Applied Mathematics E-Notes, 9(2009), 101-108 c  ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/ Existence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential Equations With Nonlocal Conditions ∗ Haribhau Laxman Tidke † , Machindra Baburao Dhakne ‡ Received 16 October 2008 Abstract In this paper, we study the existence and uniqueness of mild solutions for second order initial value problems, with nonlocal conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family. 1 Introduction Let X be a Banach space with norm ‖.‖ and throughout this paper we assume the notation J = [0,b]. Let B = C (J, X) be Banach space of all continuous functions from J into X, endowed with the norm ‖x‖ b = sup{‖x(t)‖ : x ∈ B, t ∈ J }. In the present paper we consider the following second order nonlinear integrodifferential equations with nonlocal conditions: x ′′ (t)= Ax(t)+ f ( t, x(t),  t 0 k(t, s, x(s))ds ) , t ∈ J, (1) x(0) = x 0 + q(x), (2) x ′ (0) = y 0 + p(x), (3) where A is an infinitesimal generator of a strongly continuous cosine family {C (t): t ∈ R} in Banach space X, f : J × X × X → X, k : J × J × X → X, q, p : B → X are appropriate continuous functions, and x 0 ,y 0 are given elements of X. The work on nonlocal initial value problem (IVP) was initiated by Byszewski. In [3], Byszewski, using the method of semigroups and the Banach fixed point theorem proved the existence and uniqueness of mild, strong and classical solution of first order * Mathematics Subject Classifications: 35A05, 34G20, 47D09. † Department of Mathematics, School of Mathematical Sciences, North Maharashtra University, Jalgaon, India ‡ Department of Mathematics, Dr. B. A. Marathwada University, Aurangabad-431 004, India 101