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.

Monotonicity points and the modulus of monotonicity are studied in b-Orlicz function spaces over finite nonatomic measure spaces, equipped with the Luxemburg quasi-norm. Lower strictly monotone points and lower and upper locally uniformly monotone points are characterized. For generating functions with constant intervals, upper locally uniformly monotone points may exist even when the ${\Delta }_2$ condition at infinity fails. In this case, all such points are described using the right endpoints of the constant intervals. A criterion for uniform monotonicity is established, including the necessity of the ${\Delta }_{\alpha }$ conditions. Lower and upper bounds for the modulus of monotonicity are also established. The exact modulus is computed for nonconvex generating functions given by positive linear combinations of two powers, with one exponent between zero and one and the other above one. For this family, the modulus depends only on the larger exponent. The constants in the general lower estimate are computed for a particular member of this family.