Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Yejuan Wang; Lin Yang

doi:10.3934/dcdsb.2018257

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2019, Volume 24, Issue 4: 1961-1987. Doi: 10.3934/dcdsb.2018257

This issue Previous Article Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics Next Article Well-posedness and numerical algorithm for the tempered fractional differential equations

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Yejuan Wang^, and
Lin Yang

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

* Corresponding author
* Corresponding author

Received: April 30, 2017

Revised: April 30, 2018

Early access: October 2018

Published: January 2019

This work was supported by NSF of China (Grants No. 41875084, 11571153), and the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2016-100 and lzujbky-2018- it58.

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We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

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Mathematics Subject Classification: 35B40, 35B41; Secondary: 35K57, 37L60.

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