Starting from a weak concept of stability, introduced by Berinde [1] and called “weak stability”, in [27] we develop a weaker notion, named “w-stability”. Therefore, in this paper we prove some results of this weaker stability concept for certain class of mappings and also we give some examples of w-stable but not weak stable nor stable iterations. Because of the restriction of an “approximate ” sequence, some fixed point iteration procedures are not weakly stable so if it is used a weaker type of sequence, the stability can be obtained in the meaning of a new concept. General Terms Fixed-point and coincidence theorems.

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Abstract The purpose of this paper is to introduce the random Picard-Mann hybrid iterative process. We establish the strong convergence theorems and summable almost T-stability of the random Picard-Mann hybrid iterative process and the random Mann-type iterative process generated by a generalized class of random operators in separable Banach spaces. Our results are generalizations and improvements of several well-known deterministic stability results in a stochastic version. MSC: 47H09; 47H10; 49M05; 54H25 Keywords: random Picard-Mann hybrid iterative process; random Mann-type iterative process; separable Banach spaces; generalized random contractive type operator; summable almost T-stability