Solvability of the Cauchy problem for infinite delay equations. (English) Zbl 1058.34106

By using topological methods, expressed in terms of the Kuratowski measure of noncompactness, the authors prove several existence results on mild solutions for the infinite delay functional-integral equation \[ u(t)=g(t)+\int_{\sigma}^tf(t,s,u(s),u_s)\,ds,\quad \sigma\leq t\leq T, \qquad u_\sigma=\varphi, \] and semilinear functional-differential equations of the form \[ u'(t)=Au(t)+f(t,u(t),u_t),\quad 0\leq t\leq T, \qquad u_0=\varphi, \] with \(\varphi\in {\mathcal B}\). Here, \(\mathcal{B}\) is an admissible function space, i.e., a suitable chosen subspace of functions from \((-\infty,\sigma\,]\) to \(X\), with \(X\) a Banach space, \(\varphi\in \mathcal{B}\), \(x_t(s)=x(t+s)\), \(f\) is either in \(C([\,\sigma,T\,]\times [\,\sigma,T\,]\times X\times\mathcal{B};X)\) or in \(C([\,\sigma,T\,]\times X\times\mathcal{B};X)\) and \(A:D(A)\subset X\to X\) is a linear operator. An extension to the case when \(A\) may depend on \(t\) as well is considered, and an application to a functional integro-differential equation of Schrödinger type is included.


34K30 Functional-differential equations in abstract spaces
47D62 Integrated semigroups
35Q55 NLS equations (nonlinear Schrödinger equations)
35K05 Heat equation
Full Text: DOI


