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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A class of nonlinear delay evolution equations with nonlocal initial conditions


Authors: Monica-Dana Burlică and Daniela Roşu
Journal: Proc. Amer. Math. Soc. 142 (2014), 2445-2458
MSC (2010): Primary 34K05, 34K13, 34K20, 34K30, 35K55, 35K65, 47H05
Published electronically: March 28, 2014
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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

$\displaystyle \left \{\begin {array}{ll}\displaystyle u'(t)\in Au(t)+f(t,u_t),&... ...hbb{R}_+, \\ [1mm] u(t)=g(u)(t),&\quad t\in [\,-\tau ,0\,], \end{array}\right .$    

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.

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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11969-1
PII: S 0002-9939(2014)11969-1
Keywords: Differential delay evolution equation, nonlocal delay initial condition, periodic solutions, metric fixed point arguments, nonresonance condition, nonlinear parabolic equations.
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.