Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L p or p spaces, 1 < p < ∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let T : K → K be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I − T ) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.

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The purpose of this paper is to introduce a new iterative algorithm to approximate the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Also, we show that our proposed iterative algorithm converges weakly and strongly to the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Furthermore, it is proved analytically that our new iterative algorithm converges faster than one of the leading iterative algorithms in the literature for almost contraction mappings. Some numerical examples are also provided and used to show that our new iterative algorithm has better rate of convergence than all of S, Picard-S, Thakur and M iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings. Again, we show that the proposed iterative algorithm is stable with respect to T and data dependent for almost contraction mappings. Some applications of our main results and new iterative algorithm are considered. The results in this article are improvements, generalizations and extensions of several relevant results existing in the literature.

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The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of k  strictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified iteration process.

The purpose of this article is to establish weak and strong convergence results of AI iterative scheme for fixed points of generalized α-nonexpanisve mappings in uniformly convex Banach spaces. Furthermore, we carry out a numerical experiment to compare the convergence of AI iterative scheme with several prominent iterative schemes. Finally, we use AI iteration process to find the unique solution of a functional Volterra-Fredholm integral equation with deviating argument in Banach spaces. The results of this paper are new and extend several results in the literature.

ln this paper, a new iterative algorithm for finding common ele-ments of the set of fixed points for a finite family of asymptotically quasi-nonexpansive multivalued mappings and the set of minimizers for a finite family of minimization problem is constructed. Under mild conditions on the control sequences, strong convergence of our algorithm was achieved without necessarily imposing any compactness condition on the space or the operator by using an independent approach. Our results improve, ex-tend and generalize many important results recently announced in current literature.

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