\textbf{Introduction.} Fixed point theory is central to nonlinear analysis and to the study of differential and fractional models arising in applied sciences. Generalized nonexpansive-type mappings have attracted increasing attention because they allow the treatment of problems beyond classical contractions. In this paper, we develop a new iterative framework for approximating fixed points of mappings satisfying the $(B_{\gamma,\mu,\eta})$condition, which extends several existing operator classes and iterative algorithms.

\textbf{Methods.} We propose a novel iterative scheme for mappings defined on Banach spaces under the $(B_{\gamma,\mu,\eta})$ condition. By employing tools from nonlinear functional analysis, we study the behavior of the generated sequence and establish sufficient conditions that ensure both weak and strong convergence of the iteration to a fixed point of the underlying mapping.

\textbf{Results.} The main results guarantee convergence under general $(B_{\gamma,\mu,\eta})$ assumptions, thereby broadening the applicability of classical fixed-point methods. The practicality of the scheme is demonstrated through an application to a Caputo-type fractional-order model describing the interaction between tumor stem cell proliferation and cellular crowding. In this setting, fractional derivatives incorporate memory effects in tumor dynamics, and the proposed iteration efficiently approximates equilibrium states of the model.

\textbf{Conclusions.} The developed algorithm provides a robust computational tool for generalized fixed-point problems and contributes to the mathematical modeling of cancer stem cell behavior in fractional biological systems.