The Solution by Iteration of Nonlinear Functional Equations in Banach Spaces

In a preceding note on the linear case [8], we established the following facts for linear T: (a) If X is reflexive and T is asymptotically bounded (i.e. || T\\ ^ M for some constant M and all nèzl), then the Equation (1) has a solution u for a given ƒ if and only if for any specific #o> the sequence of Picard iterates {xn} starting with x0 is bounded in X (see [2]). (b) For a general Banach space X, if T is asymptotically convergent (i.e. Tx converges strongly in X for each x in X as n—> + oo), the sequence of Picard iterates {xn} for a given x0 converges if and only if the equation (1) has a solution. (c) For a general Banach space X and T asymptotically convergent, if an infinite subsequence of the sequence {xn} converges, then the whole sequence converges to a solution of Equation (1). Our object in the present note is to give some partial extensions of these results to a general class of nonlinear operators T, and to indicate some interesting examples of the application of these nonlinear results.