Some new fixed point theorems of α-partially nonexpansive mappings

In this paper, we introduce a new class of nonlinear mappings and compare it to other classes of nonlinear mappings that have appeared in the literature. We establish various existence and convergence theorems for this class of mappings under different assumptions in Banach spaces, particularly Banach spaces with a normal structure. In addition, we provide examples to substantiate the findings presented in this study. We prove the existence of a common fixed point for a family of commuting α\alpha -partially nonexpansive self-mappings. Furthermore, we extend the results reported by Suzuki (Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088–1095), Llorens-Fuster (Partially nonexpansive mappings, Adv. Theory Nonlinear Anal. Appl. 6 (2022), no. 4, 565–573), and Dhompongsa et al. (Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350 (2009), no. 1, 12–17). Finally, we present an open problem concerning the existence of fixed points for α\alpha -partially nonexpansive mappings in the context of uniformly nonsquare Banach spaces.