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

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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 and 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.
[2]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	F. R. Chang, Optimal growth and recursive utility: Phase diagram analysis, J. Optim. Theory Appl., 80 (1994), 425-439. doi: 10.1007/BF02207773.
[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.
[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.
[14]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.
[15]	K. Deimling, Multivalued differential equations on closed sets, Differential Integral Equations, 1 (1988), 23-30.
[16]	K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.
[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.
[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.
[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.
[20]	J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
[21]	N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, John Wiley & Sons, New York, 1958.
[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.
[23]	L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica, 55 (1987), 329-355. doi: 10.2307/1913239.
[24]	L. G. Epstein and A. Hynes, The rate of time preference and dynamic economic analysis, J. Political Econom., 91 (1983), 611-635.
[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.
[26]	H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.
[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.
[28]	Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. doi: 10.2969/jmsj/01940493.
[29]	T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica, 28 (1960), 287-309. doi: 10.2307/1907722.
[30]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[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.
[32]	M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys. Math. Soc. Japan 3rd. Ser., 24 (1942), 551-559.
[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.
[34]	M. Obstfeld, Intertemporal dependence, impatience, and dynamics, J. Monetary Econom., 26 (1990), 45-75. doi: 10.3386/w3028.
[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.
[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.
[37]	H. E. Ryder, Jr. and G. M. Heal, Optimal growth with intertemporally dependent preferences, Rev. Econom. Stud., 40 (1973), 1-31.
[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.
[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.
[40]	S. Shi and L. G. Epstein, Habits and time preference, Internat. Econom. Rev., 34 (1993), 61-84. doi: 10.2307/2526950.
[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.
[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.
[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.

show all references

References:

[1]	J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[2]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	F. R. Chang, Optimal growth and recursive utility: Phase diagram analysis, J. Optim. Theory Appl., 80 (1994), 425-439. doi: 10.1007/BF02207773.
[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.
[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.
[14]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.
[15]	K. Deimling, Multivalued differential equations on closed sets, Differential Integral Equations, 1 (1988), 23-30.
[16]	K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.
[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.
[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.
[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.
[20]	J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
[21]	N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, John Wiley & Sons, New York, 1958.
[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.
[23]	L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica, 55 (1987), 329-355. doi: 10.2307/1913239.
[24]	L. G. Epstein and A. Hynes, The rate of time preference and dynamic economic analysis, J. Political Econom., 91 (1983), 611-635.
[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.
[26]	H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.
[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.
[28]	Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. doi: 10.2969/jmsj/01940493.
[29]	T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica, 28 (1960), 287-309. doi: 10.2307/1907722.
[30]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[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.
[32]	M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys. Math. Soc. Japan 3rd. Ser., 24 (1942), 551-559.
[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.
[34]	M. Obstfeld, Intertemporal dependence, impatience, and dynamics, J. Monetary Econom., 26 (1990), 45-75. doi: 10.3386/w3028.
[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.
[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.
[37]	H. E. Ryder, Jr. and G. M. Heal, Optimal growth with intertemporally dependent preferences, Rev. Econom. Stud., 40 (1973), 1-31.
[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.
[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.
[40]	S. Shi and L. G. Epstein, Habits and time preference, Internat. Econom. Rev., 34 (1993), 61-84. doi: 10.2307/2526950.
[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.
[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.
[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.

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