Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
-
Xuping Zhang
,
Pengyu Chen
Abstract
The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.
This work was supported by NNSF of China (Grant No. 11501455), NNSF of China (Grant No. 11661071) and Key Project of Gansu Provincial National Science Foundation (Grant No. 1606RJZA015).
(Communicated by Michal Fečkan)
Acknowledgement
The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading the manuscript and valuable comments and suggestions.
References
[1] Banaś, J.—Goebel, K.: Measures of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math. 60, Marcel Dekker, New York, 1980.Search in Google Scholar
[2] Bao, J.—Hou, Z.—Yuan, C.: Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc. 138 (2010), 2169–2180.10.1090/S0002-9939-10-10230-5Search in Google Scholar
[3] Byszewski, L.: Theorems about the existence and uniquenessof solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494–505.10.1016/0022-247X(91)90164-USearch in Google Scholar
[4] Byszewski, L.: Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal. 33 (1998), 413–426.10.1016/S0362-546X(97)00594-4Search in Google Scholar
[5] Chen, P.—Li, Y.: Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63 (2013), 731–744.10.1007/s00025-012-0230-5Search in Google Scholar
[6] Chen, P.—Li, Y.: Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys. 65 (2014), 711–728.10.1007/s00033-013-0351-zSearch in Google Scholar
[7] Chen, P.—Zhang, X.—Li, Y.: Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math. 14 (2017), Art. 226.10.1007/s00009-017-1029-0Search in Google Scholar
[8] Chen, P.—Zhang, X.—Li, Y.: Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl. 73 (2017), 794–803.10.1016/j.camwa.2017.01.009Search in Google Scholar
[9] Chen, P.—Li, Y.: Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math. 66 (2015), 63–76.10.1007/s13348-014-0106-ySearch in Google Scholar
[10] Chen, P.—Li, Y.—Zhang, X.: On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal. 14 (2015), 1817–1840.10.3934/cpaa.2015.14.1817Search in Google Scholar
[11] Chen, P.—Zhang, X.—Li, Y.: Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calc. Appl. Anal. 19(6) (2016), 1507–1526.10.1515/fca-2016-0078Search in Google Scholar
[12] Cui, J.—Yan, L.—Wu, X.: Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soc. 41 (2012), 279–290.10.1016/j.jkss.2011.10.001Search in Google Scholar
[13] Curtain, R. F.—Falb, P. L.: Stochastic differential equations in Hilbert space, J. Differential Equations 10 (1971), 412–430.10.1016/0022-0396(71)90004-0Search in Google Scholar
[14] Da Prato, G.—Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.10.1017/CBO9780511666223Search in Google Scholar
[15] Deimling, K.: Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.10.1007/978-3-662-00547-7Search in Google Scholar
[16] EI-Borai, M. M.—Mostafa, O. L.—Ahmed, H. M.: Asymptotic stability of some stochastic evolution equations, Appl. Math. Comput. 144 (2003), 273–286.10.1016/S0096-3003(02)00406-XSearch in Google Scholar
[17] Ezzinbi, K.—Fu, X.—Hilal, K.: Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (2007), 1613–1622.10.1016/j.na.2006.08.003Search in Google Scholar
[18] Farahi, S.—Guendouzi, T.: Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results Math. 65 (2014), 501–521.10.1007/s00025-013-0362-2Search in Google Scholar
[19] Grecksch, W.—Tudor, C.: Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.Search in Google Scholar
[20] Liang, J.—Liu, J. H.—Xiao, T. J.: Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal. 57 (2004), 183–189.10.1016/j.na.2004.02.007Search in Google Scholar
[21] Liang, J.—Liu, J. H.—Xiao, T. J.: Nonlocal impulsive problems for integro-differential equations, Math. Comput. Model. 49 (2009), 798–804.10.1016/j.mcm.2008.05.046Search in Google Scholar
[22] Liu, L.—Guo, F.—Wu, C.—Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638–649.10.1016/j.jmaa.2004.10.069Search in Google Scholar
[23] Luo, J.: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl. 342 (2008), 753–760.10.1016/j.jmaa.2007.11.019Search in Google Scholar
[24] McKibben, M.: Discovering Evolution Equations with Applications: Volume 1 – Deterministic Models, Chapman and Hall/CRC Appl. Math. Nonlinear Sci. Ser., 2011.Search in Google Scholar
[25] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983.10.1007/978-1-4612-5561-1Search in Google Scholar
[26] Ren, Y.—Zhou, Q.—Chen, L.: 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), 315–331.10.1007/s10957-010-9792-0Search in Google Scholar
[27] Sakthivel, R.—Ren, Y.—Debbouche, A.—Mahmudov, N. I.: Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal. 95 (2016), 2361–2382.10.1080/00036811.2015.1090562Search in Google Scholar
[28] Shi, H. B.— Li, W. T.— Sun, H. R.: Existence of mild solutions for abstract mixed type semilinear evolution equations, Turkish J. Math. 35 (2011), 457–472.10.3906/mat-0905-29Search in Google Scholar
[29] Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991.Search in Google Scholar
[30] Sun, J.— Zhang, X.: The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sin. 48 (2005), 439–446 (in Chinese).Search in Google Scholar
[31] Taniguchi, T.—Liu, K.—Truman, A.: Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations 181 (2002), 72–91.10.1006/jdeq.2001.4073Search in Google Scholar
[32] Vrabie, I. I.: Existence in the large for nonlinear delay evolution inclutions with nonlocal initial conditions, J. Funct. Anal. 262 (2012), 1363–1391.10.1016/j.jfa.2011.11.006Search in Google Scholar
[33] Vrabie, I. I.: Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math. 17 (2015), 1350035.10.1142/S0219199713500351Search in Google Scholar
[34] Wang, Y.—Liu, L.—Wu, Y.: Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal. 74 (2011), 3599–3605.10.1016/j.na.2011.02.043Search in Google Scholar
[35] Wang, R.–Xiao, T. J.—Liang, J.: A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lett. 24 (2011), 1435–1442.10.1016/j.aml.2011.03.026Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
