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doi: 10.3934/dcdss.2022108
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## On solution manifolds of differential systems with discrete state-dependent delays

 Universität Gießen, Mathematisches Institut, Arndtstr. 2, 35392 Gießen, Germany

For Jibin Li, on the occasion of his 80th birthday

Received  December 2021 Early access May 2022

For differential equations with state-dependent delays the associated initial value problem is well-posed, with differentiable solution operators, on submanifolds of the space $C^1_n=C^1([-r,0],\mathbb{R}^n)$, under mild smoothness assumptions. We study these solution manifolds and find that for a large class of equations their solution manifolds are nearly as simple as a graph over the subspace $X_0\subset C^1_n$ defined by $\phi'(0)=0$. The result supplements recent work on finite atlases of solution manifolds and is related to the open problem whether in some cases the constructed atlases are minimal.

Citation: Hans-Otto Walther. On solution manifolds of differential systems with discrete state-dependent delays. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022108
##### References:
 [1] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [2] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [3] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In Handbook of Differential Equations, Ordinary Differential Equations, (eds. A. Cañada, P. Drábek, and A Fonda), Elsevier, 3 (2006), 435–545. doi: 10.1016/S1874-5725(06)80009-X. [4] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topological Meth. Nonlinear Anal., 3 (1994), 101-162.  doi: 10.12775/TMNA.1994.006. [5] H. O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state dependent delay, J. Dif. Eqs., 195 (2003), 46-65.  doi: 10.1016/j.jde.2003.07.001. [6] H. O. Walther, Solution manifolds which are almost graphs, J. Dif. Eqs., 293 (2021), 226-248.  doi: 10.1016/j.jde.2021.05.024. [7] H. O. Walther, A finite atlas for solution manifolds of differential systems with discrete state-dependent delays, Dif. Int. Eqs., 35 (2022), 241-276. [8] E. Winston, Uniqueness of solutions of state dependent delay differential equations, J. Math. An. Appl., 47 (1974), 620-625.  doi: 10.1016/0022-247X(74)90013-4.

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##### References:
 [1] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [2] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [3] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In Handbook of Differential Equations, Ordinary Differential Equations, (eds. A. Cañada, P. Drábek, and A Fonda), Elsevier, 3 (2006), 435–545. doi: 10.1016/S1874-5725(06)80009-X. [4] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topological Meth. Nonlinear Anal., 3 (1994), 101-162.  doi: 10.12775/TMNA.1994.006. [5] H. O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state dependent delay, J. Dif. Eqs., 195 (2003), 46-65.  doi: 10.1016/j.jde.2003.07.001. [6] H. O. Walther, Solution manifolds which are almost graphs, J. Dif. Eqs., 293 (2021), 226-248.  doi: 10.1016/j.jde.2021.05.024. [7] H. O. Walther, A finite atlas for solution manifolds of differential systems with discrete state-dependent delays, Dif. Int. Eqs., 35 (2022), 241-276. [8] E. Winston, Uniqueness of solutions of state dependent delay differential equations, J. Math. An. Appl., 47 (1974), 620-625.  doi: 10.1016/0022-247X(74)90013-4.
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