Fixed Point Theory, Volume 4, No. 1, 2003, 49-60 http://www.math.ubbcluj.ro/ ∼ nodeacj/journal.htm LOCAL FIXED POINT THEORY FOR THE SUM OF TWO OPERATORS IN BANACH SPACES B. C. DHAGE Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist Latur, Maharashtra, India E-mail address: bcd20012001@yahoo.co.in Abstract. The present paper studies the local version of the well-known fixed point theo- rems of Krasnoselskii [5] and Nashed and Wong [5]. Some applications of newly developed local fixed point theorems to nonlinear functional integral equations of fixed type are also discussed. Keywords: fixed point theorem, functional integral equation AMS Subject Classification: 47 H10 1. Introduction Fixed point theory constitutes an important and the core part of the subject of nonlinear functional analysis and is useful for proving the existence theo- rems for nonlinear differential and integral equations. The local fixed point theory is useful for proving the existence of the local solution of the prob- lems governed by nonlinear differential or integral equations. In the present paper we shall obtain the local versions of the well- known fixed point theo- rems of Krasnoselskii [5] and Nashed and Wong [7] and discuss some of their applications to functional integral equations. Throughout this paper let X denote a Banach space with a norm ‖·‖. Let a ∈ X and let r be a positive real number. Then by B r (a) and B r (a) we respectively denote an open and a closed ball in X centered at the point a ∈ X and of radius r. A mapping T : X → X is called a contraction if there exists a constant 0 ≤ α< 1 such that (1.1) ‖Tx − Ty‖≤ α‖x − y‖ 49