-
Previous Article
Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts
- DCDS-B Home
- This Issue
-
Next Article
On discrete-time semi-Markov processes
Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.
References:
[1] |
P. Acquistapace,
Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.
|
[2] |
P. Acquistapace and B. Terreni,
A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107.
|
[3] |
H. Amann,
Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.
doi: 10.1016/0022-0396(88)90156-8. |
[4] |
J. |
[5] |
J. Bao, Z. Hou and C. Yuan,
Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.
doi: 10.1090/S0002-9939-10-10230-5. |
[6] |
P. Chen and Y. Li,
Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.
doi: 10.1007/s00025-012-0230-5. |
[7] |
P. Chen, Y. Li and X. Zhang,
On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.
doi: 10.3934/cpaa.2015.14.1817. |
[8] |
P. Chen, X. Zhang and Y. Li,
Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803.
doi: 10.1016/j.camwa.2017.01.009. |
[9] |
P. Chen, X. Zhang and Y. Li,
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.
doi: 10.3934/cpaa.2018094. |
[10] |
P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2019), 17pp.
doi: 10.1007/s11784-019-0719-6. |
[11] |
P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), 14pp.
doi: 10.1007/s00009-019-1384-0. |
[12] |
P. Chen, X. Zhang and Y. Li,
Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.
doi: 10.1007/s10883-018-9423-x. |
[13] |
P. Chen, X. Zhang and Y. Li,
Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.
doi: 10.1007/s11868-018-0257-9. |
[14] |
R. F. Curtain and P. L. Falb,
Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.
doi: 10.1016/0022-0396(71)90004-0. |
[15] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[16] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[17] |
X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15pp. |
[18] |
X. Fu,
Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.
doi: 10.3934/eect.2017026. |
[19] |
W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Mathematical Research, 85, Akademie-Verlag, Berlin, 1995. |
[20] |
D. J. Guo,
Solutions of nonlinear integrodifferential equations of mixed type in Banach spaces, J. Appl. Math. Simulation, 2 (1989), 1-11.
doi: 10.1155/S1048953389000018. |
[21] |
H. P. Heinz,
On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371.
doi: 10.1016/0362-546X(83)90006-8. |
[22] |
V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, 2, Pergamon Press, Oxford-New York, 1981.
![]() |
[23] |
J. Liang, J. H. Liu and T.-J. Xiao,
Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.
doi: 10.3934/cpaa.2006.5.529. |
[24] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: 10.1201/9781420034820. |
[25] |
L. Liu, F. Guo, C. Wu and Y. Wu,
Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638-649.
doi: 10.1016/j.jmaa.2004.10.069. |
[26] |
L. Liu, C. Wu and F. Guo,
Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13-22.
doi: 10.1016/S0898-1221(04)90002-8. |
[27] |
J. Luo,
Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.
doi: 10.1016/j.jmaa.2007.11.019. |
[28] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, Chichester, 1997. |
[29] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[30] |
Y. Ren, Q. Zhou and L. Chen,
Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.
doi: 10.1007/s10957-010-9792-0. |
[31] |
K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3712-6. |
[32] |
J. X. Sun and X. Y. Zhang,
A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.), 48 (2005), 439-446.
|
[33] |
T. Taniguchi, K. Liu and A. Truman,
Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[34] |
J. Wang,
Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.
doi: 10.1016/j.amc.2014.12.155. |
[35] |
R. N. Wang, K. Ezzinbi and P.-X. Zhu,
Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.
doi: 10.1216/JIE-2014-26-2-275. |
[36] |
R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013) 180–191.
doi: 10.1016/j.na.2013.02.026. |
[37] |
X. Zhang, P. Chen, A. Abdelmonem and Y. Li,
Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.
doi: 10.1080/17442508.2018.1466885. |
[38] |
X. Zhang, P. Chen, A. Abdelmonem and Y. Li,
Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.
doi: 10.1515/ms-2017-0207. |
show all references
References:
[1] |
P. Acquistapace,
Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.
