2012, 11(6): 2351-2369. doi: 10.3934/cpaa.2012.11.2351

Flow invariance for nonautonomous nonlinear partial differential delay equations

1. 

Department of Mathematics, Razi University, Kermanshah, Iran

2. 

Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany

Received  March 2011 Revised  May 2011 Published  April 2012

Several fundamental results on existence and flow-invariance of solutions to the nonlinear nonautonomous partial differential delay equation $ \dot{u}(t) + B(t)u(t) \ni F(t; u_t), 0 \leq s \leq t, u_s = \varphi, $ with $ B(t)\subset X\times X$ $\omega-$accretive, are developed for a general Banach space $X.$ In contrast to existing results, with the history-response $F(t;\cdot)$ globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with $F(t;\cdot)$ defined on -- possibly -- thin subsets of the initial-history space $E$ only, and are applied to place several classes of population models in their natural $L^1-$setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations with $B(\cdot)\equiv 0,$ and (c) the semilinear case.
Citation: Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
References:
[1]

H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems,, J. Math. Anal. Appl., 65 (1978), 432. doi: 10.1016/0022-247X(78)90192-0.

[2]

P. Bénilan and M. G. Crandall, Completely accretive operators,, In, (1991), 41.

[3]

P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators,, Monograph, ().

[4]

D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Equ., 3 (2003), 375. doi: 10.1007/s00028-003-0099-5.

[5]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions,, J. Evol. Equ., 5 (2005), 227. doi: 10.1007/s00028-005-0185-z.

[6]

D. W. Brewer, A nonlinear semigroup for a functional differential equation,, Trans. Amer. Math. Soc., 236 (1978), 173. doi: 10.1090/S0002-9947-1978-0466838-2.

[7]

D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups,, Illinois J. Math., 26 (1982), 374.

[8]

M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. doi: 10.1007/BF02761448.

[9]

J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223.

[10]

J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations,, Proc. Royal Soc. Edinburgh, 82A (1979), 171. doi: 10.1017/S030821050001115X.

[11]

J. Dyson and R. Villella-Bressan, Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions,, Nonlinear Diff. Eqns. Appl., 3 (1996), 127. doi: 10.1007/BF01194220.

[12]

L. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. doi: 10.1007/BF03007654.

[13]

S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations,, J. Math. Anal. Appl., 345 (2008), 854. doi: 10.1016/j.jmaa.2008.04.041.

[14]

S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations,, in preparation., ().

[15]

J. K. Hale, Functional differential equations with infinite delays,, J. Math. Anal. Appl., 48 (1974), 276. doi: 10.1016/0022-247X(74)90233-9.

[16]

J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems,, J. Math. Anal. Appl., 118 (1986), 455. doi: 10.1016/0022-247X(86)90273-8.

[17]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.

[18]

A. G. Kartsatos and M. E. Parrott, Global solutions of functional evolution equations involving locally defined Lipschitzian perturbations,, J. London Math. Soc., 27 (1983), 306. doi: 10.1112/jlms/s2-27.2.306.

[19]

A. G. Kartsatos and M. E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations,, J. Diff. Eqns., 47 (1983), 358. doi: 10.1016/0022-0396(83)90041-4.

[20]

A. G. Kartsatos and M. E. Parrott, The weak solution of a functional differential equation in a general Banach space,, J. Diff. Eqns., 75 (1988), 290. doi: 10.1016/0022-0396(88)90140-4.

[21]

V. Lakshmikhantam, S. Leela and V. Moauro, Existence and uniqueness of solutions of delay differential equations on a closed subset of a Banach space,, Nonlinear Analysis TMA, 2 (1978), 311. doi: 10.1016/0362-546X(78)90020-2.

[22]

S. Leela and V. Moauro, Existence of solutions in a closed set for delay differential equations in Banach space,, Nonlinear Analysis TMA, 2 (1978), 47. doi: 10.1016/0362-546X(78)90040-8.

[23]

J. H. Lightbourne III, Function space flow-invariance for functional differential equations of retarded type,, Proc. Amer. Math. Soc., 77 (1979), 91. doi: 10.1090/S0002-9939-1979-0539637-7.

