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

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.

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.

[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. BeckerJ. 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. BoukasA. 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. CarlsonA. 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. ChenK. 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. DiestelW. 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. ErolC. 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. PalivosP. 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. BeckerJ. 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. BoukasA. 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. CarlsonA. 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. ChenK. 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. DiestelW. 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. ErolC. 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. PalivosP. 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.

[1]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[2]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837

[3]

Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527

[4]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[5]

Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349

[6]

Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038

[7]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[8]

Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28

[9]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[10]

P. M. Jordan, Louis Fishman. Phase space and path integral approach to wave propagation modeling. Conference Publications, 2001, 2001 (Special) : 199-210. doi: 10.3934/proc.2001.2001.199

[11]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[12]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663

[13]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[14]

Jędrzej Śniatycki. Integral curves of derivations on locally semi-algebraic differential spaces. Conference Publications, 2003, 2003 (Special) : 827-833. doi: 10.3934/proc.2003.2003.827

[15]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

[16]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155

[17]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[18]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791

[19]

Baiyu Liu, Li Ma. Blow up threshold for a parabolic type equation involving space integral and variational structure. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2169-2183. doi: 10.3934/cpaa.2015.14.2169

[20]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (16)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]