| Summary: | Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of <i>b</i>-metric spaces. First, the concept of a convex <i>b</i>-metric space by means of a convex structure is introduced and Mann’s iteration algorithm is extended to this space. Next, by the help of Mann’s iteration scheme, strong convergence theorems for two types of contraction mappings in convex <i>b</i>-metric spaces are obtained. Some examples supporting our main results are also presented. Moreover, the problem of the <i>T</i>-stability of Mann’s iteration procedure for the above mappings in complete convex <i>b</i>-metric spaces is considered. As an application, we apply our main result to approximating the solution of the Fredholm linear integral equation.
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