A characterization of well-posedness for abstract Cauchy problems with finite delay

C. Lizama, F. Poblete

Research output: Contribution to journalArticle

Abstract

Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay, {u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equation G(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A. © 2017 Elsevier Inc.
LanguageEnglish
Pages410-435
Number of pages26
JournalJournal of Mathematical Analysis and Applications
Volume457
Issue number1
DOIs
Publication statusPublished - 2018

Fingerprint

Abstract Cauchy Problem
Well-posedness
Functional equation
Mathematical operators
First-order
Closed Operator
C0-semigroup
Banach spaces
Bounded Operator
Bounded Linear Operator
Phase Space
Banach space
Family

Keywords

  • C0-semigroups
  • Cauchy problem
  • Finite delay
  • Functional equations
  • Well posedness

Cite this

A characterization of well-posedness for abstract Cauchy problems with finite delay. / Lizama, C.; Poblete, F.

In: Journal of Mathematical Analysis and Applications, Vol. 457, No. 1, 2018, p. 410-435.

Research output: Contribution to journalArticle

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title = "A characterization of well-posedness for abstract Cauchy problems with finite delay",
abstract = "Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay, {u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equation G(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A. {\circledC} 2017 Elsevier Inc.",
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note = "Export Date: 6 April 2018 Correspondence Address: Lizama, C.; Universidad de Santiago de Chile, Facultad de Ciencias, Departamento de Matem{\'a}tica y Ciencia de la Computaci{\'o}n, Casilla 307, Correo 2, Chile; email: carlos.lizama@usach.cl References: Abadias, L., Lizama, C., Miana, P.J., Sharp extensions and algebraic properties for solution families of vector-valued differential equations (2016) Banach J. Math. Anal., 10 (1), pp. 169-208; Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems (2011) Monogr. Math., 96. , second edition Birkh{\"a}user/Springer Basel AG, Basel; Ashyralyev, A., Agirseven, D., Well-posedness of delay parabolic equations with unbounded operators acting on delay terms (2014) Bound. Value Probl., 126, pp. 1-15; B{\'a}tkai, A., Piazzera, S., Semigroups for Delay Equations (2005) Res. Notes Math., 10. , A K Peters, Ltd. Wellesley, MA xii+259 pp; Diekmann, O., Gyllenberg, M., Equations with infinite delay: blending the abstract and the concrete (2012) J. Differential Equations, 252 (2), pp. 819-851; Engel, K.J., Nagel, R., One-Parameter Semigroups for Linear Evolution Equations (2000) Grad. Texts in Math., 194. , Springer New York; Fitzgibbon, W.F., Stability for abstract nonlinear Volterra equations involving finite delay (1977) J. Math. Anal. Appl., 60 (2), pp. 429-434; Hale, J.K., Functional Differential Equations (1971), Springer-Verlag New York; Hille, E., Phillips, R.S., Functional Analysis and Semi-groups (1957) Amer. Math. Soc. Colloq. Publ., 31. , Amer. Math. Soc. Providence, RI; Jiang, W., Guo, F., Huang, F., Well-posedness of linear partial differential equations with unbounded delay operators (2004) J. Math. Anal. Appl., 293 (1), pp. 310-328; Liu, K., Retarded stationary Ornstein–Uhlenbeck processes driven by L{\'e}vy noise and operator selfdecomposability (2010) Potential Anal., 33 (3), pp. 291-312; Liu, K., On regularity property of retarded Ornstein–Uhlenbeck processes in Hilbert spaces (2012) J. Theoret. Probab., 25 (2), pp. 565-593; Liu, K., On stationarity of stochastic retarded linear equations with unbounded drift operators (2016) Stoch. Anal. Appl., 34 (4), pp. 547-572; Liu, K., Hu, L., Luo, J., Stability property and essential spectrum of linear retarded functional differential equations (2013) J. Comput. Appl. Math., 244, pp. 19-35; Lizama, C., Poblete, F., On a functional equation associated with (a, k)-regularized resolvent families (2012) Abstr. Appl. Anal.; Petzeltov{\'a}, H., Solution semigroup and invariant manifolds for functional equations with infinite delay (1993) Math. Bohem., 118 (2), pp. 175-193; Petzeltov{\'a}, H., Milota, J., Resolvent operator for abstract functional-differential equations with infinite delay (1987) Numer. Funct. Anal. Optim., 9 (7-8), pp. 779-807; Travis, C.C., Webb, G.F., Existence and stability for partial functional differential equations (1974) Trans. Amer. Math. Soc., 200, pp. 395-418; Travis, C.C., Webb, G.F., Existence, stability, and compactness in the α-norm for partial functional differential equations (1978) Trans. Amer. Math. Soc., 240, pp. 129-143; Webb, G., Functional differential equations and nonlinear semigroups in Lp-spaces (1976) J. Differential Equations, 29, pp. 71-89",
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T1 - A characterization of well-posedness for abstract Cauchy problems with finite delay

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AU - Poblete, F.