[24]

R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", Wiley, (1976).

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans, 321 (1990), 1. doi: 10.2307/2001590.

[26]

R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay,, in, (1991), 259.

[27]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,, J. reine angew. Math., 413 (1991), 1.

[28]

I. Miyadera, "Nonlinear Semigroups,", Transl. of Math. Monographs 109, (1992).

[29]

S. Murakami, Stable equilibrium point of some diffusive functional differential equations,, Nonlinear Analysis TMA, 25 (1995), 1037. doi: 10.1016/0362-546X(95)00097-F.

[30]

M. E. Parrott, Representation and approximation of generalized solutions of a nonlinear functional differential equation,, Nonlinear Analysis TMA, 6 (1982), 307. doi: 10.1016/0362-546X(82)90018-9.

[31]

N. H. Pavel, "Differential Equations, Flow Invariance and Applications,", Research Notes Math. 113, (1984).

[32]

N. Pavel, "Nonlinear Evolution Operators and Semigroups,", Lecture Notes Math. 1260, (1260).

[33]

N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type,, Israel J. Math., 28 (1977), 254. doi: 10.1007/BF02759812.

[34]

M. Pierre, Invariant closed subsets for nonlinear semigroups,, Nonlinear Analysis TMA, 2 (1978), 107. doi: 10.1016/0362-546X(78)90046-9.

[35]

A. T. Plant, Nonlinear semigroups of translations in Banach space generated by functional differential equations,, J. Math. Anal. Appl., 60 (1977), 67. doi: 10.1016/0022-247X(77)90048-8.

[36]

J. Prüss, On semilinear parabolic equations on closed sets,, J. Math. Anal. Appl., 77 (1980), 513. doi: 10.1016/0022-247X(80)90245-0.

[37]

W. M. Ruess, The evolution operator approach to functional differential equations with delay,, Proc. Amer. Math. Soc., 119 (1993), 783.

[38]

W. M. Ruess, Existence of solutions to partial functional differential equations with delay,, in, (1996), 259.

[39]

W. M. Ruess, Existence of solutions to partial functional evolution equations with delay,, in, (1996), 377.

[40]

W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay,, Adv. Differential Equations, 4 (1999), 843.

[41]

W. M. Ruess, Flow invariance for nonlinear partial differential delay equations,, Trans. Amer. Math. Soc., 361 (2009), 4367. doi: 10.1090/S0002-9947-09-04833-8.

[42]

W. M. Ruess and W. H. Summers, Operator semigroups for functional differential equations with delay,, Trans. Amer. Math. Soc., 341 (1994), 695. doi: 10.2307/2154579.

[43]

W. M. Ruess and W. H. Summers, Linearized stability for abstract differential equations with delay,, J. Math. Anal. Appl., 198 (1996), 310. doi: 10.1006/jmaa.1996.0085.

[44]

A. Schiaffino, On a diffusion Volterra equation,, Nonlinear Analysis TMA, 3 (1979), 595. doi: 10.1016/0362-546X(79)90088-9.

[45]

G. Seifert, Positively invariant closed sets for systems of delay differential equations,, J. Differential Equations, 22 (1976), 292. doi: 10.1016/0022-0396(76)90029-2.

[46]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3.

[47]

C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable,, J. Math. Anal. Appl., 56 (1976), 397. doi: 10.1016/0022-247X(76)90052-4.

[48]

G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups,, J. Math. Anal. Appl., 46 (1974), 1. doi: 10.1016/0022-247X(74)90277-7.

[49]

G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations,, Proc. Amer. Math. Soc., 54 (1976), 225. doi: 10.1090/S0002-9939-1976-0402237-0.

[50]

P. Wittbold, "Absorptions nonlinéaires,", Thèse Doctorat, (1994).

[51]

P. Wittbold, Nonlinear diffusion with absorption,, in, (1996), 142.

[52]

P. Wittbold, Nonlinear diffusion with absorption,, Potential Anal., 7 (1997), 437. doi: 10.1023/A:1017998221347.

[53]

K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology,, Hiroshima Math. J., 12 (1982), 321.

show all references

References:
[1]

H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems,, J. Math. Anal. Appl., 65 (1978), 432. doi: 10.1016/0022-247X(78)90192-0.

