In this paper, we present a series of fixed point results for Mann’s iteration scheme in the framework of $G_b$-metric spaces. First, we introduce the concept of convex $G_b$-metric space by means of a convex structure and Mann’s iteration algorithm is extended to this space. Furthermore, using Mann’s iteration scheme, we prove some fixed point results for several mappings satisfying various suitable conditions on complete convex $G_b$-metric spaces. Some examples supporting our main results are also presented. We also discuss the well-posedness of the fixed point problems and the P property for given mappings. Moreover, as an application, we apply our main result to prove the existence of the solutions to integral equations.

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces $L^{\Phi }$ generated by the b-Orlicz function $\Phi$ equipped with a Luxemburg quasi-norm contain both classical spaces $L^p(p\ge 1)$ and $L^p(0<p<1)$. The Orlicz spaces $L^{\Phi }$ are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given.