An iterative method for approximating fixed points of Presić nonexpansive mappings

Abstract

Some fixed point theorems of Presić type for nonexpansive mappings\(f: X^k\rightarrow X\), where \(k\geq 1\) is an integer, are obtained.The main result of the paper unifies two important fixed point theorems published in the same year, 1965, the first one discovered independently by Browder [F.E. Browder, Nonexpansive nonlinear operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041-1044], Göhde [D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr., 30(1965), 251-258] and Kirk [W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly,72 (1965), 1004-1006], while the second one is due to Presić [S.B. Presić, Sur une classe d' inéquationsaux differences finite et sur la convergence de certaines suites, Publ. Inst. Math. (Beograd)(N.S.), 5(19) (1965),75-78]. In this way we show how amazingly two apparently different beautiful results in mathematics can meet after almost half a century! This appears to be the first attempt to study multi-step iterative methods by means of the fixed point theory of nonexpansive mappings. Several related results in literature are extended, unified and generalized.

Keywords

• Banach space
• Presić type contraction condition
• fixed point
• \(k\)-step iteration procedure
• nonexpansive type operator