The convergence of the Mann iteration scheme to fixed points of any class of mappings in any space is guaranteed only on the existence of a convexity structure defined therein.

For metric spaces, one notion of convexity is that propounded by Takahashi, and it has been generalized to the framework of metric spaces, establishing convex metric spaces as defined by Takahashi. For these particular spaces, a result has been published that establishes the condition for which the Mann iteration converges to fixed points of Banach contraction mappings defined therein.

On consideration of this result, the author of this work was inspired to formulate an analogous result that establishes the condition for which the Mann iteration converges to fixed points of Hardy–Rogers contraction mappings in the same framework. This result is novel in the sense that no such result has been published by any author. Additionally, it extends the cited existing result to a bigger class of contraction mappings.

Also, since Banach contraction mappings, Kennan contraction mappings and Chatterjee contraction mappings are special cases of Hardy–Rogers contraction mappings, this result produces three corollaries, each establishing a condition for the convergence of the Mann iteration for only one of the mentioned classes of contraction mappings.

Finally, this result is applied to the existence and approximation of solutions of Fredolm-type integral equations of the second kind; these equations arise in many problems in signal processing, theory of imaging and fluid mechanics, among other fields in the physical sciences and engineering.

In sum, the result to be presented extends a result on the convergence of the Mann iteration in convex Gb metric spaces and is applied to the existence and approximation of solutions of Fredolm integral equations, which find expressions in many models in science and engineering.