N1 - Export Date: 6 April 2018 Correspondence Address: Lizama, C.; Universidad de Santiago de Chile, Facultad de Ciencias, Departamento de Matemática y Ciencia de la Computación, Casilla 307, Correo 2, Chile; email: carlos.lizama@usach.cl References: Abadias, L., Lizama, C., Miana, P.J., Sharp extensions and algebraic properties for solution families of vector-valued differential equations (2016) Banach J. Math. Anal., 10 (1), pp. 169-208; Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems (2011) Monogr. Math., 96. , second edition Birkhäuser/Springer Basel AG, Basel; Ashyralyev, A., Agirseven, D., Well-posedness of delay parabolic equations with unbounded operators acting on delay terms (2014) Bound. Value Probl., 126, pp. 1-15; Bátkai, A., Piazzera, S., Semigroups for Delay Equations (2005) Res. Notes Math., 10. , A K Peters, Ltd. Wellesley, MA xii+259 pp; Diekmann, O., Gyllenberg, M., Equations with infinite delay: blending the abstract and the concrete (2012) J. Differential Equations, 252 (2), pp. 819-851; Engel, K.J., Nagel, R., One-Parameter Semigroups for Linear Evolution Equations (2000) Grad. Texts in Math., 194. , Springer New York; Fitzgibbon, W.F., Stability for abstract nonlinear Volterra equations involving finite delay (1977) J. Math. Anal. Appl., 60 (2), pp. 429-434; Hale, J.K., Functional Differential Equations (1971), Springer-Verlag New York; Hille, E., Phillips, R.S., Functional Analysis and Semi-groups (1957) Amer. Math. Soc. Colloq. Publ., 31. , Amer. Math. Soc. Providence, RI; Jiang, W., Guo, F., Huang, F., Well-posedness of linear partial differential equations with unbounded delay operators (2004) J. Math. Anal. Appl., 293 (1), pp. 310-328; Liu, K., Retarded stationary Ornstein–Uhlenbeck processes driven by Lévy noise and operator selfdecomposability (2010) Potential Anal., 33 (3), pp. 291-312; Liu, K., On regularity property of retarded Ornstein–Uhlenbeck processes in Hilbert spaces (2012) J. Theoret. Probab., 25 (2), pp. 565-593; Liu, K., On stationarity of stochastic retarded linear equations with unbounded drift operators (2016) Stoch. Anal. Appl., 34 (4), pp. 547-572; Liu, K., Hu, L., Luo, J., Stability property and essential spectrum of linear retarded functional differential equations (2013) J. Comput. Appl. Math., 244, pp. 19-35; Lizama, C., Poblete, F., On a functional equation associated with (a, k)-regularized resolvent families (2012) Abstr. Appl. Anal.; Petzeltová, H., Solution semigroup and invariant manifolds for functional equations with infinite delay (1993) Math. Bohem., 118 (2), pp. 175-193; Petzeltová, H., Milota, J., Resolvent operator for abstract functional-differential equations with infinite delay (1987) Numer. Funct. Anal. Optim., 9 (7-8), pp. 779-807; Travis, C.C., Webb, G.F., Existence and stability for partial functional differential equations (1974) Trans. Amer. Math. Soc., 200, pp. 395-418; Travis, C.C., Webb, G.F., Existence, stability, and compactness in the α-norm for partial functional differential equations (1978) Trans. Amer. Math. Soc., 240, pp. 129-143; Webb, G., Functional differential equations and nonlinear semigroups in Lp-spaces (1976) J. Differential Equations, 29, pp. 71-89

PY - 2018

Y1 - 2018

N2 - Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay, {u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equation G(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A. © 2017 Elsevier Inc.

AB - Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay, {u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equation G(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A. © 2017 Elsevier Inc.

KW - C0-semigroups

KW - Cauchy problem

KW - Finite delay

KW - Functional equations

KW - Well posedness

U2 - 10.1016/j.jmaa.2017.08.023

DO - 10.1016/j.jmaa.2017.08.023

M3 - Article

VL - 457

SP - 410

EP - 435

JO - Journal of Mathematical Analysis and Applications

T2 - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -