# American Institute of Mathematical Sciences

April  2019, 24(4): 1961-1987. doi: 10.3934/dcdsb.2018257

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

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

* Corresponding author

Dedicated to the memory of professor V.S. Mel'nik

Received  May 2017 Revised  May 2018 Published  October 2018

Fund Project: 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.

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.

Citation: Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
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