J. Korean Math. Soc. 48 (2011), No. 3, pp. 585–597 DOI 10.4134/JKMS.2011.48.3.585 REGULARITY OF SOLUTIONS OF ABSTRACT QUASILINEAR DELAY INTEGRODIFFERENTIAL EQUATIONS Dong Gun Park, Krishnan Balachandran, and Francis Paul Samuel Abstract. We prove the existence and uniqueness of classical solutions for a quasilinear delay integrodifferential equation in Banach spaces. The result is established by using the semigroup theory and the Banach fixed point theorem. 1. Introduction Abstract quasilinear integrodifferential equations arise in many areas of sci- ence such as population dynamics, mathematical physics, heat conduction the- ory of material with memory etc. For this reason, this type of equations have received much attention in recent years. The literature related to quasilinear differential and integrodifferential equations is very extensive. A general theory of quasilinear evolution equations has been developed by Kato [14, 15]. Using the method of semigroup, existence and uniqueness of mild and classical solu- tions of quasilinear evolution equations have been discussed by Pazy [20]. The problem of existence of solutions of quasilinear evolution equations in Banach spaces has been studied by several authors [2, 6, 15, 16, 17, 18]. Pazy [20] considered the following quasilinear equation of the form u ′ (t)+ A(t, u)u(t) = 0, 0 <t ≤ T, u(0) = u 0 , and discussed the mild and classical solutions by using the fixed point argument. The existence of classical solution has been studied to the nonhomogeneous quasilinear evolution equation u ′ (t)+ A(t, u)u(t) = f (t, u), 0 <t ≤ T, u(0) = u 0 , Received February 8, 2010; Revised May 14, 2010. 2010 Mathematics Subject Classification. 34G20, 47D03, 47H10. Key words and phrases. contraction principle, mild and classical solution, semigroup theory. This Paper is supported by Dong-A University Research Foundation. c ⃝2011 The Korean Mathematical Society 585