A class of nonlinear delay evolution equations with nonlocal initial conditions
HTML articles powered by AMS MathViewer
- by Monica-Dana Burlică and Daniela Roşu PDF
- Proc. Amer. Math. Soc. 142 (2014), 2445-2458 Request permission
Abstract:
We establish a sufficient condition for the existence, uniqueness and global uniform asymptotic stability of a $C^0$-solution for the nonlinear delay differential evolution equation \begin{equation*}\left \{\begin {array}{ll} \displaystyle u’(t)\in Au(t)+f(t,u_t),&\quad t\in \mathbb {R}_+, \\[1mm] u(t)=g(u)(t),&\quad t\in [ -\tau ,0 ], \end{array}\right .\end{equation*} where $\tau >0$, $X$ is a real Banach space, $A$ is the infinitesimal generator of a nonlinear semigroup of contractions, $f:\mathbb {R}_+\times C([ -\tau ,0 ];\overline {D(A)})\to X$ is continuous and $g:C_b([ -\tau ,+\infty );\overline {D(A)})\to C([ -\tau ,0 ];\overline {D(A)})$ is nonexpansive.References
- S. Aizicovici, N. H. Pavel, and I. I. Vrabie, Anti-periodic solutions to strongly nonlinear evolution equations in Hilbert spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 44 (1998), no. 2, 227–234 (2000). Dedicated to Professor C. Corduneanu on the occasion of his 70th birthday. MR 1783202
- Viorel Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Monographs in Mathematics, Springer, New York, 2010. MR 2582280, DOI 10.1007/978-1-4419-5542-5
- Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan 25 (1973), 565–590. MR 336050, DOI 10.2969/jmsj/02540565
- Ludwik Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), no. 2, 494–505. MR 1137634, DOI 10.1016/0022-247X(91)90164-U
- Radu Caşcaval and Ioan I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mat. Univ. Complut. Madrid 7 (1994), no. 2, 325–338. MR 1297518
- M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357, DOI 10.2307/2373376
- Keng Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), no. 2, 630–637. MR 1249842, DOI 10.1006/jmaa.1993.1373
- Jesús García-Falset and Simeon Reich, Integral solutions to a class of nonlocal evolution equations, Commun. Contemp. Math. 12 (2010), no. 6, 1031–1054. MR 2748284, DOI 10.1142/S021919971000410X
- Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
- Hernán R. Henríquez and Carlos Lizama, Periodic solutions of abstract functional differential equations with infinite delay, Nonlinear Anal. 75 (2012), no. 4, 2016–2023. MR 2870895, DOI 10.1016/j.na.2011.10.002
- Norimichi Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), no. 1, 185–192. MR 1174494, DOI 10.1090/S0002-9939-1994-1174494-8
- Yongxiang Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal. 261 (2011), no. 5, 1309–1324. MR 2807101, DOI 10.1016/j.jfa.2011.05.001
- M. McKibben, Discovering Evolution Equations with Applications. Vol. I. Deterministic Models, Appl. Math. Nonlinear Sci. Ser., Chapman & Hall/CRC, 2011.
- Angela Paicu, Periodic solutions for a class of nonlinear evolution equations in Banach spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 55 (2009), no. 1, 107–118. MR 2510715
- Angela Paicu and Ioan I. Vrabie, A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear Anal. 72 (2010), no. 11, 4091–4100. MR 2606769, DOI 10.1016/j.na.2010.01.041
- Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653–661. MR 1015686, DOI 10.1090/S0002-9939-1990-1015686-4
- I. I. Vrabie, Compactness methods for nonlinear evolutions, 2nd ed., Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. With a foreword by A. Pazy. MR 1375237
- Ioan I. Vrabie, Differential equations, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. An introduction to basic concepts, results and applications. MR 2839525, DOI 10.1142/8044
- Ioan I. Vrabie, Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions, Nonlinear Anal. 74 (2011), no. 18, 7047–7060. MR 2833693, DOI 10.1016/j.na.2011.07.025
- Ioan I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012), no. 4, 1363–1391. MR 2873844, DOI 10.1016/j.jfa.2011.11.006
- Ioan I. Vrabie, Nonlinear retarded evolution equations with nonlocal initial conditions, Dynam. Systems Appl. 21 (2012), no. 2-3, 417–439. MR 2918389
- Ioan I. Vrabie, Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions, Set-Valued Var. Anal. 20 (2012), no. 3, 477–497. MR 2949638, DOI 10.1007/s11228-012-0203-6
Additional Information
- Monica-Dana Burlică
- Affiliation: Department of Mathematics and Informatics, “G. Asachi” Technical University, Bvd. Carol I, no. 11 A, Iaşi, 700506, Romania
- Email: monicaburlica@yahoo.com
- Daniela Roşu
- Affiliation: Department of Mathematics and Informatics, “G. Asachi” Technical University, Bvd. Carol I, no. 11 A, Iaşi, 700506, Romania
- Email: rosudaniela100@yahoo.com
- Received by editor(s): June 14, 2012
- Received by editor(s) in revised form: July 30, 2012
- Published electronically: March 28, 2014
- Additional Notes: This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0052.
- Communicated by: Yingfei Yi
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2445-2458
- MSC (2010): Primary 34K05, 34K13, 34K20, 34K30, 35K55, 35K65, 47H05
- DOI: https://doi.org/10.1090/S0002-9939-2014-11969-1
- MathSciNet review: 3195766