Sufficient conditions for the convergence of iterations to points of attraction in Banach space

. In this study we provide weak sufficient conditions for the convergence of iterations to points of attraction in Banach space. Our results are used to show existence and uniqueness of solutions of nonlinear equations. We complete our study with an example that shows that our convergence results apply where earlier ones do not. (1991) A.M.S. Subject Classification Codes. 65J15, 47H17, 49D15. Key Words and Phrases. Banach space, points of attraction, nonlinear equation. I. INTRODUCTION In this study, we are concerned with approximating a locally unique solution x of the nonlinear equation (1) F (x; h) = 0 for all h 2 E 0 ; where F : D \Theta E 0 ` E m+1 ! E is a continuous operator, E is a Banach space, D, E 0 are subsets of E and E m respectively and m 2 N [ f0g. We introduce the iteration (2) xn+1 = G(xn ; hn ) (n 0); to approximate the solution x of equation (1), where (3) G(x; h) = x \Gamma F (x; h) for all (x; h) 2 D \Theta E 0 : Cameron University, Lawton, Oklahom..