• Previous Article
    Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity
  • PROC Home
  • This Issue
  • Next Article
    Optimal control of a linear stochastic Schrödinger equation
2013, 2013(special): 447-456. doi: 10.3934/proc.2013.2013.447

Quasi-subdifferential operators and evolution equations

1. 

Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555

Received  August 2012 Revised  December 2012 Published  November 2013

We introduce the concept of a quasi-subdifferential operator and that of a quasi-subdifferential evolution equation. We prove the existence of solutions to related problems and give applications to variational and quasi-variational inequalities.
Citation: Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447
References:
[1]

T. Aiki, Mathematical models including a hysteresis operator,, in, 71 (2006), 1.   Google Scholar

[2]

H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite,, Ann. Mat. Pura Appl. 120 (1979), 120 (1979), 35.   Google Scholar

[3]

H. Attouch, P. Bénilan, A. Damlamian, C. Picard, Equations d'évolution avec condition unilatérale,, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 607.   Google Scholar

[4]

C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities",, Wiley-Interscience, (1984).   Google Scholar

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité,, Ann. Inst. Fourier, 18 (1968), 115.   Google Scholar

[6]

H. Brézis, Problèmes unilatéraux,, J. Math. Pure Appl. IX. Ser., 51 (1972), 1.   Google Scholar

[7]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert",, North-Holland, (1973).   Google Scholar

[8]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Funct. Anal., 11 (1972), 251.   Google Scholar

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint,, J. Math. Anal. Appl., 256 (2001), 668.   Google Scholar

[10]

J.-L. Joly and U. Mosco, À propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles,, J. Funct. Anal., 34 (1979), 107.   Google Scholar

[11]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (2009), 583.   Google Scholar

[12]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints,, Adv. Math. Sci. Appl., 19 (2009), 565.   Google Scholar

[13]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175.   Google Scholar

[14]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304.   Google Scholar

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Educ., 30 (1981), 1.   Google Scholar

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in, (2007).   Google Scholar

[17]

N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities,, Nonlinear Anal., 34 (1998), 665.   Google Scholar

[18]

N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media,, in ''Free Boundary Value Problems, 95 (1990), 127.   Google Scholar

[19]

M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials,, Funkc. Ekvacioj, 32 (1989), 301.   Google Scholar

[20]

M. Kubo, A filtration model with hysteresis,, J. Differ. Equations, 201 (2004), 75.   Google Scholar

[21]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints,, Adv. Math. Sci. Appl., 15 (2005), 60.   Google Scholar

[22]

M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints,, Discrete Contin. Dyn. Syst., 19 (2007), 335.   Google Scholar

[23]

M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations,, J. Math. Anal. Appl., 387 (2012), 490.   Google Scholar

[24]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators , J. Differential Equations, 46 (1982), 268.   Google Scholar

[25]

M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383.   Google Scholar

[26]

A. Visintin, "Differential models of hysteresis",, Springer-Verlag, (1994).   Google Scholar

[27]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete Contin. Dyn. Syst., 2005 (2005), 920.   Google Scholar

[28]

Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci., 23 (1976), 491.   Google Scholar

show all references

References:
[1]

T. Aiki, Mathematical models including a hysteresis operator,, in, 71 (2006), 1.   Google Scholar

[2]

H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite,, Ann. Mat. Pura Appl. 120 (1979), 120 (1979), 35.   Google Scholar

[3]

H. Attouch, P. Bénilan, A. Damlamian, C. Picard, Equations d'évolution avec condition unilatérale,, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 607.   Google Scholar

[4]

C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities",, Wiley-Interscience, (1984).   Google Scholar

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité,, Ann. Inst. Fourier, 18 (1968), 115.   Google Scholar

[6]

H. Brézis, Problèmes unilatéraux,, J. Math. Pure Appl. IX. Ser., 51 (1972), 1.   Google Scholar

[7]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert",, North-Holland, (1973).   Google Scholar

[8]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Funct. Anal., 11 (1972), 251.   Google Scholar

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint,, J. Math. Anal. Appl., 256 (2001), 668.   Google Scholar

[10]

J.-L. Joly and U. Mosco, À propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles,, J. Funct. Anal., 34 (1979), 107.   Google Scholar

[11]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (2009), 583.   Google Scholar

[12]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints,, Adv. Math. Sci. Appl., 19 (2009), 565.   Google Scholar

[13]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175.   Google Scholar

[14]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304.   Google Scholar

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Educ., 30 (1981), 1.   Google Scholar

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in, (2007).   Google Scholar

[17]

N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities,, Nonlinear Anal., 34 (1998), 665.   Google Scholar

[18]

N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media,, in ''Free Boundary Value Problems, 95 (1990), 127.   Google Scholar

[19]

M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials,, Funkc. Ekvacioj, 32 (1989), 301.   Google Scholar

[20]

M. Kubo, A filtration model with hysteresis,, J. Differ. Equations, 201 (2004), 75.   Google Scholar

[21]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints,, Adv. Math. Sci. Appl., 15 (2005), 60.   Google Scholar

[22]

M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints,, Discrete Contin. Dyn. Syst., 19 (2007), 335.   Google Scholar

[23]

M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations,, J. Math. Anal. Appl., 387 (2012), 490.   Google Scholar

[24]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators , J. Differential Equations, 46 (1982), 268.   Google Scholar

[25]

M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383.   Google Scholar

[26]

A. Visintin, "Differential models of hysteresis",, Springer-Verlag, (1994).   Google Scholar

[27]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete Contin. Dyn. Syst., 2005 (2005), 920.   Google Scholar

[28]

Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci., 23 (1976), 491.   Google Scholar

[1]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046

[2]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[3]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[4]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360

[5]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

[6]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

[7]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[8]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[9]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[10]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[11]

Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021006

[12]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[13]

Gernot Holler, Karl Kunisch. Learning nonlocal regularization operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021003

[14]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[15]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[16]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[17]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304

[18]

Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394

[19]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[20]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

 Impact Factor: 

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]