May  2017, 37(5): 2455-2482. doi: 10.3934/dcds.2017106

Existence of solutions for a class of abstract neutral differential equations

1. 

Departamento de Matemática, Universidad de Santiago, USACH, Casilla 307, correo 2, Santiago, Chile

2. 

Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740, Brazil

3. 

Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Valdivia, Chile

4. 

Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile

* Corresponding author

1This author was partially supported by CONICYT under grant FONDECYT 1130144 and DICYT-USACH.
2This author was partially supported by CNPq/Brazil under Grant 478053/2013-4.
3This author was supported by CONICYT under grant FONDECYT Grant 3140103.
4This author was supported by project DIUFRO: DI17-0071.

Received  August 2014 Revised  January 2017 Published  February 2017

This paper is devoted to studying the existence of solutions for a general class of abstract neutral functional differential equations of first order with finite delay. Specifically, we distinguish among mild, strong and classical solutions, and we characterize in terms of the forcing function of the equation the existence of solutions of each one of these types.

Citation: Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106
References:
[1]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Birkhäuser, Boston, 2007. Google Scholar

[2]

E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, 2nd edition, World Scientific, New Jersey, 2001. doi: 10.1142/4745. Google Scholar

[3]

M. C. Delfour and S. K. Mitter, Hereditary differential systems with constant delays. Ⅰ. General case, J. Differential Equations, 12 (1972), 213-235. doi: 10.1016/0022-0396(72)90030-7. Google Scholar

[4]

M. C. Delfour, State theory of linear hereditary differential systems, J. Math. Anal. Appl., 60 (1977), 8-35. doi: 10.1016/0022-247X(77)90044-0. Google Scholar

[5]

M. C. Delfour and J. Karrakchou, State space theory of linear time invariant systems with delays in state, control, and observation variables, Ⅰ, Ⅱ., J. Math. Anal. Appl., 125 (1987), 361-450. doi: 10.1016/0022-247X(87)90099-0. Google Scholar

[6]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar

[7] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar
[8] H. R. Henríquez, Introducción a la Integración Vectorial, Editorial Académica Española, Saarbrücken, 2012. Google Scholar
[9]

E. Hernández and D. O'Regan, On a new class of abstract neutral differential equations, J. Funct. Anal., 261 (2011), 3457-3481. Google Scholar

[10] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publ., Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar
[11] A. Lunardi, Analytic Semigroup and Optimal Regularity in Parabolic Problems, Birkhäuser-Verlag, Basel, 1995. Google Scholar
[12] W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems, SIAM, Philadelphia, 2007. doi: 10.1137/1.9780898718645. Google Scholar
[13]

J. A. Nohel, Nonlinear Volterra equations for heat flow in material with memory, in Integral and Functional Differential Equations (eds. T. L. Herdman, S. M. Rankin Ⅲ, and H. W. Stech), Marcel Dekker, 67 (1981), 3-82. Google Scholar

[14] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar
[15] D. Salamon, Control and Observation of Neutral Systems, Chapman & Hall/CRC, Boston, 1984. Google Scholar
[16] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar

show all references

References:
[1]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Birkhäuser, Boston, 2007. Google Scholar

[2]

E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, 2nd edition, World Scientific, New Jersey, 2001. doi: 10.1142/4745. Google Scholar

[3]

M. C. Delfour and S. K. Mitter, Hereditary differential systems with constant delays. Ⅰ. General case, J. Differential Equations, 12 (1972), 213-235. doi: 10.1016/0022-0396(72)90030-7. Google Scholar

[4]

M. C. Delfour, State theory of linear hereditary differential systems, J. Math. Anal. Appl., 60 (1977), 8-35. doi: 10.1016/0022-247X(77)90044-0. Google Scholar

[5]

M. C. Delfour and J. Karrakchou, State space theory of linear time invariant systems with delays in state, control, and observation variables, Ⅰ, Ⅱ., J. Math. Anal. Appl., 125 (1987), 361-450. doi: 10.1016/0022-247X(87)90099-0. Google Scholar

[6]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar

[7] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar
[8] H. R. Henríquez, Introducción a la Integración Vectorial, Editorial Académica Española, Saarbrücken, 2012. Google Scholar
[9]

E. Hernández and D. O'Regan, On a new class of abstract neutral differential equations, J. Funct. Anal., 261 (2011), 3457-3481. Google Scholar

[10] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publ., Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar
[11] A. Lunardi, Analytic Semigroup and Optimal Regularity in Parabolic Problems, Birkhäuser-Verlag, Basel, 1995. Google Scholar
[12] W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems, SIAM, Philadelphia, 2007. doi: 10.1137/1.9780898718645. Google Scholar
[13]

J. A. Nohel, Nonlinear Volterra equations for heat flow in material with memory, in Integral and Functional Differential Equations (eds. T. L. Herdman, S. M. Rankin Ⅲ, and H. W. Stech), Marcel Dekker, 67 (1981), 3-82. Google Scholar

[14] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar
[15] D. Salamon, Control and Observation of Neutral Systems, Chapman & Hall/CRC, Boston, 1984. Google Scholar
[16] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar
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