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Existence, Uniqueness and Qualitative Properties of Global Solutions of Abstract Differential Equations with State-Dependent Delay

Part of: Miscellaneous topics - Partial differential equations Groups and semigroups of linear operators, their generalizations and applications Functional-differential and differential-difference equations

Published online by Cambridge University Press:  30 January 2019

Eduardo Hernández  and
Jianhong Wu
Eduardo Hernández
Affiliation:
Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto Universidade de São Paulo, CEP 14040-901 Ribeirão Preto, SP, Brazil ([email protected])
Jianhong Wu
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada ([email protected])
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Abstract

We study the existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Results on the existence of almost periodic-type solutions (including, periodic, almost periodic, asymptotically almost periodic and almost automorphic solutions) are proved. Some examples of partial differential equations with state-dependent delay arising in population dynamics are presented.

Keywords

state-dependent delay equationsglobal solutionsstrict solutionsanalytic semigrouppartial differential equations

MSC classification

Primary: 34K30: Equations in abstract spaces
Secondary: 35R10: Partial functional-differential equations 47D06: One-parameter semigroups and linear evolution equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 771 - 788
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1.Aiello, W., Freedman, H. I. and Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math. 52(3) (1992), 855–869.Google Scholar
2.Andrade, F., Cuevas, C. and Henríquez, H., Periodic solutions of abstract functional differential equations with state-dependent delay, Math. Methods Appl. Sci. 39(13) (2016), 3897–3909.Google Scholar
3.Bartha, M., Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Anal. 53(6) (2003), 839–857.Google Scholar
4.Büger, M. and Martin, M., The escaping disaster: a problem related to state-dependent delays, Z. Angew. Math. Phys. 55(4) (2004), 547–574.Google Scholar
5.Chueshov, I. and Rezounenko, A., Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Anal. Theory Methods Appl. 123 (2015), 126–149.Google Scholar
6.Chueshov, I. and Rezounenko, A., Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pure Appl. Anal. 14(5) (2015), 1685–1704.Google Scholar
7.Driver, R. D., A functional-differential system of neutral type arising in a two-body problem of classical electrodynamics, International symposium on nonlinear differential equations and nonlinear mechanics (eds. LaSalle, J. and Lefschtz, S.), pp. 474–484 (Academic Press, New York, 1963).Google Scholar
8.Driver, R. D., A neutral system with state-dependent delay, J. Differ. Equ. 54 (1984), 73–86.Google Scholar
9.Hartung, F., Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differ. Equ. 23(4) (2011), 843–884.Google Scholar
10.Hartung, F., On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, Ann. Mat. Pura Appl. (4) 192(1) (2013), 17–47.Google Scholar
11.Hartung, F. and Turi, J., On differentiability of solutions with respect to parameters in state-dependent delay equations, J. Differ. Equ. 135(2) (1997), 192–237.Google Scholar
12.Hartung, F., Krisztin, T., Walther, H-O. and Wu, J., Functional differential equations with state-dependent delays: theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, Volume III, pp. 435–545 (Elsevier, 2006).Google Scholar
13.Hernandez, E. and Pelicer, M., Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations, Appl. Math. Lett. 18(11) (2005), 1265–1272.Google Scholar
14.Hernández, E., Prokopczyk, A. and Ladeira, L., A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real World Appl. 7(4) (2006), 510–519.Google Scholar
15.Hernández, E., Henríquez, H. and Diagana, T., Almost automorphic mild solutions to some partial neutral functional differential equations and applications, Nonlinear Anal. 69 (2008), 1485–1493.Google Scholar
16.Hernandez, E., Pierri, M. and Wu, J., C1 + α-strict solutions and wellposedness of abstract differential equations with state dependent delay, J. Differ. Equ. 261(12) (2016), 6856–6882.Google Scholar
17.Hino, Y. and Murakami, S., Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl 286 (2003), 741–752.Google Scholar
18.Hutchinson, G. E., Circular causal systems in ecology, Ann. N. Y. Acad. Sci. 50(4) (1948), 221–246.Google Scholar
19.Kennedy, B., Multiple periodic solutions of an equation with state-dependent delay, J. Dynam. Differ. Equ. 23(2) (2011), 283–313.Google Scholar
20.Kosovalic, N., Magpantay, F. M. G., Chen, Y. and Wu, J., Abstract algebraic-delay differential systems and age structured population dynamics, J. Differ. Equ. 255(3) (2013), 593–609.Google Scholar
21.Kosovalic, N., Chen, Y. and Wu, J., Algebraic-delay differential systems: C 0-extendable submanifolds and linearization, Trans. Amer. Math. Soc. 369(5) (2017), 3387–3419.Google Scholar
22.Krisztin, T. and Rezounenkob, A., Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold, J. Differ. Equ. 260(5) (2016), 4454–4472.Google Scholar
23.Li, Y. and Kuang, Y., Periodic solutions in periodic state-dependent delay equations and population models, Proc. Amer. Math. Soc. 130(5) (2002), 1345–1353.Google Scholar
24.Li, X. and Li, Z., Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay, Commun. Pure Appl. Anal. 10(2) (2011), 687–700.Google Scholar
25.Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and Their Applications, Volume 16 (Birkhäauser Verlag, Basel, 1995).Google Scholar
26.Lv, Y, Rong, Y. and Yongzhen, P., Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equ. 260 (2016), 6201–6231.Google Scholar
27.Mackey, M. C. and Glass, L., Oscillation and chaos in physiological control systems, Science 197 (1977), 287–289.Google Scholar
28.Magpantay, F. M. G., Kosovalic, N. and Wu, J., An age-structured population model with state-dependent delay: derivation and numerical integration, SIAM J. Numer. Anal. 52(2) (2014), 735–756.Google Scholar
29.Mallet-Paret, J. and Nussbaum, R. D., Stability of periodic solutions of state-dependent delay-differential equations, J. Differ. Equ. 250(11) (2011), 4085–4103.Google Scholar
30.Murray, J. D., Mathematical biology. I. An introduction, 3rd edn. Interdisciplinary Applied Mathematics, Volume 17 (Springer-Verlag, New York, 2002).Google Scholar
31.Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Volume 44 (Springer-Verlag, New York–Berlin, 1983).Google Scholar
32.Rezounenko, A., Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70(11) (2009), 3978–3986.Google Scholar
33.Rezounenko, A., Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal. 73(6) (2010), 1707–171.Google Scholar
34.Rezounenko, A., A condition on delay for differential equations with discrete state-dependent delay, J. Math. Anal. Appl. 385(1) (2012), 506–516.Google Scholar
35.Rezounenko, A. and Wu, J., A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, J. Comput. Appl. Math. 190(1–2) (2006), 99–113.Google Scholar
36.So, J. W. H. and Yang, Y., Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differ. Equ. 150(2) (1998), 317–348.Google Scholar
37.So, J. W. H., Wu, J. and Yang, Y., Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput. 111(1) (2000), 33–51.Google Scholar
38.Torrejón, R., Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. 20(12) (1993), 1383–1416.Google Scholar
39.Walther, H-O., The solution manifold and C 1-smoothness for differential equations with state-dependent delay, J. Differ. Equ. 195(1) (2003), 46–65.Google Scholar
40.Walther, H-O., A periodic solution of a differential equation with state-dependent delay, J. Differ. Equ. 244(8) (2008), 1910–1945.Google Scholar
41.Wang, X. and Li, Z., Dynamics for a type of general reaction-diffusion model, Nonlinear Anal. 67(9) (2007), 2699–2711.Google Scholar
42.Wu, J. and Zou, X., Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differ. Equ. 13(3) (2001), 651–687.Google Scholar
43.Yoshizawa, T., Stability theory and the existence of periodic solutions and almost periodic solutions, Applied Mathematical Sciences, Volume 14 (Springer-Verlag, New York–Heidelberg, 1975).Google Scholar
44.Zaidman, S., A non linear abstract differential equation with almost-periodic solution, Riv. Mat. Univ. Parma 4 (1984), 331–336.Google Scholar
45.Zaidman, S., Almost-periodic functions in abstract spaces, Pitman Research Notes in Mathematics, Volume 126 (Advanced Publishing Program, Boston, MA, 1985).Google Scholar