Hernández M., Eduardo; McKibben, Mark A. On state-dependent delay partial neutral functional–differential equations. (English) Zbl 1119.35106 Appl. Math. Comput. 186, No. 1, 294-301 (2007). Summary: The purpose of this article is establish the existence of mild solutions for a class of abstract neutral functional-differential equations with state-dependent delay described by the form \[ \frac{d}{dt}D(u_t)=AD(u_t)+F (t,x_{\rho(t,x_t)}), \quad t\in I=[0,a],\tag{1} \]\[ x_0=\varphi\in{\mathcal B}, \tag{2} \] where \(A\) is the infinitesimal generator of a compact \(C_0\)-semigroup of bounded linear operators \((T(t))_{t\geq 0}\) on a Banach space \(X\); the function \(x_s:(-\infty,0]\to X\), \(x_s(\theta)=x(s+ \theta)\), belongs to some abstract phase space \({\mathcal B}\) described axiomatically; \(F,G\) are appropriate functions; and \(D\psi=\psi(0)-G(t, \psi)\), where \(\psi\) is in \({\mathcal B}\). Cited in 27 Documents MSC: 35R10 Partial functional-differential equations 47D03 Groups and semigroups of linear operators Keywords:abstract Cauchy problem; neutral equations; state-dependent delay; semigroup of linear operators; unbounded delay PDFBibTeX XMLCite \textit{E. Hernández M.} and \textit{M. A. McKibben}, Appl. Math. Comput. 186, No. 1, 294--301 (2007; Zbl 1119.35106) Full Text: DOI References: [1] Arino, Ovide; Boushaba, Khalid; Boussouar, Ahmed, A mathematical model of the dynamics of the phytoplankton-nutrient system: spatial heterogeneity in ecological models (Alcalá de Henares, 1998), Nonlinear Anal. RWA, 1, 1, 69-87 (2000) · Zbl 0984.92032 [2] Aiello, Walter; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 3, 855-869 (1992) · Zbl 0760.92018 [3] Bartha, Maria, Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Anal. TMA, 53, 6, 839-857 (2003) · Zbl 1028.34062 [4] Cao, Yulin; Fan, Jiangping; Gard, Thomas C., The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear Anal. TMA, 19, 2, 95-105 (1992) · Zbl 0777.92014 [5] Alexander, Domoshnitsky; Drakhlin, Michael; Litsyn, Elena, On equations with delay depending on solution, Nonlinear Anal. TMA, 49, 5, 689-701 (2002) · Zbl 1012.34066 [6] Li, Yongkun; Zhu, Lifei, Positive periodic solutions for a class of higher-dimensional state-dependent delay functional differential equations with feedback control, Appl. Math. Comput., 159, 3, 783-795 (2004) · Zbl 1161.34346 [7] Zaghrout, A. A.S.; Attalah, S. H., Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay, Appl. Math. Comput., 77, 2-3, 185-194 (1996) · Zbl 0848.92017 [8] Hartung, Ferenc, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study, Proceedings of the Third World Congress of Nonlinear Analysis, Part 7 (Catania, 2000). Proceedings of the Third World Congress of Nonlinear Analysis, Part 7 (Catania, 2000), Nonlinear Anal. TMA, 47, 7, 4557-4566 (2001) · Zbl 1042.34582 [9] Hartung, Ferenc; Herdman, Terry L.; Turi, Janos, Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear Anal. TMA, 39, 3, 305-325 (2000) · Zbl 0955.34067 [10] Hartung, Ferenc; Turi, Janos, Identification of parameters in delay equations with state-dependent delays, Nonlinear Anal. TMA, 29, 11, 1303-1318 (1997) · Zbl 0894.34071 [11] Kuang, Y.; Smith, H. L., Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Anal. TMA, 19, 9, 855-872 (1992) · Zbl 0774.34054 [12] Driver, R. D., A neutral system with state-dependent delay, J. Differen. Equat., 54, 1, 73-86 (1984) · Zbl 0543.34053 [13] Yang, Zhihui; Cao, Jinde, Existence of periodic solutions in neutral state-dependent delays equations and models, J. Comput. Appl. Math., 174, 1, 179-199 (2005) · Zbl 1107.34059 [14] Hernández, E.; Prokopczyk, A.; Ladeira, Luiz, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7, 4, 510-519 (2006) · Zbl 1109.34060 [15] Rezounenko, A. V.; Wu, Jianhong, A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, J. Comput. Appl. Math., 190, 1-2, 99-113 (2006) · Zbl 1082.92039 [16] Hale, J. K.; Lunel, Verduyn, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99 (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002 [17] Adimy, M.; Ezzinbi, K., A class of linear partial neutral functional-differential equations with nondense domain, J. Differen. Equat., 147, 2, 285-332 (1998) · Zbl 0915.35109 [18] Hale, J. K., Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39, 4, 339-344 (1994) · Zbl 0817.35119 [19] Wu, J.; Xia, H., Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differen. Equat., 124, 1, 247-278 (1996) · Zbl 0840.34080 [20] Wu, Jianhong, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, vol. 119 (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0870.35116 [21] Hernández, E.; Henríquez, H., Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 452-475 (1998) · Zbl 0915.35110 [22] Hernández, E.; Henríquez, H., Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221, 2, 499-522 (1998) · Zbl 0926.35151 [23] Hernández, E., Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292, 1, 194-210 (2004) · Zbl 1056.45012 [24] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44 (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023 [25] Hino, Yoshiyuki; Murakami, Satoru; Naito, Toshiki, Functional-differential equations with infinite delay, Lecture Notes in Mathematics, vol. 1473 (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0732.34051 [26] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1025.47002 [27] Martin, R. H., Nonlinear Operators and Differential Equations in Banach Spaces (1987), Robert E. Krieger Publ. Co.: Robert E. Krieger Publ. Co. Florida 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.