Document Type : Research Paper

**Author**

Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran.

**Abstract**

Let $ H$ be a Hilbert space and $C$ be a closed, convex and nonempty subset of $H$. Let $T:C \rightarrow H$ be a non-self and non-expansive mapping. V. Colao and G. Marino with particular choice of the sequence $\{\alpha_{n}\}$ in Krasonselskii-Mann algorithm, ${x}_{n+1}={\alpha}_{n}{x}_{n}+(1-{\alpha}_{n})T({x}_{n}),$ proved both weak and strong converging results. In this paper, we generalize their algorithm and result, imposing some conditions upon the set $C$ and finite many mappings from $C$ in to $H$, to obtain a converging sequence to a common fixed point for these non-self and non-expansive mappings.

**Keywords**

[1] J. Ayaragarnchanakul, *A common fixed-point iterative process with errors for quasi-nonexpansive nonself-mappings in Banach spaces*, Thai J. Math., 6 (2008), pp. 323-330.

[2] H.H. Bauschke, PL. Combettes, *A weak-to-strong convergence principle for monotone methods in Hilbert spaces*, Math. Oper. Res., 26 (2001), pp. 248-264 .

[3] F.E. Browder, *Convergence theorems for sequences of nonlinear operators in Banach spaces*, Math. Z. 100 (1967), pp. 201-225.

[4] V. Colao and G. Marino, *Krasnoselskii-Mann method for non-self mappings*, Fixed Point Theory Appl., 39 (2015), pp. 1-7.

[5] C. Chidume, *Geometric properties of Banach spaces and nonlinear iterations*, Lecture Notes in Mathematics, Vol. 1965, Springer, Berlin, 2009.

[6] M. Edelstein, RC. O'Brien, *Nonexpansive mappings, asymptotic regularity and successive approximations*, J. London. Math. Soc., 2 (1978), pp. 547-554.

[7] C.W. Groetsch, *A note on segmenting Mann iterates*, J. Math. Anal. Appl., 40 (1972), pp. 369-372.

[8] M. Guo, X. Li and Y. Su, *On an open question of V. Colao and G. Marino presented in the paper “Krasnoselskii–Mann method for nonself mappings”*, SpringerPlus, 5 (2016), pp. 1-9.

[9] TL. Hicks and JD. Kubicek, *On the Mann iteration process in a Hilbert space*, J. Math. Anal. Appl., 59 (1977), pp. 498-504.

[10] B.P. Hillam, *A generalization of Krasnoselski’s theorem on the real line*, Math. Mag., 48 (1975), pp. 167-168.

[11] W.R. Mann, *Mean value methods in iteration*, Proc. Amer. Math. Soc., 4 (1953), pp. 506-510.

[12] G. Marino and G. Trombetta, *On approximating fixed points for nonexpansive mappings*, Indian J. Math., 34 (1992), pp. 91-98.

[13] S. Reich, *Weak con in Banach spaces*, J. Math. Anal. Appl., 67 (1979), pp. 274-276.

[14] L. Wang, *Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings*, J. Math. Anal. Appl., 323 (2006), 550-557.

[15] H.K. Xu, *Approximating curves of nonexpansive nonself-mappings in Banach spaces*, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), pp. 151-156.

[16] H.K. Xu and X.M. Yin, *Strong convergence theorems for nonexpansive nonself-mappings*, Nonlinear Anal., 24(1995), pp. 223-228.