increasing returns and long-run growth,, J. Political Economy, 94 (1986), 1002. doi: 10.1086/261420.
[24]	A. Rustichini, Functional differential equations of mixed type: the linear autonomous case,, J. Dyn. Diff. Eq., 11 (1989), 121. doi: 10.1007/BF01047828.
[25]	A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type,, J. Dyn. Diff. Eq., 11 (1989), 145. doi: 10.1007/BF01047829.
[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. doi: 10.1086/258100.
[27]	J. Tirole, Asset bubbles and overlapping generations,, Econometrica, 53 (1985), 1071. doi: i:10.2307/1911012.
[28]	A. Venditti and K. Nishimura, Indeterminacy in discrete-time infinite-horizon models,, in, 1 (2006), 273.

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