American Institute of Mathematical Sciences

Advanced
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

[1]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[2]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109
[3]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[4]

Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021001
[5]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[6]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[7]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[8]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[9]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028
[10]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
[11]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[12]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286
[13]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[14]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[15]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[16]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
[17]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048
[18]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[19]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[20]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (80)
  • HTML views (151)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences