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doi: 10.3934/jimo.2022017
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Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

Department of Mathematics, Soochow University, Suzhou 215006, China

*Corresponding author: Pengcheng Wu

Received  August 2021 Revised  December 2021 Early access February 2022

Fund Project: This research was supported by grants from National Natural Science Foundation of China (11771319) and (11971339)

In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem $ {{\rm{(VIP)}}} $ by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem $ {{\rm{(VIP)}}} $. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.

Citation: Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022017
References:
[1]

G. J. BaoT. T. Wang and G. F. Li, On very weak solutions to a class of double obstacle problems, J. Math. Anal. Appl., 402 (2013), 702-709.  doi: 10.1016/j.jmaa.2013.01.065.

[2]

F. E. Browder, On a principle of H. Brézis and its applications, J. Functional Analysis, 25 (1977), 356-365.  doi: 10.1016/0022-1236(77)90044-1.

[3]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.  doi: 10.1016/0022-1236(72)90070-5.

[4]

S. Carl, Existence and extremal solutions of parabolic variational-hemivariational inequalities, Monatsh. Math., 172 (2013), 29-54.  doi: 10.1007/s00605-013-0502-5.

[5]

S. Carl and V. K. Le, On systems of parabolic variational inequalities with multivalued terms, Monatsh. Math., 194 (2021), 227-260.  doi: 10.1007/s00605-020-01477-6.

[6]

S. Carl and V. K. Le, Quasilinear parabolic variational inequalities with multi-valued lower-order terms, Z. Angew. Math. Phys., 65 (2014), 845-864.  doi: 10.1007/s00033-013-0357-6.

[7]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[8]

P. DrábekA. Kufner and F. Nicolosi, On the solvability of degenerated quasilinear elliptic equations of higher order, J. Differential Equations, 109 (1994), 325-347.  doi: 10.1006/jdeq.1994.1053.

[9]

Y. R. DuanS. Wang and Y. Y. Zhou, A power penalty approach to a mixed quasilinear elliptic complementarity problem, J. Global Optim., 81 (2021), 901-918.  doi: 10.1007/s10898-021-01000-7.

[10]

M. FrentzK. NyströmA. Pascucci and S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options, Math. Ann., 347 (2010), 805-838.  doi: 10.1007/s00208-009-0456-z.

[11]

F. Giannessi and A. Maugeri, Variational Analysis and Applications, Springer, New York, 2005.

[12]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.

[13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, Inc. New York-London, 1980. 
[14]

J. L. Lions, The work of G. Stampacchia in variational inequalities, Variational Analysis and Applications, Springer, New York, 79 (2005), 3–30. doi: 10.1007/0-387-24276-7_1.

[15]

L. LuZ. H. Liu and X. F. Guo, Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions, Topol. Methods Nonlinear Anal., 55 (2020), 429-449.  doi: 10.12775/tmna.2019.088.

[16]

D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff Publishers, 1978.

[17]

M. Théra, Existence results for the nonlinear complementarity problem and application to nonlinear analysis, J. Math. Anal. Appl., 154 (1991), 572-584.  doi: 10.1016/0022-247X(91)90059-9.

[18] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.  doi: 10.1007/978-1-4899-3614-1.
[19]

S. Wang and C. S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Anal., 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014.

[20]

S. WangX. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from american option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3.

[21]

Y. B. XiaoN. J. Huang and J. Lu, A system of time-dependent hemivariational inequalities with volterra integral terms, J. Optim. Theory Appl., 165 (2015), 837-853.  doi: 10.1007/s10957-014-0602-y.

[22]

K. Zhang and K. L. Teo, Convergence analysis of power penalty method for American bond option pricing, J. Global Optim., 56 (2013), 1313-1323.  doi: 10.1007/s10898-012-9843-1.

[23]

K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435.

[24]

K. ZhangX. Q. Yang and K. L. Teo, A power penalty approach to American option pricing with jump diffusion processes, J. Ind. Manag. Optim., 4 (2008), 783-799.  doi: 10.3934/jimo.2008.4.783.

[25]

Y. Y. Zhou, Some results about perturbed maximal monotone mappings, Comput. Math. Appl., 51 (2006), 487-496.  doi: 10.1016/j.camwa.2005.09.003.

[26]

Y. Y. ZhouS. Wang and X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, J. Global Optim., 60 (2014), 531-550.  doi: 10.1007/s10898-013-0122-6.

[27]

E. Zeilder, Nonlinear Functional Analysis and Its Applications Ⅱ/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

G. J. BaoT. T. Wang and G. F. Li, On very weak solutions to a class of double obstacle problems, J. Math. Anal. Appl., 402 (2013), 702-709.  doi: 10.1016/j.jmaa.2013.01.065.

[2]

F. E. Browder, On a principle of H. Brézis and its applications, J. Functional Analysis, 25 (1977), 356-365.  doi: 10.1016/0022-1236(77)90044-1.

[3]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis, 11 (1972), 251-294.  doi: 10.1016/0022-1236(72)90070-5.

[4]

S. Carl, Existence and extremal solutions of parabolic variational-hemivariational inequalities, Monatsh. Math., 172 (2013), 29-54.  doi: 10.1007/s00605-013-0502-5.

[5]

S. Carl and V. K. Le, On systems of parabolic variational inequalities with multivalued terms, Monatsh. Math., 194 (2021), 227-260.  doi: 10.1007/s00605-020-01477-6.

[6]

S. Carl and V. K. Le, Quasilinear parabolic variational inequalities with multi-valued lower-order terms, Z. Angew. Math. Phys., 65 (2014), 845-864.  doi: 10.1007/s00033-013-0357-6.

[7]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[8]

P. DrábekA. Kufner and F. Nicolosi, On the solvability of degenerated quasilinear elliptic equations of higher order, J. Differential Equations, 109 (1994), 325-347.  doi: 10.1006/jdeq.1994.1053.

[9]

Y. R. DuanS. Wang and Y. Y. Zhou, A power penalty approach to a mixed quasilinear elliptic complementarity problem, J. Global Optim., 81 (2021), 901-918.  doi: 10.1007/s10898-021-01000-7.

[10]

M. FrentzK. NyströmA. Pascucci and S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options, Math. Ann., 347 (2010), 805-838.  doi: 10.1007/s00208-009-0456-z.

[11]

F. Giannessi and A. Maugeri, Variational Analysis and Applications, Springer, New York, 2005.

[12]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.

[13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, Inc. New York-London, 1980. 
[14]

J. L. Lions, The work of G. Stampacchia in variational inequalities, Variational Analysis and Applications, Springer, New York, 79 (2005), 3–30. doi: 10.1007/0-387-24276-7_1.

[15]

L. LuZ. H. Liu and X. F. Guo, Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions, Topol. Methods Nonlinear Anal., 55 (2020), 429-449.  doi: 10.12775/tmna.2019.088.

[16]

D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff Publishers, 1978.

[17]

M. Théra, Existence results for the nonlinear complementarity problem and application to nonlinear analysis, J. Math. Anal. Appl., 154 (1991), 572-584.  doi: 10.1016/0022-247X(91)90059-9.

[18] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.  doi: 10.1007/978-1-4899-3614-1.
[19]

S. Wang and C. S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Anal., 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014.

[20]

S. WangX. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from american option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3.

[21]

Y. B. XiaoN. J. Huang and J. Lu, A system of time-dependent hemivariational inequalities with volterra integral terms, J. Optim. Theory Appl., 165 (2015), 837-853.  doi: 10.1007/s10957-014-0602-y.

[22]

K. Zhang and K. L. Teo, Convergence analysis of power penalty method for American bond option pricing, J. Global Optim., 56 (2013), 1313-1323.  doi: 10.1007/s10898-012-9843-1.

[23]

K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435.

[24]

K. ZhangX. Q. Yang and K. L. Teo, A power penalty approach to American option pricing with jump diffusion processes, J. Ind. Manag. Optim., 4 (2008), 783-799.  doi: 10.3934/jimo.2008.4.783.

[25]

Y. Y. Zhou, Some results about perturbed maximal monotone mappings, Comput. Math. Appl., 51 (2006), 487-496.  doi: 10.1016/j.camwa.2005.09.003.

[26]

Y. Y. ZhouS. Wang and X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, J. Global Optim., 60 (2014), 531-550.  doi: 10.1007/s10898-013-0122-6.

[27]

E. Zeilder, Nonlinear Functional Analysis and Its Applications Ⅱ/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

Figure 1.  Computed solution for Example 1 at different t
Figure 2.  The left-hand side one is a figure of the solution uε to the problem (VIP) in Example 5.2, the right-hand side one is a figure of the solution uε and bounds u* and u*
Figure 3.  This is a figure of the solution uun to the unconstrained problem L(u) + Au = f(u), and bounds u* and u*
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