[1] Acquistapace, P., Abstract linear nonautonomous parabolic equations: a survey, Differential equations in Banach spaces, (Dore, G.; Favini, A.; Obrecht, E.; Venni, A., Proc. Bologna, 1991. Proc. Bologna, 1991, Lecture Notes in Pure and Applied Mathematics, Vol. 148 (1993), Dekker: Dekker New York), 1-19 · Zbl 0833.35060
[2] Adimy, M.; Bouzahir, H.; Ezzinbi, K., Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal. Ser. A: Theory Methods, 46, 91-112 (2001) · Zbl 1003.34068
[3] Banas, J.; Goebel, K., Measures of noncompactness, Lecture Notes in Pure and Applied Mathematics, Vol. 60 (1980), Dekker: Dekker New York · Zbl 0441.47056
[4] Banks, H. T.; Burns, J. A., Hereditary control problemsnumerical methods based on averaging approximations, SIAM J. Control Optim., 16, 169-208 (1978) · Zbl 0379.49025
[6] Burns, J. A.; Herdman, T. L., Adjoint semigroup theory for a class of functional differential equations, SIAM J. Math. Anal., 5, 729-745 (1976) · Zbl 0336.45002
[7] Burns, J. A.; Herdman, T. L.; Stech, H. W., Linear functional differential equations as semigroups on product spaces, SIAM J. Math. Anal., 14, 98-116 (1983) · Zbl 0528.34062
[8] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional-Differential Equations (1985), Academic Press: Academic Press Orlando, FL · Zbl 0635.34001
[9] Corduneanu, C.; Lakshmikantham, V., Equations with unbounded delaya survey, Nonlinear Anal. TMA, 4, 831-877 (1979) · Zbl 0449.34048
[10] Cushing, J. M., Integrodifferential equations and delay models in population dynamics, Lecture Notes in Biomathematics, Vol. 20 (1977), Springer: Springer Berlin · Zbl 0363.92014
[11] Da Prato, G.; Lunardi, A., Hopf bifurcation for nonlinear integro-differential equations in Banach spaces with infinite delay, Indiana Univ. Math. J., 36, 241-255 (1987) · Zbl 0634.45013
[12] Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin, New York · Zbl 0559.47040
[13] Deimling, K., Multivalued Differential Equations (1992), Walter de Gruyter: Walter de Gruyter Berlin, New York · Zbl 0760.34002
[14] deLaubenfels, R., Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Mathematics, Vol. 1570 (1994), Springer: Springer Berlin, New York · Zbl 0811.47034
[15] Delfour, M. C.; Mitter, S. K., Hereditary differential systems with constant delays. I. General case, J. Differential Equations, 12, 213-235 (1972) · Zbl 0242.34055
[16] Di Blasio, G., Delay differential equations with unbounded operators acting on delay terms, Nonlinear Anal., 52, 1-18 (2003) · Zbl 1034.34095
[17] Di Blasio, G.; Kunisch, K.; Sinestrari, E., \(L^2\)-regularity for parabolic integrodifferential equations with delay in the highest order dervatives, J. Math. Anal. Appl., 102, 38-57 (1984) · Zbl 0538.45007
[18] Diekmann, O.; van Gils, S.; Verduyn Lunel, S. M.; Walther, H. O., Delay equations, functional-, complex-, and nonlinear analysis, Applied Mathematical Science, Vol. 110 (1995), Springer: Springer Berlin, New York · Zbl 0826.34002
[19] Engel, K.-J; Nagel, R., One-parameter semigroups for linear evolution equations, GTM, Vol. 194 (2000), Springer: Springer Berlin, New York · Zbl 0952.47036
[20] Fattorini, H. O., The Cauchy problem, Encyclopedia of Mathematics and its Applications, Vol. 18 (1983), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0493.34005
[21] Goldstein, J. A., Abstract evolution equations, Trans. Amer. Math. Soc., 141, 159-185 (1969) · Zbl 0181.42602
[22] Goldstein, J. A., Semigroups of Linear Operators and Applications (1985), Oxford University Press: Oxford University Press New York · Zbl 0592.47034
[23] Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 11-41 (1978) · Zbl 0383.34055
[24] Hale, J. K.; Lunel, S. M.V, Introduction to Functional-Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002
[25] Heinz, H. P., On the behavior of measures of noncompactness with respect to differentiation and integration of vector-valued function, Nonlinear Anal. TMA, 7, 1351-1371 (1983) · Zbl 0528.47046
[26] Henriquez, H. R., Regularity of solutions of abstract retarded functional-differential equations with unbounded delay, Nonlinear Anal. TMA, 28, 513-531 (1997) · Zbl 0864.35112
[27] Hino, Y.; Murakami, S.; Naito, T., Functional-differential equations with infinite delay, Lecture Notes in Mathematics, Vol. 1473 (1991), Springer: Springer Berlin · Zbl 0732.34051
[28] Hörmander, L., Estimates for translation invariant operator in \(L^p\) spaces, Acta Math., 104, 93-140 (1960) · Zbl 0093.11402
[29] Kappel, F.; Schappacher, W., Nonlinear functional differential equations and abstract integral equations, Proc. Roy. Soc. Edingburgh A, 84, 71-91 (1979) · Zbl 0455.34057
[31] Liang, J.; Xiao, T. J., Solutions of abstract functional differential equations with infinite delay, Acta Math. Sinica, 34, 631-644 (1991) · Zbl 0744.34065
[32] Liang, J.; Xiao, T. J., The Cauchy problem for nonlinear abstract functional differential equations with infinite delay, Comput. Math. Appl., 40, 693-703 (2000) · Zbl 0960.34067
[33] Liang, J.; Xiao, T. J.; van Casteren, J., A note on semilinear abstract functional differential and integrodifferential equations with infinite delay, Appl. Math. Lett., 17, 473-477 (2004) · Zbl 1082.34543
[34] Liu, J. H., Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 247, 627-644 (2000) · Zbl 1056.34085
[35] Milota, J., Stability and saddle point property for a linear autonomous functional parabolic equation, Comment. Math. Univ. Carolina, 27, 87-101 (1986) · Zbl 0606.35081
[37] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer Berlin · Zbl 0516.47023
[38] Prüß, J., Evolutionary Integral Equations and Applications (1993), Birkhäuser: Birkhäuser Basel · Zbl 0793.45014
[39] Schumacher, K., Existence and continuous dependence for functional differential equations with unbounded delay, Arch. Rational Mech. Anal., 67, 315-335 (1978) · Zbl 0383.34052
[40] Shin, J. S.; Naito, T., Existence and continuous dependence of mild solutions to semilnear functional differential equations in Banach spaces, Tohoku Math. J., 51, 555-583 (1999) · Zbl 0964.34068
[41] Tanabe, H., Equations of Evolution (1979), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA, London
[42] Tanaka, N.; Okazawa, N., Local \(C\)-semigroups and local integrated semigroups, Proc. London Math. Soc., 61, 63-90 (1990) · Zbl 0703.47031
[43] 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
[44] Wu, J., Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, Vol. 119 (1996), Springer: Springer New York
[45] Xiao, T. J.; Liang, J., The Cauchy problem for higher order abstract differential equations, Lecture Notes in Mathematics, Vol. 1701 (1998), Springer: Springer Berlin, New York
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