DEMONSTRATIO MATHEMATICA Vol. XLIII No 3 2010 H. L. Tidke, M. B. Dhakne EXISTENCE AND UNIQUENESS OF MILD AND STRONG SOLUTIONS OF NONLINEAR VOLTERRA INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES Abstract. In this paper we prove the existence and uniqueness of mild and strong solutions of a nonlinear Volterra integrodifferential equation with nonlocal condition. Our analysis is based on semigroup theory and Banach fixed point theorem and inequalities are established by Gronwall and B. G. Pachpatte. 1. Introduction Let X be a Banach space with norm ‖·‖. Let B = C ([t 0 ,t 0 + β ]; X ) be Banach space of all continuous functions from [t 0 ,t 0 + β ] into X , endowed with the norm ‖x‖ B = sup{‖x(t)‖ : x ∈ B}, 0 ≤ t 0 ≤ t 0 + β. In this paper we study the existence and uniqueness of mild and strong solutions of a nonlocal problem of the type: x ′ (t)+ Ax(t)= f  t, x(t), t  t 0 k(t, s, x(s))ds  , t ∈ [t 0 ,t 0 + β ], (1.1) x(t 0 )+ g(x)= x 0 , (1.2) where −A is the infinitesimal generator of a C 0 semigroup T (t), t ≥ 0, on a Banach space X and the nonlinear operators f :[t 0 ,t 0 + β ] × X × X → X , g : B → X , k :[t 0 ,t 0 + β ] × [t 0 ,t 0 + β ] × X → X are continuous and x 0 is a given element of X. In this paper, B r = {z : ‖z ‖≤ r} denotes the closed ball Key words and phrases : Volterra integrodifferential, strong solutions, continuous de- pendence, Pachpatte’s inequality, nonlocal condition. 2000 Mathematics Subject Classification : 45J05, 34G20, 47H10. Unauthenticated Download Date | 7/25/18 12:23 AM 10.1515/dema-2013-0253