[2]

P. Bénilan and M. G. Crandall, Completely accretive operators,, In, (1991), 41.

[3]

P. Bénilan, M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators,, Monograph, ().

[4]

D. Bothe, Nonlinear evolutions with Carathéodory forcing,, J. Evol. Equ., 3 (2003), 375. doi: 10.1007/s00028-003-0099-5.

[5]

D. Bothe, Flow invariance for nonlinear accretive evolutions under range conditions,, J. Evol. Equ., 5 (2005), 227. doi: 10.1007/s00028-005-0185-z.

[6]

D. W. Brewer, A nonlinear semigroup for a functional differential equation,, Trans. Amer. Math. Soc., 236 (1978), 173. doi: 10.1090/S0002-9947-1978-0466838-2.

[7]

D. W. Brewer, Locally Lipschitz continuous functional differential equations and nonlinear semigroups,, Illinois J. Math., 26 (1982), 374.

[8]

M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Israel J. Math., 11 (1972), 57. doi: 10.1007/BF02761448.

[9]

J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators,, Proc. Royal Soc., 75A (): 223.

[10]

J. Dyson and R. Villella-Bressan, Semigroups of translation associated with functional and functional differential equations,, Proc. Royal Soc. Edinburgh, 82A (1979), 171. doi: 10.1017/S030821050001115X.

[11]

J. Dyson and R. Villella-Bressan, Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions,, Nonlinear Diff. Eqns. Appl., 3 (1996), 127. doi: 10.1007/BF01194220.

[12]

L. Evans, Nonlinear evolution equations in an arbitrary Banach space,, Israel J. Math., 26 (1977), 1. doi: 10.1007/BF03007654.

[13]

S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations,, J. Math. Anal. Appl., 345 (2008), 854. doi: 10.1016/j.jmaa.2008.04.041.

[14]

S. M. Ghavidel, Flow invariance for solutions to nonlinear nonautonomous evolution equations,, in preparation., ().

[15]

J. K. Hale, Functional differential equations with infinite delays,, J. Math. Anal. Appl., 48 (1974), 276. doi: 10.1016/0022-247X(74)90233-9.

[16]

J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems,, J. Math. Anal. Appl., 118 (1986), 455. doi: 10.1016/0022-247X(86)90273-8.

[17]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.

[18]

A. G. Kartsatos and M. E. Parrott, Global solutions of functional evolution equations involving locally defined Lipschitzian perturbations,, J. London Math. Soc., 27 (1983), 306. doi: 10.1112/jlms/s2-27.2.306.

[19]

A. G. Kartsatos and M. E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations,, J. Diff. Eqns., 47 (1983), 358. doi: 10.1016/0022-0396(83)90041-4.

[20]

A. G. Kartsatos and M. E. Parrott, The weak solution of a functional differential equation in a general Banach space,, J. Diff. Eqns., 75 (1988), 290. doi: 10.1016/0022-0396(88)90140-4.

[21]

V. Lakshmikhantam, S. Leela and V. Moauro, Existence and uniqueness of solutions of delay differential equations on a closed subset of a Banach space,, Nonlinear Analysis TMA, 2 (1978), 311. doi: 10.1016/0362-546X(78)90020-2.

[22]

S. Leela and V. Moauro, Existence of solutions in a closed set for delay differential equations in Banach space,, Nonlinear Analysis TMA, 2 (1978), 47. doi: 10.1016/0362-546X(78)90040-8.

[23]

J. H. Lightbourne III, Function space flow-invariance for functional differential equations of retarded type,, Proc. Amer. Math. Soc., 77 (1979), 91. doi: 10.1090/S0002-9939-1979-0539637-7.

[24]

R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", Wiley, (1976).

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans, 321 (1990), 1. doi: 10.2307/2001590.

[26]

R. H. Martin and H. L. Smith, Convergence in Lotka-Volterra systems with diffusion and delay,, in, (1991), 259.

[27]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,, J. reine angew. Math., 413 (1991), 1.

