Fixed points of nonexpansive and quasi-nonexpansive mappings

In the paper Krasnoselskii-Mann method for non-self mappings in the journal of Fixed Point Theory and Applications, Colao and Marino proved strong convergence of Krasnoselskii-Mann algorithm defined by xn+1=alpha nxn+(1-alpha n)Txn for a non-expansive non-self mapping in a Hilbert space and they proposed three open questions. In this paper we have proved theorems that are answers to all the open questions raised in that paper by relaxing the space, involved map and inward condition to be uniformly convex Banach space, quasi-nonexpansive and weakly inward condition respectively. An application of non-linear parabolic partial differential equation is discussed.We also provide numerical example to verify our main result.