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Existence results for partial neutral functional differential equations with unbounded delay. (English) Zbl 0915.35110
The authors prove existence of mild solutions of the Cauchy problem $${d\over dt} (x(t)+ F(t, x_t))= Ax(t)+ G(t, x_t),\quad t\ge\sigma,\quad x(\sigma)= \varphi\in \Omega,\tag 1$$ where $\Omega$ is an open subset of the phase space, $F,G: [0,a]\times \Omega\mapsto X$ are continuous functions, $0\le\sigma< a$ and $A$ is the infinitesimal generator of an analytic semigroup $T(.)$ of bounded linear operators on $X$. In the case $\sigma= 0$ the existence of strong solutions of the Cauchy problem (1) is proved.
Reviewer: D.Bainov (Sofia)

35R10Partial functional-differential equations
34K40Neutral functional-differential equations
Full Text: DOI
[1] Ananjevskii, I. M.; Kolmanovskii, V. B.: Stabilization of some nonlinear hereditary mechanical systems. Nonlinear anal. 15, 101-114 (1990)
[2] Corduneanu, C.; Lakshmikantham, V.: Equations with unbounded delay. Nonlinear anal. 4, 831-877 (1980) · Zbl 0449.34048
[3] Da Prato, G.; Lunardi, A.: Solvability on the real line of a class of linear Volterra integrodifferential equations of parabolic type. Ann. mat. Pura appl. (4) 150, 67-117 (1988) · Zbl 0646.45013
[4] Dhakne, M. B.; Pachpatte, B. G.: On a general class of abstract functional integrodifferential equations. Indian J. Pure appl. Math. 19, 728-746 (1988) · Zbl 0663.45009
[5] Fattorini, H. O.: Second order linear differential equations in Banach spaces. North-holland mathematics studies 108 (1985) · Zbl 0564.34063
[6] Goldstein, J.: Semigroups of linear operators and applications. (1985) · Zbl 0592.47034
[7] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[8] Gopalsamy, K.; Weng, P. -X.: On the stability of a neutral integro-partial differential equation. Bull. inst. Math. acad. Sinica 20, 267-284 (1992) · Zbl 0759.45007
[9] Grabosch, A.; Moustakas, U.: A semigroup approach to retarded differential equations. Lecture notes in math. 1184 (1986)
[10] Györi, I.; Ladas, G.: Oscillation theory of delay differential equations. (1991) · Zbl 0780.34048
[11] Hale, J.; Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. ekvac. 21 (1978) · Zbl 0383.34055
[12] Hale, J.: Asymptotic behavior of dissipative systems. (1988) · Zbl 0642.58013
[13] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[14] Henriquez, H. R.: Periodic solutions of quasi-linear partial functional differential equations with unbounded delay. Funkcial. ekvac. 37, 329-343 (1994) · Zbl 0814.35141
[15] Henriquez, H. R.: Approximation of abstract functional differential equations with unbounded delay. Indian J. Pure appl. Math. 27, 357-386 (1996) · Zbl 0853.34072
[16] Henriquez, H. R.: Regularity of solutions of abstract retarded functional differential equations with unbounded delay. Nonlinear anal. 28, 513-531 (1997)
[17] Hille, E.; Phillips, R. S.: Functional analysis and semi-groups. Amer. math. Soc. colloq. Publ. 31 (1957) · Zbl 0078.10004
[18] Hino, Y.; Murakami, S.; Naito, T.: Functional differential equations with infinite delay. Lecture notes in math. 1473 (1991) · Zbl 0732.34051
[19] Kwon, W. H.; Lee, G. W.; Kim, S. W.: Performance improvement using time delays in multivariable controller design. Internat. J. Control 52, 1455-1473 (1990) · Zbl 0708.93024
[20] Kartsatos, A. G.; Parrott, M. E.: The weak solution of a functional differential equation in a general Banach space. J. differential equations 75, 219-232 (1988) · Zbl 0666.34076
[21] Komura, J.: Differentiability of nonlinear semigroups. J. math. Soc. Japan 21, 375-402 (1969) · Zbl 0193.11004
[22] Liang, J.; Xiao, T.: Functional differential equations with infinite delay in Banach spaces. Internat. J. Math. math. Sci. 14, 497-508 (1991) · Zbl 0743.34082
[23] Marle, C. -M.: Mesures et probabilités. (1974)
[24] Mcbride, A. C.: Semigroups of linear operators: an introduction. Pitman res. Notes math. Ser. 156 (1987) · Zbl 0635.47035
[25] Nagel, R.: One-parameter semigroups of positive operators. Lecture notes in math. 1184 (1986) · Zbl 0585.47030
[26] Parrott, M. E.: Linearized stability and irreducibility for a functional differential equation. SIAM J. Math. anal. 23, 649-661 (1992) · Zbl 0763.34058
[27] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023
[28] Petzeltova, H.: Solution semigroup and invariant manifolds for functional equations with infinite delay. Math. bohemica 118, 175-192 (1993)
[29] Sadovskii, B. N.: On a fixed point principle. Funct. anal. Appl. 1, 74-76 (1967) · Zbl 0165.49102
[30] Salamon, D.: Neutral functional differential equations and semigroups of operators. Lecture notes in control and inform. Sci. 54 (1983) · Zbl 0518.93035
[31] Travis, C. C.; Webb, G. F.: Second order differential equations in Banach spaces. (1978) · Zbl 0455.34044
[32] Travis, C. C.; Webb, G. F.: Existence and stability for partial functional differential equations. Trans. amer. Math. soc. 200, 395-418 (1974) · Zbl 0299.35085
[33] Wiener, J.; Debnath, L.: The Fourier method for partial differential equations with piecewise continuous delay. Contemporary mathematics 129 (1992) · Zbl 0826.35131