[28]

I. Miyadera, "Nonlinear Semigroups,", Transl. of Math. Monographs 109, (1992).

[29]

S. Murakami, Stable equilibrium point of some diffusive functional differential equations,, Nonlinear Analysis TMA, 25 (1995), 1037. doi: 10.1016/0362-546X(95)00097-F.

[30]

M. E. Parrott, Representation and approximation of generalized solutions of a nonlinear functional differential equation,, Nonlinear Analysis TMA, 6 (1982), 307. doi: 10.1016/0362-546X(82)90018-9.

[31]

N. H. Pavel, "Differential Equations, Flow Invariance and Applications,", Research Notes Math. 113, (1984).

[32]

N. Pavel, "Nonlinear Evolution Operators and Semigroups,", Lecture Notes Math. 1260, (1260).

[33]

N. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarded type,, Israel J. Math., 28 (1977), 254. doi: 10.1007/BF02759812.

[34]

M. Pierre, Invariant closed subsets for nonlinear semigroups,, Nonlinear Analysis TMA, 2 (1978), 107. doi: 10.1016/0362-546X(78)90046-9.

[35]

A. T. Plant, Nonlinear semigroups of translations in Banach space generated by functional differential equations,, J. Math. Anal. Appl., 60 (1977), 67. doi: 10.1016/0022-247X(77)90048-8.

[36]

J. Prüss, On semilinear parabolic equations on closed sets,, J. Math. Anal. Appl., 77 (1980), 513. doi: 10.1016/0022-247X(80)90245-0.

[37]

W. M. Ruess, The evolution operator approach to functional differential equations with delay,, Proc. Amer. Math. Soc., 119 (1993), 783.

[38]

W. M. Ruess, Existence of solutions to partial functional differential equations with delay,, in, (1996), 259.

[39]

W. M. Ruess, Existence of solutions to partial functional evolution equations with delay,, in, (1996), 377.

[40]

W. M. Ruess, Existence and stability of solutions to partial functional differential equations with delay,, Adv. Differential Equations, 4 (1999), 843.

[41]

W. M. Ruess, Flow invariance for nonlinear partial differential delay equations,, Trans. Amer. Math. Soc., 361 (2009), 4367. doi: 10.1090/S0002-9947-09-04833-8.

[42]

W. M. Ruess and W. H. Summers, Operator semigroups for functional differential equations with delay,, Trans. Amer. Math. Soc., 341 (1994), 695. doi: 10.2307/2154579.

[43]

W. M. Ruess and W. H. Summers, Linearized stability for abstract differential equations with delay,, J. Math. Anal. Appl., 198 (1996), 310. doi: 10.1006/jmaa.1996.0085.

[44]

A. Schiaffino, On a diffusion Volterra equation,, Nonlinear Analysis TMA, 3 (1979), 595. doi: 10.1016/0362-546X(79)90088-9.

[45]

G. Seifert, Positively invariant closed sets for systems of delay differential equations,, J. Differential Equations, 22 (1976), 292. doi: 10.1016/0022-0396(76)90029-2.

[46]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3.

[47]

C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable,, J. Math. Anal. Appl., 56 (1976), 397. doi: 10.1016/0022-247X(76)90052-4.

[48]

G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups,, J. Math. Anal. Appl., 46 (1974), 1. doi: 10.1016/0022-247X(74)90277-7.

[49]

G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations,, Proc. Amer. Math. Soc., 54 (1976), 225. doi: 10.1090/S0002-9939-1976-0402237-0.

[50]

P. Wittbold, "Absorptions nonlinéaires,", Thèse Doctorat, (1994).

[51]

P. Wittbold, Nonlinear diffusion with absorption,, in, (1996), 142.

[52]

P. Wittbold, Nonlinear diffusion with absorption,, Potential Anal., 7 (1997), 437. doi: 10.1023/A:1017998221347.

[53]

K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology,, Hiroshima Math. J., 12 (1982), 321.

[1]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533

[2]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633

[3]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[4]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[5]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[6]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[7]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[8]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451

[9]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345

[10]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007

[11]

Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165

[12]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

[13]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115

[14]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906

[15]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395

[16]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[17]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301

[18]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685

[19]

Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133

[20]

István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]