American Institute of Mathematical Sciences

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December  2018, 11(6): 1219-1232. doi: 10.3934/dcdss.2018069

Recursive variational problems in nonreflexive Banach spaces with an infinite horizon: An existence result

Nobusumi Sagara

Faculty of Economics, Hosei University, 4342, Aihara, Machida, Tokyo, 194-0298, Japan

Received  March 2017 Revised  July 2017 Published  June 2018

Fund Project: This research is supported by JSPS KAKENHI Grant Number JP18K01518 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

We investigate variational problems with recursive integral functionals governed by infinite-dimensional differential inclusions with an infinite horizon and present an existence result in the setting of nonreflexive Banach spaces. We find an optimal solution in a Sobolev space taking values in a Banach space under the Cesari type condition. We also investigate sufficient conditions for the existence of solutions to the initial value problem for the differential inclusion.

Keywords: Recursive integral functional, Sobolev space, Bochner integral, differential inclusion, viability.
Mathematics Subject Classification: Primary: 49J53, 49J45; Secondary: 28B05, 91B62.
Citation: Nobusumi Sagara. Recursive variational problems in nonreflexive Banach spaces with an infinite horizon: An existence result. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1219-1232. doi: 10.3934/dcdss.2018069
References:
[1]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[3]

E. J. Balder, Existence of optimal solutions for control and variational problems with recursive objectives, J. Math. Anal. Appl., 178 (1993), 418-437.  doi: 10.1006/jmaa.1993.1316.  Google Scholar

[4]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4$^{th}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[5]

R. A. Becker and J. H. Boyd III, Recursive utility and optimal capital accumulation. Ⅱ. Sensitivity and duality theory, Econom. Theory, 2 (1992), 547-563.  doi: 10.1007/BF01212476.  Google Scholar

[6]

R. A. Becker, J. H. Boyd III and B. Y. Sung, Recursive utility and optimal capital accumulation. Ⅰ. Existence, J. Econom. Theory, 47 (1989), 76-100.  doi: 10.1016/0022-0531(89)90104-X.  Google Scholar

[7]

E. K. Boukas, A. Haurie and P. Michel, An optimal control problem with a random stopping time, J. Optim. Theory Appl., 64 (1990), 471-480.  doi: 10.1007/BF00939419.  Google Scholar

[8]

D. A. Carlson, Infinite horizon optimal controls for problems governed by a Volterra integral equation with state dependent discount factor, J. Optim. Theory Appl., 66 (1990), 311-336.  doi: 10.1007/BF00939541.  Google Scholar

[9]

D. A. Carlson, A. Haurie and A. Jabrane, Existence of overtaking solutions to infinite dimensional control problems on unbounded time intervals, SIAM J. Control Optim., 25 (1987), 1517-1541.  doi: 10.1137/0325084.  Google Scholar

[10]

D. A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, 2$^{nd}$ edition, Springer, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

[11]

F. R. Chang, Optimal growth and recursive utility: Phase diagram analysis, J. Optim. Theory Appl., 80 (1994), 425-439.  doi: 10.1007/BF02207773.  Google Scholar

[12]

B.-L. Chen, K. Nishimura and K. Shimomura, Time preference and two-country trade, Internat. J. Econom. Theory, 4 (2008), 29-52.  doi: 10.1111/j.1742-7363.2007.00067.x.  Google Scholar

[13]

M. Das, Optimal growth with decreasing marginal impatience, J. Econom. Dynam. Control, 27 (2003), 1881-1898.  doi: 10.1016/S0165-1889(02)00088-X.  Google Scholar

[14]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[15]

K. Deimling, Multivalued differential equations on closed sets, Differential Integral Equations, 1 (1988), 23-30.   Google Scholar

[16]

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.  Google Scholar

[17]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(μ,X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.  Google Scholar

[18]

J.-P. Drugeon, Impatience and long-run growth, J. Econom. Dynam. Control, 20 (1996), 281-313.  doi: 10.1016/0165-1889(94)00852-3.  Google Scholar

[19]

J.-P. Drugeon and B. Wigniolle, On time preference, rational addiction and utility satiation, J. Math. Econom., 43 (2007), 279-286.  doi: 10.1016/j.jmateco.2006.06.010.  Google Scholar

[20]

J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.  Google Scholar

[21]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, John Wiley & Sons, New York, 1958. Google Scholar

[22]

L. G. Epstein, A simple dynamic general equilibrium model, J. Econom. Theory, 41 (1987), 68-95.  doi: 10.1016/0022-0531(87)90006-8.  Google Scholar

[23]

L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica, 55 (1987), 329-355.  doi: 10.2307/1913239.  Google Scholar

[24]

L. G. Epstein and A. Hynes, The rate of time preference and dynamic economic analysis, J. Political Econom., 91 (1983), 611-635.   Google Scholar

[25]

S. Erol, C. Le Van and C. Saglam, Existence, optimality and dynamics of equilibria with endogenous time preference, J. Math. Econom., 47 (2011), 170-179.  doi: 10.1016/j.jmateco.2010.12.006.  Google Scholar

[26]

H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.  Google Scholar

[27]

K. Iwai, Optimal economic growth and stationary ordinal utility-a Fisherian approach, J. Econom. Theory, 5 (1972), 121-151.  doi: 10.1016/0022-0531(72)90122-6.  Google Scholar

[28]

Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940493.  Google Scholar

[29]

T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica, 28 (1960), 287-309.  doi: 10.2307/1907722.  Google Scholar

[30]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[31]

T. Maruyama, A generalization of the weak convergence theorem in Sobolev spaces with applications to differential inclusions in a Banach space, Proc. Japan Acad. Ser. A Math. Sci., 77 (2001), 5-10.  doi: 10.3792/pjaa.77.5.  Google Scholar

[32]

M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys. Math. Soc. Japan 3rd. Ser., 24 (1942), 551-559.   Google Scholar

[33]

A. Naiary, Asymptotic behavior and optimal properties of a consumption-investment model with variable time preference, J. Econom. Dynam. Control, 7 (1984), 283-313.  doi: 10.1016/0165-1889(84)90021-6.  Google Scholar

[34]

M. Obstfeld, Intertemporal dependence, impatience, and dynamics, J. Monetary Econom., 26 (1990), 45-75.  doi: 10.3386/w3028.  Google Scholar

[35]

T. Palivos, P. Wang and J. Zhang, On the existence of balanced growth equilibrium, Internat. Econom. Rev., 38 (1997), 205-224.  doi: 10.2307/2527415.  Google Scholar

[36]

M. Petrakis and J. J. Uhl, Jr., Differentiation in Banach spaces, in Proceedings of the Analysis Conference, Singapore 1986, (eds. S. T. L. Choy, J. P. Jesudason and P. Y. Lee), North-Holland, 150 (1988), 219-241. doi: 10.1016/S0304-0208(08)71340-9.  Google Scholar

[37]

H. E. Ryder, Jr. and G. M. Heal, Optimal growth with intertemporally dependent preferences, Rev. Econom. Stud., 40 (1973), 1-31.   Google Scholar

[38]

N. Sagara, Optimal growth with recursive utility: An existence result without convexity assumptions, J. Optim. Theory Appl., 109 (2001), 371-383.  doi: 10.1023/A:1017518523055.  Google Scholar

[39]

N. Sagara, Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation, J. Math. Anal. Appl., 327 (2007), 203-219.  doi: 10.1016/j.jmaa.2006.04.012.  Google Scholar

[40]

S. Shi and L. G. Epstein, Habits and time preference, Internat. Econom. Rev., 34 (1993), 61-84.  doi: 10.2307/2526950.  Google Scholar

[41]

G. Sorger, Maximum principle for control problems with uncertain horizon and variable discount rate, J. Optim. Theory Appl., 70 (1991), 607-618.  doi: 10.1007/BF00941305.  Google Scholar

[42]

H. Uzawa, Time preferences, the consumption function, and optimum asset holdings, in Value, Capital, and Growth: Papers in Honour of Sir John Hicks (ed. J. N. Wolfe), Edinburgh University Press, (1989), 485-504. doi: 10.1017/CBO9780511664496.005.  Google Scholar

[43]

A. J. Zaslavski, Existence and structure of optimal solutions of infinite-dimensional control problems, Appl. Math. Optim., 42 (2000), 291-313.  doi: 10.1007/s002450010011.  Google Scholar

show all references

References:
[1]

J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[3]

E. J. Balder, Existence of optimal solutions for control and variational problems with recursive objectives, J. Math. Anal. Appl., 178 (1993), 418-437.  doi: 10.1006/jmaa.1993.1316.  Google Scholar

[4]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4$^{th}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[5]

R. A. Becker and J. H. Boyd III, Recursive utility and optimal capital accumulation. Ⅱ. Sensitivity and duality theory, Econom. Theory, 2 (1992), 547-563.  doi: 10.1007/BF01212476.  Google Scholar

[6]

R. A. Becker, J. H. Boyd III and B. Y. Sung, Recursive utility and optimal capital accumulation. Ⅰ. Existence, J. Econom. Theory, 47 (1989), 76-100.  doi: 10.1016/0022-0531(89)90104-X.  Google Scholar

[7]

E. K. Boukas, A. Haurie and P. Michel, An optimal control problem with a random stopping time, J. Optim. Theory Appl., 64 (1990), 471-480.  doi: 10.1007/BF00939419.  Google Scholar

[8]

D. A. Carlson, Infinite horizon optimal controls for problems governed by a Volterra integral equation with state dependent discount factor, J. Optim. Theory Appl., 66 (1990), 311-336.  doi: 10.1007/BF00939541.  Google Scholar

[9]

D. A. Carlson, A. Haurie and A. Jabrane, Existence of overtaking solutions to infinite dimensional control problems on unbounded time intervals, SIAM J. Control Optim., 25 (1987), 1517-1541.  doi: 10.1137/0325084.  Google Scholar

[10]

D. A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, 2$^{nd}$ edition, Springer, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

[11]

F. R. Chang, Optimal growth and recursive utility: Phase diagram analysis, J. Optim. Theory Appl., 80 (1994), 425-439.  doi: 10.1007/BF02207773.  Google Scholar

[12]

B.-L. Chen, K. Nishimura and K. Shimomura, Time preference and two-country trade, Internat. J. Econom. Theory, 4 (2008), 29-52.  doi: 10.1111/j.1742-7363.2007.00067.x.  Google Scholar

[13]

M. Das, Optimal growth with decreasing marginal impatience, J. Econom. Dynam. Control, 27 (2003), 1881-1898.  doi: 10.1016/S0165-1889(02)00088-X.  Google Scholar

[14]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[15]

K. Deimling, Multivalued differential equations on closed sets, Differential Integral Equations, 1 (1988), 23-30.   Google Scholar

[16]

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.  Google Scholar

[17]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^1(μ,X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.  Google Scholar

[18]

J.-P. Drugeon, Impatience and long-run growth, J. Econom. Dynam. Control, 20 (1996), 281-313.  doi: 10.1016/0165-1889(94)00852-3.  Google Scholar

[19]

J.-P. Drugeon and B. Wigniolle, On time preference, rational addiction and utility satiation, J. Math. Econom., 43 (2007), 279-286.  doi: 10.1016/j.jmateco.2006.06.010.  Google Scholar

[20]

J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.  Google Scholar

[21]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, John Wiley & Sons, New York, 1958. Google Scholar

[22]

L. G. Epstein, A simple dynamic general equilibrium model, J. Econom. Theory, 41 (1987), 68-95.  doi: 10.1016/0022-0531(87)90006-8.  Google Scholar

