Liang, Jin; Xiao, Ti-Jun; van Casteren, J. A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. (English) Zbl 1082.34543 Appl. Math. Lett. 17, No. 4, 473-477 (2004). By a fixed-point argument, the authors prove the existence and uniqueness of the solution to the following abstract integrodifferential equation \[ u(t)=g(t)+\int_0^tE(t-s)f(s,u(s),u_s)\,ds,\,\,0\leq t\leq T,\quad u_0=\phi . \] As an application, conditions are derived implying the existence and uniqueness of the solution to the abstract semilinear functional-integrodifferential equation with infinite delay \[ u'(t)=A\left[u(t)+\int_0^tF(t-s)u(s)\,ds\right]+f(t,u(t),u_t),\,\,0\leq t\leq T,\quad u_0=\phi . \] Reviewer: Adelaziz Rhandi (Marrakesh) Cited in 21 Documents MSC: 34K30 Functional-differential equations in abstract spaces 45J05 Integro-ordinary differential equations Keywords:infinite delay; semilinear abstract functional-differential equations; semilinear abstract functional-integrodifferential equations PDF BibTeX XML Cite \textit{J. Liang} et al., Appl. Math. Lett. 17, No. 4, 473--477 (2004; Zbl 1082.34543) Full Text: DOI References: [1] Liang, J; Huang, F.L; Xiao, T.J, Exponential stability for abstract linear autonomous functional differential equations with infinite delay, Inter. J. math. math. sci., 21, 255-260, (1998) [2] Liang, J; Xiao, T.J, Functional differential equations with infinite delay in Banach spaces, Inter. J. math. and math. sci., 14, 497-508, (1991) · Zbl 0743.34082 [3] Liang, J; Xiao, T.J, Solutions of abstract functional differential equations with infinite delay, Acta math. sinica, 34, 631-644, (1991) · Zbl 0744.34065 [4] Liang, J; Xiao, T.J, The Cauchy problem for nonlinear abstract functional differential equations with infinite delay, Computers math. applic., 40, 6/7, 693-703, (2000) · Zbl 0960.34067 [5] Hale, J.K; Kato, J, Phase space for retarded equations with infinite delay, Funkcialaj ekvacioj, 21, 11-41, (1978) · Zbl 0383.34055 [6] Huang, F.L, On linear autonomous functional differential equations with infinite delay, Ann. of diff. eqs., 3, 3, 275-292, (1987) · Zbl 0629.34071 [7] Pazy, A, Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023 [8] Diekmann, O; van Gils, S; Lunel, S; Walther, H.O, Delay equations. functional-, complex-, and nonlinear analysis, () [9] Engel, K.-J; Nagel, R, One-parameter semigroups for linear evolution equations, () · Zbl 0616.34060 [10] Goldstein, J.A, () [11] Grimmer, R, Resolvent operators for integral equations in Banach space, Trans. amer. math. soc., 273, 333-349, (1982) · Zbl 0493.45015 [12] Liang, J; Xiao, T.J, Functional differential equations with infinite delay in Fréchet spaces, J. sichuan univ. (sichuan daxue xuebao), 26, 382-390, (1989) · Zbl 0725.34084 [13] deLaubenfels, R, Existence families, functional calculi and evolution equations, () · Zbl 0811.47034 [14] Xiao, T.J; Liang, J, The Cauchy problem for higher order abstract differential equations, () [15] Schumacher, K, Existence and continuous dependence for functional differential equations with unbounded delay, Arch. rat. mech. ana., 67, 315-335, (1978) · Zbl 0383.34052 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.