Existence and regularity of solutions to nonlocal retarded differential equations Shruti A. Dubey a, * , Dhirendra Bahuguna b a LAAS-CNRS, Université de Toulouse, Toulouse, France b Indian Institute of Technology, Kanpur, India article info Keywords: Retarded differential equation Nonlocal history conditions C 0 semigroup Mild Strong and classical solutions abstract In this paper the existence and uniqueness of different types of solutions to a class of semi- linear retarded differential equations with nonlocal history conditions are obtained by a fixed point argument. Also finite dimensional approximation of these solutions in a Hilbert space is established. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Many real life problems in biosciences, economics, medicine, material sciences, that actually involve a significant memory effect have been better represented by mathematical model in the form of retarded differential equations. For example, mod- el of population dynamics [8, cf. p. 434], model for the growth of the tumor with time delays in cell proliferation [9] are amongst the several application areas. The study of such equations is required as they display better consistency with the nature of the underlying process and predictive results. In this work, we investigate the existence and uniqueness of solution of nonlocal retarded differential equation in Banach space. Let X be a Banach space and C t :¼ Cð½s; t; XÞ; s > 0; 0 6 t 6 T < 1, be a Banach space of all continuous functions from ½s; t into X endowed with the norm kwk t :¼ sup s6g6t kwðgÞk X ; w 2 C t ; where kk X is the norm in X. Now, consider the following retarded differential equation in a Banach space X: u 0 ðtÞþ AuðtÞ¼ f ðt; uðtÞ; uðbðtÞÞÞ; 0 < t 6 T ; gðuÞ¼ x;  ð1:1Þ where A is a linear operator defined from DðAÞ X into X such that A is the infinitesimal generator of a C 0 -semigroup fSðtÞ : t P 0g of bounded linear operators in X, the nonlinear map f is defined from ½0; T  X  X into X satisfying a type of local Lipschitz-like condition, g is a map defined from C 0 :¼ Cð½s; 0; XÞ into X and x 2 X. A few examples of g are the following. (I) Let k 2 L 1 ð0; sÞ such that j :¼ R s 0 kðsÞds – 0. Let 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.036 * Corresponding author. E-mail addresses: shrurkd@gmail.com (S.A. Dubey), dhiren@iitk.ac.in (D. Bahuguna). Applied Mathematics and Computation 215 (2009) 2413–2424 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc