ON THE SOLVABILITY OF THE NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES

The purpose of this paper is to study the solvability of the equation (E). In Section 2, we give preliminary definitions. In Section 3, we prove related two results (Theorem 1 and Corollary 1) concerning the closedness property of accretive operators in the class of spaces whose nonempty bounded closed convex subsets have the fixed point property for nonexpansive self-mapping. Using therem 1, we derive a result (Theorem 2) on the range of accetive operators in (.pi.)$_{1}$ spaces with a view to establishing a new result, which improves a result of Kartsatos [8] and Webb [15]. Further, we give an interesting consequence (Corollary 3) of Theorem 2. In section 4, we apply Corollary 1 to obtain two results (Theorem 3 and 4) for the range of sums of two accretive operators, which generalize two results of Reich [12].