[23]

L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica, 55 (1987), 329-355.  doi: 10.2307/1913239.  Google Scholar

[24]

L. G. Epstein and A. Hynes, The rate of time preference and dynamic economic analysis, J. Political Econom., 91 (1983), 611-635.   Google Scholar

[25]

S. Erol, C. Le Van and C. Saglam, Existence, optimality and dynamics of equilibria with endogenous time preference, J. Math. Econom., 47 (2011), 170-179.  doi: 10.1016/j.jmateco.2010.12.006.  Google Scholar

[26]

H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.  Google Scholar

[27]

K. Iwai, Optimal economic growth and stationary ordinal utility-a Fisherian approach, J. Econom. Theory, 5 (1972), 121-151.  doi: 10.1016/0022-0531(72)90122-6.  Google Scholar

[28]

Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940493.  Google Scholar

[29]

T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica, 28 (1960), 287-309.  doi: 10.2307/1907722.  Google Scholar

[30]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[31]

T. Maruyama, A generalization of the weak convergence theorem in Sobolev spaces with applications to differential inclusions in a Banach space, Proc. Japan Acad. Ser. A Math. Sci., 77 (2001), 5-10.  doi: 10.3792/pjaa.77.5.  Google Scholar

[32]

M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys. Math. Soc. Japan 3rd. Ser., 24 (1942), 551-559.   Google Scholar

[33]

A. Naiary, Asymptotic behavior and optimal properties of a consumption-investment model with variable time preference, J. Econom. Dynam. Control, 7 (1984), 283-313.  doi: 10.1016/0165-1889(84)90021-6.  Google Scholar

[34]

M. Obstfeld, Intertemporal dependence, impatience, and dynamics, J. Monetary Econom., 26 (1990), 45-75.  doi: 10.3386/w3028.  Google Scholar

[35]

T. Palivos, P. Wang and J. Zhang, On the existence of balanced growth equilibrium, Internat. Econom. Rev., 38 (1997), 205-224.  doi: 10.2307/2527415.  Google Scholar

[36]

M. Petrakis and J. J. Uhl, Jr., Differentiation in Banach spaces, in Proceedings of the Analysis Conference, Singapore 1986, (eds. S. T. L. Choy, J. P. Jesudason and P. Y. Lee), North-Holland, 150 (1988), 219-241. doi: 10.1016/S0304-0208(08)71340-9.  Google Scholar

[37]

H. E. Ryder, Jr. and G. M. Heal, Optimal growth with intertemporally dependent preferences, Rev. Econom. Stud., 40 (1973), 1-31.   Google Scholar

[38]

N. Sagara, Optimal growth with recursive utility: An existence result without convexity assumptions, J. Optim. Theory Appl., 109 (2001), 371-383.  doi: 10.1023/A:1017518523055.  Google Scholar

[39]

N. Sagara, Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation, J. Math. Anal. Appl., 327 (2007), 203-219.  doi: 10.1016/j.jmaa.2006.04.012.  Google Scholar

[40]

S. Shi and L. G. Epstein, Habits and time preference, Internat. Econom. Rev., 34 (1993), 61-84.  doi: 10.2307/2526950.  Google Scholar

[41]

G. Sorger, Maximum principle for control problems with uncertain horizon and variable discount rate, J. Optim. Theory Appl., 70 (1991), 607-618.  doi: 10.1007/BF00941305.  Google Scholar

[42]

H. Uzawa, Time preferences, the consumption function, and optimum asset holdings, in Value, Capital, and Growth: Papers in Honour of Sir John Hicks (ed. J. N. Wolfe), Edinburgh University Press, (1989), 485-504. doi: 10.1017/CBO9780511664496.005.  Google Scholar

[43]

A. J. Zaslavski, Existence and structure of optimal solutions of infinite-dimensional control problems, Appl. Math. Optim., 42 (2000), 291-313.  doi: 10.1007/s002450010011.  Google Scholar

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