|
[2] |
P. Acquistapace and B. Terreni,
A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107.
|
[3] |
H. Amann,
Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.
doi: 10.1016/0022-0396(88)90156-8. |
[4] |
J. |
[5] |
J. Bao, Z. Hou and C. Yuan,
Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.
doi: 10.1090/S0002-9939-10-10230-5. |
[6] |
P. Chen and Y. Li,
Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.
doi: 10.1007/s00025-012-0230-5. |
[7] |
P. Chen, Y. Li and X. Zhang,
On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.
doi: 10.3934/cpaa.2015.14.1817. |
[8] |
P. Chen, X. Zhang and Y. Li,
Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803.
doi: 10.1016/j.camwa.2017.01.009. |
[9] |
P. Chen, X. Zhang and Y. Li,
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.
doi: 10.3934/cpaa.2018094. |
[10] |
P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2019), 17pp.
doi: 10.1007/s11784-019-0719-6. |
[11] |
P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), 14pp.
doi: 10.1007/s00009-019-1384-0. |
[12] |
P. Chen, X. Zhang and Y. Li,
Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.
doi: 10.1007/s10883-018-9423-x. |
[13] |
P. Chen, X. Zhang and Y. Li,
Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.
doi: 10.1007/s11868-018-0257-9. |
[14] |
R. F. Curtain and P. L. Falb,
Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.
doi: 10.1016/0022-0396(71)90004-0. |
[15] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[16] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[17] |
X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15pp. |
[18] |
X. Fu,
Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.
doi: 10.3934/eect.2017026. |
[19] |
W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Mathematical Research, 85, Akademie-Verlag, Berlin, 1995. |
[20] |
D. J. Guo,
Solutions of nonlinear integrodifferential equations of mixed type in Banach spaces, J. Appl. Math. Simulation, 2 (1989), 1-11.
doi: 10.1155/S1048953389000018. |
[21] |
H. P. Heinz,
On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371.
doi: 10.1016/0362-546X(83)90006-8. |
[22] |
V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, 2, Pergamon Press, Oxford-New York, 1981.
![]() |
[23] |
J. Liang, J. H. Liu and T.-J. Xiao,
Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.
doi: 10.3934/cpaa.2006.5.529. |
[24] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: 10.1201/9781420034820. |
[25] |
L. Liu, F. Guo, C. Wu and Y. Wu,
Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638-649.
doi: 10.1016/j.jmaa.2004.10.069. |
[26] |
L. Liu, C. Wu and F. Guo,
Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13-22.
doi: 10.1016/S0898-1221(04)90002-8. |
[27] |
J. Luo,
Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.
doi: 10.1016/j.jmaa.2007.11.019. |
[28] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, Chichester, 1997. |
[29] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[30] |
Y. Ren, Q. Zhou and L. Chen,
Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.
doi: 10.1007/s10957-010-9792-0. |
[31] |
K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3712-6. |
[32] |
J. X. Sun and X. Y. Zhang,
A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.), 48 (2005), 439-446.
|
[33] |
T. Taniguchi, K. Liu and A. Truman,
Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[34] |
J. Wang,
Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.
doi: 10.1016/j.amc.2014.12.155. |
[35] |
R. N. Wang, K. Ezzinbi and P.-X. Zhu,
Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.
doi: 10.1216/JIE-2014-26-2-275. |
[36] |
R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013) 180–191.
doi: 10.1016/j.na.2013.02.026. |
[37] |
X. Zhang, P. Chen, A. Abdelmonem and Y. Li,
Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.
doi: 10.1080/17442508.2018.1466885. |
[38] |
X. Zhang, P. Chen, A. Abdelmonem and Y. Li,
Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.
doi: 10.1515/ms-2017-0207. |
[1] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[2] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[3] |
Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 |
[4] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[5] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[6] |
Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021004 |
[7] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[8] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[9] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[10] |
Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 |
[11] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[12] |
Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021040 |
[13] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
[14] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[15] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[16] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[17] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[18] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[19] |
Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 |
[20] |
Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]