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Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay
Journal of Inequalities and Applications volume 2007, Article number: 049125 (2007)
Abstract
We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.
References
Adimy M, Bouzahir H, Ezzinbi K: Local existence and stability for some partial functional differential equations with infinite delay. Nonlinear Analysis. Theory, Methods & Applications. Series A 2002,48(3):323–348. 10.1016/S0362-546X(00)00184-X
Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj. Serio Internacia 1978,21(1):11–41.
Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317.
Brendle S, Nagel R: PFDE with nonautonomous past. Discrete and Continuous Dynamical Systems. Series A 2002,8(4):953–966.
Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.
Adimy M, Bouzahir H, Ezzinbi K: Existence for a class of partial functional differential equations with infinite delay. Nonlinear Analysis. Theory, Methods & Applications. Series A 2001,46(1):91–112. 10.1016/S0362-546X(99)00447-2
Benkhalti R, Bouzahir H, Ezzinbi K: Existence of a periodic solution for some partial functional-differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2001,256(1):257–280. 10.1006/jmaa.2000.7321
Bouzahir H, Ezzinbi K: Global attractor for a class of partial functional differential equations with infinite delay. In Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun.. Volume 29. Edited by: Faria T, Freitas P. American Mathematical Society, Providence, RI, USA; 2001:63–71.
Ruess WM: Linearized stability for nonlinear evolution equations. Journal of Evolution Equations 2003,3(2):361–373.
Ruess WM: Existence of solutions to partial functional evolution equations with delay. In Functional Analysis (Trier, 1994). Edited by: Dierolf S, Dineen S, Domański P. de Gruyter, Berlin, Germany; 1996:377–387.
Ruess WM: Existence and stability of solutions to partial functional-differential equations with delay. Advances in Differential Equations 1999,4(6):843–876.
Ruess WM: Existence of solutions to partial functional-differential equations with delay. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:259–288.
Ruess WM, Summers WH: Linearized stability for abstract differential equations with delay. Journal of Mathematical Analysis and Applications 1996,198(2):310–336. 10.1006/jmaa.1996.0085
Ruess WM, Summers WH: Operator semigroups for functional-differential equations with delay. Transactions of the American Mathematical Society 1994,341(2):695–719. 10.2307/2154579
Ruess WM, Summers WH: Almost periodicity and stability for solutions to functional-differential equations with infinite delay. Differential and Integral Equations 1996,9(6):1225–1252.
Kartsatos AG, Parrott ME: The weak solution of a functional-differential equation in a general Banach space. Journal of Differential Equations 1988,75(2):290–302. 10.1016/0022-0396(88)90140-4
Thieme HR: Semiflows generated by Lipschitz perturbations of non-densely defined operators. Differential and Integral Equations 1990,3(6):1035–1066.
Adimy M, Laklach M, Ezzinbi K: Non-linear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain. Acta Mathematica Sinica (English Series) 2004,20(5):933–942. 10.1007/s10114-004-0341-3
Laklach M: Contribution à l'étude des équations aux dérivées partielles à retard et de type neutre, Thèse de doctorat.
Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.
Naito T, Minh NV, Shin JS: Spectrum and (almost) periodic solutions of functional differential equations. Vietnam Journal of Mathematics 2002,30(supplement):577–589.
Naito T, Minh NV, Shin JS: The spectrum of evolution equations with infinite delay. unpublished unpublished
Travis CC, Webb GF: Existence and stability for partial functional differential equations. Transactions of the American Mathematical Society 1974, 200: 395–418.
Yosida K: Functional Analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501.
Crandall MG, Liggett TM: Generation of semi-groups of nonlinear transformations on general Banach spaces. American Journal of Mathematics 1971, 93: 265–298. 10.2307/2373376
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Bouzahir, H. Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay. J Inequal Appl 2007, 049125 (2007). https://doi.org/10.1155/2007/49125
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DOI: https://doi.org/10.1155/2007/49125