Louihi, M.; Hbid, M. L.; Arino, O. Semigroup properties and the Crandall Liggett approximation for a class of differential equations with state-dependent delays. (English) Zbl 1006.34061 J. Differ. Equations 181, No. 1, 1-30 (2002). The following differential equation with state-dependent delays is considered \[ x'(t)= f(x(t- r(x_t))),\quad t\geq 0,\tag{1} \] with the initial condition \[ x_0= \varphi.\tag{2} \] It is assumed that \(f: \mathbb{R}\to \mathbb{R}\) is Lipschitzian, \(r\) is a functional acting from the function space \(C= C([- M,0],\mathbb{R})\) into \([0,M]\) and by \(x_t\) we denote the function from the space \(C\) defined by the equatility \(x_t(\Theta)= x(t+\Theta)\) for \(\Theta\in [-M,0]\).Here, it is shown that problem (1)–(2) generates a strongly continuous semigroup in a closed subset of the space \(C_{0,1}\) consisting of functions \(x: [-M,0]\to \mathbb{R}\) being Lipschitz continuous. The infinitesimal generator of that semigroup is characterized in terms of its domain. Moreover, an approximation of the Crandall-Liggett type of that semigroup is obtained in a dense subset of \(C\). Reviewer: J.Banaś (Rzeszów) Cited in 13 Documents MSC: 34K05 General theory of functional-differential equations Keywords:Crandall Liggett approximation; state-dependent delays PDFBibTeX XMLCite \textit{M. Louihi} et al., J. Differ. Equations 181, No. 1, 1--30 (2002; Zbl 1006.34061) Full Text: DOI References: [1] Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 855-869 (1992) · Zbl 0760.92018 [2] Arino, O.; Hbid, M. L.; Bravo, R., A mathematical model of population of fish in the larval stage: Density-dependence effects, Math. Biosc., 150, 1-20 (1998) · Zbl 0938.92028 [3] Arino, O.; Hadeler, K. P.; Hbid, M. L., Existence of periodic solutions for delay differential equation with state-depending delay, J. Differential Equations, 144, 263-301 (1998) · Zbl 0913.34057 [4] Bélair, J., Population models with state-dependent delays, (Arino, O.; Axelrod, D.; Kimmel, M., Mathematical Population Dynamics. Mathematical Population Dynamics, Lecture Notes in Pure and Appl. Math., 131 (1991), Dekker: Dekker New York), 165-177 · Zbl 0749.92014 [5] Crandall, M. G.; Liggett, T. M., Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93, 265-298 (1971) · Zbl 0226.47038 [6] Dyson, J.; Villella-Bressan, R., A semigroup approach to non-autonomous neutral functional differential equations, J. Differential Equations, 59, 206-228 (1985) · Zbl 0564.34067 [7] Kuang, Y.; Smith, H. L., Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Anal., 19, 872-885 (1992) · Zbl 0774.34054 [8] Mallet-Paret, J.; Nussbaum, R. D.; Paraskevopoulos, P., Periodic solutions for functional differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis (1994), J. Juliusz Schauder Center, p. 101-162 · Zbl 0808.34080 [9] Neerven, J. V., The Asymptotic Behavior of a Semigroup of Linear Operators (1996), Birkhäuser: Birkhäuser Basel [10] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1987), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0516.47023 [11] Plant, A. T., Fluid Mechanics Research Institute, Report (1976) [12] Sidki, O., Une Approche par la Théorie des Semigroupes Non Linéaire de la Résolution d’une Classe d’Equations Différentielles Fonctionnelles de Type Neutre. Application à une Equation de dynamique de Population (1994), Université de Pau [13] Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics. Theory of Nonlinear Age-Dependent Population Dynamics, Pure and Applied Mathematics, 89 (1985), Dekker: Dekker New York · Zbl 0555.92014 [14] Webb, G. F., Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl., 46, 1-12 (1974) · Zbl 0277.34070 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.