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 - 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.

[1]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27

[2]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

[3]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929

[4]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

[5]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016

[6]

Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033

[7]

John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056

[8]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[9]

Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191

[10]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029

[11]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[12]

R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435

[13]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[14]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[15]

Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697

[16]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167

[17]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[18]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416

[19]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[20]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

[Back to Top]